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+{"text":"\\section*{\\centering\\LARGE{Acknowledgment}}\n\\thispagestyle{empty}\nI would like to acknowledge, and thank, the \\textit{Coordena\\c{c}\\~ao de Aperfei\\c{c}oamento de Pessoal de N\\'ivel Superior} (CAPES - Coordination for Improvement of Higher Education Personnel) for the PhD scholarship with process number $88882.329848\/2019-01$, given through the \\textit{Programa de Excel\\^encia Acad\\^emica} (PROEX - Academic Excellence Program), without which this work would not have been possible.\n\\vspace{5mm}\n\nI thank Professor Marcelo E. Coniglio for his advisement, for which I am forever grateful. While at times he only gently steered things in a better direction, be it a definition to be clearer or a theorem to be broader, at others he delved with me into research, when doing so seemed necessary. I firmly believe that, despite the thesis being my own, some developments that it lead to are as his as they are mine, probably more.\n\\vspace{5mm}\n\nOf utmost importance where the contributions by Professors Walter A. Carnielli and Hugo Mariano, who saw an earlier version of this work in my Ph.D. qualifying exam and suggested several improvements, as well as directions the ongoing research could take. If this thesis has not followed all of these directions, and it hasn't, it was only for lack of more time.\n\\vspace{5mm}\n\nMany other ameliorations and corrections, all of which I tried to incorporate into the final text, were brought forward in my thesis defense by Professors Walter Carnielli, Itala L. D'Ottaviano, H{\\'e}rcules A. Feitosa, Hitoshi Omori, Darllan C. Pinto and Anna Zamansky, who all in addition suggested invaluable ideas for future work.\n\\vspace{5mm}\n\nAnd to all those who have otherwise influenced this work, including all students, professors and staff of the \\textit{Centro de L\\'ogica, Epistemologia e Hist\\'oria da Ci\\^encia} (CLE - Center for Logic, Epistemology and History of Science), the \\textit{Instituto de Filosofia e Ci\\^encias Humanas} (IFCH - Institute of Philosophy and Human Sciences), and the \\textit{Instituto de Matem\\'atica e Estat\\'istica da Universidade de S\\~ao Paulo} (IME-USP - Institute of Mathematics and Statistics of the University of S\\~ao Paulo), I am also deeply grateful.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{\\fill}\n\\begin{flushright}\n\\textit{``But man is not made for defeat.\\\\\nA man can be destroyed but not defeated.``\\\\\n(\\textit{The Old Man and the Sea}, by Ernest Hemingway)}\n\\end{flushright}\n\\newpage\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{\\centering\\LARGE{Abstract}}\n\\thispagestyle{empty}\n\nThis work is divided between two main areas: in the theory of multialgebras, we focus mostly on a new definition of what a freely generated object should be in their category, and on how this category is equivalent to another with partially ordered algebras as objects; we then use non-deterministic semantics, specially those we have named restricted Nmatrices, on paraconsistent logics and some systems dealing with a new presentation of the natural concept of incompatibility, which generalizes inconsistency.\n\nIn algebra, we will focus on the non-deterministic ones, also known as multialgebras, whose operations return non-empty subsets of their universes. While the category of algebras over a signature has freely generated objects, which in a sense permit for the unique extension of functions to homomorphisms, the category of multialgebras over a given signature does not have elements with comparable properties. To circumvent this problem, we widen our understanding of algebras of formulas: if a multialgebra generalizes an algebra by having multiple results for a given operation, a multialgebra of formulas should generalize an algebra of formulas by having multiple possibilities for applying a connective to given formulas. Concerning the category of multialgebras itself, we offer an equivalence between it and a category avoiding non-determinism altogether, relying instead on ordered Boolean-like algebras as objects.\n\nOn the part devoted to logic, our goals are again roughly twofold: firstly, some logics of formal inconsistency, \\textit{exempli gratia} those found in da Costa's hierarchy, cannot be characterized by finite Nmatrices. In what is a very natural development, a restricted Nmatrix (or RNmatrix) restricts those homomorphisms to be taken into consideration when evaluating the validity of a deduction according to an Nmatrix. We show how this distinction gives far greater expressiveness to finite RNmatrices, enough to adequately characterize da Costa's systems and provide decision methods for those logics, both based on row-branching, row-eliminating truth tables, and tableau semantics. In another direction, we generalize logics of formal inconsistency to new systems built around the notion of incompatibility: the \\textit{Leitmotiv} being that having two incompatible formulas to simultaneously hold trivializes a deduction, and as a special case, a formula is consistent when it is incompatible with its negation. We show how this notion extends that of inconsistency in a non-trivial way, presenting conservative translations for many simple inconsistent systems into logics of incompatibility; we also provide semantics built on RNmatrices for these new logics, and prove that they can not be characterized by more standard methods.\n\n\\vspace*{\\fill}\n\\textbf{Keywords}: Non-classical mathematical logic; Paraconsistent logic; Algebraic logic; Inconsistency (Logic); Universal algebra.\n\n\\newpage\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{\\centering\\LARGE{Resumo}}\n\\thispagestyle{empty}\n\n\\foreignlanguage{portuguese}{\nEste trabalho est\\'a dividido entre duas grandes \\'areas: na teoria de multi\\'algebras, focamos majoritariamente em uma nova defini\\c{c}\\~ao do que um objeto livremente gerado deveria ser em sua categoria e em como esta categoria \\'e equivalente a outra com \\'algebras parcialmente ordenadas como objetos; ent\\~ao usamos sem\\^anticas n\\~ao-determin\\'isticas, especialmente aquela que nomeamos Nmatrizes restritas, nas l\\'ogicas paraconsistentes e em alguns sistemas lidando com uma nova apresenta\\c{c}\\~ao do conceito natural de incompatibilidade, que generaliza o conceito de inconsist\\^encia.}\n\n\\foreignlanguage{portuguese}{\nEm \\'algebra, nos focaremos nas n\\~ao-determin\\'isticas, tamb\\'em conhecidas como multi\\'algebras, cujas opera\\c{c}\\~oes retornam subconjuntos n\\~ao vazios de seus universos. Enquanto a categoria de \\'algebras sobre uma assinatura possui objetos livremente gerados, os quais permitem em certo sentido a extens\\~ao \\'unica de fun\\c{c}\\~oes a homomorfismos, a categoria de multi\\'algebras sobre uma assinatura dada n\\~ao possui elementos com propriedades compar\\'aveis. Para contornar este problema, estendemos o significado de uma \\'algebra de f\\'ormulas: se uma multi\\'algebra generaliza uma \\'algebra ao ter m\\'ultiplos resultados para uma dada opera\\c{c}\\~ao, uma multi\\'algebra de f\\'ormulas generaliza uma \\'algebra de f\\'ormulas ao ter m\\'ultiplas possibilidade para a aplica\\c{c}\\~ao de um conectivo a f\\'ormulas dadas. Quanto \\`a categoria de multi\\'algebras propriamente dita, oferecemos uma equival\\^encia entre ela e uma categoria livre de n\\~ao-determinismo, que alternativamente possui \\'algebras ordenadas, semelhantes a \\'algebras de Boole, como objetos.}\n\n\\foreignlanguage{portuguese}{\nNa parte dedicada \\`a l\\'ogica, nossos objetivos s\\~ao novamente dois: primeiramente, algumas l\\'ogicas de inconsist\\^encia formal, \\textit{exempli gratia} aquelas da hierarquia de da Costa, n\\~ao podem ser caracterizadas por Nmatrizes finitas. No que \\'e um desenvolvimento muito natural, uma Nmatriz restrita, ou RNmatriz, restringe aquelas homomorfismos que devem ser considerados quando testamos a validade de uma dedu\\c{c}\\~ao por uma Nmatriz. Mostramos como esta distin\\c{c}\\~ao prov\\^e as RNmatrizes finitas com poder expressivo muito superior, suficientente para adequadamente caracterizar os sistemas de da Costa e dar a eles m\\'etodos de decis\\~ao, tanto baseados em tabelas de verdade quanto em sem\\^anticas de tableaux. Em outra dire\\c{c}\\~ao, generalizamos as l\\'ogicas de inconsist\\^encia formal a sistemas constru\\'idos em torno da no\\c{c}\\~ao de incompatibilidade: o \\textit{Leitmotiv} sendo que duas f\\'ormulas incompat\\'iveis simultaneamente verdadeiras trivializam uma dedu\\c{c}\\~ao, e como um caso especial, uma f\\'ormula \\'e consistente quando \\'e incompat\\'ivel com sua nega\\c{c}\\~ao. Mostramos como essa no\\c{c}\\~ao estende aquela de inconsist\\^encia de maneira n\\~ao-trivial, apresentando tradu\\c{c}\\~oes conservativas para muitos dos sistemas inconsistentes mais simples em l\\'ogicas de incompatilidade, apresentamos sem\\^anticas constru\\'idas com RNmatrizes para essas novas l\\'ogicas e mostramos que elas n\\~ao podem ser caracterizadas por m\\'etodos mais usuais.}\n\n\\vspace*{\\fill}\n\\foreignlanguage{portuguese}{\n\\textbf{Palavras-chave}: L\\'ogica matem\\'atica n\\~ao cl\\'assica; L\\'ogica paraconsistente; L\\'ogica alg\\'ebrica; Inconsist\\^encia (L\\'ogica); \\'Algebra universal.}\n\n\\pagestyle{empty}\n\\addtocontents{toc}{\\protect\\thispagestyle{empty}}\n\\shorttoc{Abridged Table of Contents}{1}\n\\thispagestyle{empty}\n\\newpage\n\n\n\\shorttoc{Extended Table of Contents}{3}\n\\thispagestyle{empty}\n\\newpage\n\n\n\\tableofcontents\n\\cleardoublepage\n\n\n\n\\pagenumbering{arabic}\n\\setcounter{page}{16}\n\n\n\\newpage\n\\thispagestyle{plain}\n\\hspace{0pt}\n\\vfill\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{.\/Image1}\n\\caption*{Specimen of \\textit{Eupetomena macroura} resting, Monte Alegre do Sul, Brazil.\\\\Photographed by Guilherme Vicentin de Toledo, all rights reserved.}\n\\end{figure}\n\\vfill\n\\hspace{0pt}\n\n\n\n\n\n\n\\pagestyle{fancy}\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\chapter*{Introduction}\\label{Chapter0}\n\\chaptermark{Introduction}\n\\addcontentsline{toc}{chapter}{Introduction} \n\n \nThe research that makes up the bulk of this work may be divided in two grand areas, what explains our division of the thesis in two parts: algebra, corresponding to Part \\ref{Part1}, and logic, to Part \\ref{Part2}. \n\nIn algebra, we focus on the subject of multialgebras: we present a new concept of \\textit{weakly free multialgebras}, which we designed in order to offer a generalization of free algebras; furthermore, we show how the category of multialgebras, as usually defined in the context of non-deterministic semantics, can be presented alternatively as a category of algebras equipped with orders that are compatible with the underlying operations.\n\nIn logic, we will delve into paraconsistent logics, together with their generalizations, and non-deterministic semantics: we generalize non-deterministic matrices to \\textit{restricted non-deterministic matrices} (also known as RNmatrices), a semantical tool that uses both multialgebras and restrictions over the set of valuations to be considered, and apply this methodology to give an extend analysis of da Costa's hierarchy that includes new decision methods for its logics; we also give a new formalization of a natural concept, that of incompatibility, and treat the resulting logical systems again with RNmatrices.\n\n\\subsection*{Multialgebras}\n\nIn Part \\ref{Part1} we will deal mainly with multialgebras, also known as hyperalgebras or\\\\ non-deterministic algebras. Chapter \\ref{Chapter1} gives a brief introduction to the subject, broaching the definition of multialgebras, as well as those of homomorphisms between multialgebras, submultialgebras, the interpretation of formulas, and so on. We spend a few pages over alternative definitions for a homomorphism of multialgebras that allow for stronger representation theorems, and discuss briefly how these, sadly, are not very practical to use in non-deterministic semantics. Given its frequent use in our work, we also use this chapter to define, for completeness sake, lattices, Boolean algebras and Heyting algebras.\n\nChapter \\ref{Chapter2} offers an alternative solution to a classical problem: the category of algebras, on a given signature, possesses objects satisfying the universal mapping property, namely free algebras; meanwhile, the category of multialgebras over a signature $\\Sigma$, denoted by $\\textbf{MAlg}(\\Sigma)$, does not. Algebras of formulas over a signature $\\Sigma$ and variables $\\mathcal{V}$, that we denote by $\\textbf{F}(\\Sigma, \\mathcal{V})$, are then extended to structures that admit non-determinism: after all, if a multialgebra generalizes an algebra by allowing multiple results to an operation, a multialgebra of formulas should allow many formulas to be obtained from the application of a connective to given formulas. This new concept is shown to have many desirable characterizations, some inspired by linear algebra, others graph-theoretic in nature, motivating us to coin the nomenclature of weakly free multialgebras for them. They are shown to greatly simplify the proof that $\\textbf{MAlg}(\\Sigma)$ does not have free objects, and some other categorical considerations are then developed by the end of the chapter.\n\nConcerning the category $\\textbf{MAlg}(\\Sigma)$ itself, Chapter \\ref{Chapter3} proves this category is equivalent to another that avoids non-determinism altogether by considering ordered, Boolean-like algebras. Intuitively, one wants to capture both the operations and order of the natural algebra over the powerset of the universe $A$ of a multialgebra $\\mathcal{A}$, where: the order is the usual order for a powerset; and, given non-empty subsets $A_{1}$ trough $A_{n}$ of $A$, an $n$-ary operation $\\sigma$ on $(A_{1}, \\dotsc , A_{n})$ is given by the union of $\\sigma(a_{1}, \\dotsc , a_{n})$ for $(a_{1}, \\dotsc , a_{n})$ in $A_{1}\\times\\cdots\\times A_{n}$. This presents an alternative for those logicians wishing to stick to deterministic semantics without any loss of expressiveness, and can be generalized to offer equivalences for other categories with non-deterministic algebras, such as partial multialgebras.\n\n\n\n\\subsection*{Paraconsistent Logic}\n\nPart \\ref{Part2} involves paraconsistent logics, as well as non-deterministic semantics and broader systems. It starts with Chapter \\ref{Chapter4}, which provides a brief introduction to formal logic, and semantics of logical matrices, non-deterministic matrices (also known as Nmatrices) and restricted matrices. The high point of the chapter, however, is the definition of restricted non-deterministic matrices, also known as restricted Nmatrices or RNmatrices, which very naturally combine restricted and non-deterministic matrices: some theoretical considerations are made about these semantics, as well as a brief analysis of its previous, unrecognized uses in the literature. In essence, an RNmatrix is a triple $(\\mathcal{A}, D, \\mathcal{F})$, with $\\mathcal{A}$ a $\\Sigma$-multialgebra, $D$ a subset of its universe and $\\mathcal{F}$ a set of homomorphisms $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$. Their motivation is to give finite semantics to logical systems that are not characterizable by finite Nmatrices, such as the logics of formal inconsistency ($\\textbf{LFI}'s$) between $\\textbf{mbCcl}$ and $\\textbf{Cila}$, and the whole hierarchy of da Costa.\n\nThe $C$-systems of da Costa, due to their complexity, are treated separately in Chapter \\ref{Chapter5}: in it, we start by defining these logics first devised in order to formalize the notion of inconsistency, or paraconsistency. We then proceed to provide RNmatrices of $n+2$ elements capable of characterizing each and every $C_{n}$ of the hierarchy, starting from $C_{2}$ to fix ideas: the intuition is that an RNmatrix for $C_{n}$ must have two classical values, standing from true and false, as well as $n$ inconsistent values, each standing for a different degree of inconsistency achieved in the logic; of course, this suggests that the $n$-th logic in the hierarchy could be regarded as an $n+2$-valued, non-deterministic logic. Furthermore, we show how these finite RNmatrices can be made into decision methods for da Costa's systems through both row-branching, row-eliminating truth tables, and tableau semantics.\n\nChapter \\ref{Chapter6} extends the RNmatrices for $C_{n}$ from the previous chapter to allow for model-theoretic considerations: while we started with $n+2$-valued restricted non-deterministic matrices, these can be shown to be constructed from the two-valued Boolean algebra as swap structures; the next logical step is to construct these swap structures over any Boolean algebras, leading to a class of RNmatrices capable itself of characterizing $C_{n}$. Further results include a brief combinatorial description of the snapshots found in these swap structures, as well as the construction of a category of swap structures for $C_{n}$, which is then proven to be isomorphic to the category of non-trivial Boolean algebras; this has important applications to the model theory of da Costa's hierarchy.\n\n\nChapter \\ref{Chapter7} introduces logics of incompatibility: a first attempt of formalizing the notion of incompatibility from natural language would be to declare two formulas incompatible if, and only if, together they trivialize a deduction. As it is done in logics of formal inconsistency, we weaken that condition to state that two incompatible formulas, a concept which may be primitive, are capable of trivializing an argument; we will denote by $\\alpha\\uparrow\\beta$ the fact that $\\alpha$ and $\\beta$ are incompatible. We start by defining some very basic systems of incompatibility, as of yet without negation, characterize then with RNmatrices and provide decision methods: of $\\bI^{-}$ we ask nothing, while $\\bI$ must have a commutative incompatibility connective and $\\bIpr$ propagates incompatibility under some conditions. We finish the chapter by discussing axioms that collapse $\\uparrow$ to its classical interpretation, and finally comparing our approach to a preexisting formalization of incompatibility, that of Brandom.\n\nIt is natural, once we have a connective for incompatibility, to consider its interplay with negation: the systems of Chapter \\ref{Chapter7} are devoid of negation, by design, but Chapter \\ref{Chapter8} consider those logics now equipped with negation and some axioms governing its interaction with incompatibility, most of them heavily inspired by the most common axioms for paraconsistent systems: while $\\nbI$ only adds a negation to $\\bI$ satisfying \\textit{tertium non datur}, $\\nbIciw$, $\\nbIci$ and $\\nbIcl$ generalize the logics of paraconsistency $\\mbCciw$, $\\mbCci$ and $\\mbCcl$, respectively. Of course, we then provide these systems with characterizing RNmatrices, as well as decision methods through row-branching, row-eliminating truth tables and tableau calculi, and explain why these semantics, instead of more classical ones, are necessary: not only our basic incompatible systems are not algebraizable by Blok and Pigozzi, they are also not characterizable by either finite Nmatrices or finite restricted matrices. We finish by analyzing the generalizations of logical matrices we have broached here, how do they relate to each other and to other possible generalizations.\n\n\n\nFinally, Chapter \\ref{Chapter9} studies an equivalence merely implied in previous chapters: in logics of formal inconsistency, a formula $\\alpha$ being consistent is recurrently expressed by $\\circ\\alpha$, where the connective $\\circ$ stands precisely for consistency. When dealing with incompatibility, the fact consistency is adequately reintroduced as incompatibility with negation becomes apparent: that is, $\\circ\\alpha$ may be viewed as $\\alpha\\uparrow\\neg\\alpha$. Accordingly, we define a function from the logics of formal inconsistency into the logics of incompatibility that is not only a translation, but a conservative one nonetheless. This seems to imply that our interpretation of incompatibility in logic strictly extends the notion of inconsistency in a non-trivial way.\n\n\n\\subsection*{Publications}\n\nA considerable portion of this thesis was made available as preprints in\n\n\\begin{enumerate}\n\n\\item \\fullcite{AbsFreeHyp};\n\n\\item \\fullcite{CostaRNmatrix};\n\n\\item \\fullcite{RestrictedSwap};\n\n\\item \\fullcite{Frominconsistency},\n\n\\end{enumerate}\n\nand published in\n\n\\begin{enumerate}\n\n\\item \\fullcite{WeaklyFreeMultialgebras};\n\n\\item \\fullcite{TwoDecisionProcedures}.\n\n\\end{enumerate}\n\n\n\n\\subsection*{Presentations}\n\nThe work found in this thesis has also lead to the following presentations.\n\n\\begin{enumerate}\n\n\\item ``\\textbf{RNmatrices for da Costa's hierarchy}'', in \\textit{I Enc(ue-o)ntro de L\\'ogica Brasil\\\\ Col(o-\\^o)mbia} (1st Meeting Brazil-Colombia in Logic), December of 2021.\n\n\\item ``\\textbf{Uma categoria de \\'algebras ordenadas equivalente a uma categoria de hiper\\'algebras}'', in \\textit{I Encontro Brasileiro em Teoria das Categorias}, January of 2021.\n\n\\item ``\\textbf{Sem\\^anticas n\\~ao-determin\\'isticas para l\\'ogicas n\\~ao-cl\\'assicas: Uma abordagem da\\\\ perspectiva de Teoria de Modelos e de \\'Algebra Universal}'', in \\textit{Encontro Brasileiro de L\\'ogica 2019} (19th Brazilian Logic Conference), May of 2019.\n\n\\item ``\\textbf{Sem\\^anticas n\\~ao-determin\\'isticas para l\\'ogicas n\\~ao-cl\\'assicas: Uma abordagem da\\\\ perspectiva de Teoria de Modelos e de \\'Algebra Universal}'', in \\textit{XVIII Encontro Nacional da ANPOF}, October of 2018.\n\n\\item ``\\textbf{Nondeterministic Semantics for Nonclassical Logics: An Approach from the\\\\ Perspective of Model Theory and Universal Algebra}'', in \\textit{Fourth Workshop CLE-Buenos Aires Logic Group}, April of 2018.\n\\end{enumerate}\n\n\n\\end{refsegment}\n\n\\part{Multialgebras}\\label{Part1}\n\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{0}\n\\chapter{$\\Sigma$-Multialgebras and lattices}\\label{Chapter1}\\label{Chapter 1}\n\nA multialgebra, or hyperalgebra\\index{Hyperalgebra}, is a generalization of the notion of algebra usually found in the context of universal algebras, see \\cite{Burris} for a standard approach to universal algebra. The main objective of such a generalization is to address the possibility that the outcome of an operation may be diffuse, that is, non-deterministic: we may have an idea of what the outcome should be, but not be certain about it. The first known appearance of multialgebras in literature may be found in \\cite{Marty}.\n\nA collection of disjoint sets $\\Sigma=\\{\\Sigma_{n}\\}_{n\\in\\mathbb{N}}$ indexed by $\\mathbb{N}$ will be called a signature\\index{Signature}\\label{signature}; the elements of the sets $\\Sigma_{n}$ will be called functional symbols of arity $n$, or $n$-ary functional symbols. For simplicity, we will denote $\\bigcup_{n\\in\\mathbb{N}}\\Sigma_{n}$ also by $\\Sigma$.\n\nA pair $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is said to be, where $\\mathcal{P}(A)$\\label{powerset} denotes the powerset (also known as power set) of $A$:\n\\begin{enumerate}\n\\item a $\\Sigma$-algebra\\index{Algebra} if, for every $\\sigma\\in\\Sigma_{n}$, $\\sigma_{\\mathcal{A}}$ is a function of the form\n\\[\\sigma_{\\mathcal{A}}:A^{n}\\rightarrow A;\\]\n\\item a $\\Sigma$-multialgebra\\index{Multialgebra} if $A\\neq\\emptyset$ and, for every $\\sigma\\in\\Sigma_{n}$, $\\sigma_{\\mathcal{A}}$ is a function of the form\n\\[\\sigma_{\\mathcal{A}}:A^{n}\\rightarrow \\mathcal{P}(A)\\setminus\\{\\emptyset\\};\\]\n\\end{enumerate}\n\nThe set $A$ is called the universe\\index{Universe} of $\\mathcal{A}$. \n\nGiven a $\\Sigma$-algebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$, one can always define the $\\Sigma$-multialgebra $\\overline{\\mathcal{A}}=(A, \\{\\sigma_{\\overline{\\mathcal{A}}}\\}_{\\sigma\\in\\Sigma})$ such that, for $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$, \n\\[\\sigma_{\\overline{\\mathcal{A}}}(a_{1}, \\dotsc  , a_{n})=\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\};\\]\nit is clear that $\\overline{\\mathcal{A}}$ carries the same information that $\\mathcal{A}$, and so one can see $\\Sigma$-algebras as $\\Sigma$-multialgebras. For most of our studies here we will focus mainly on multialgebras.\n\n\n\n\n\n\\section{Homomorphisms}\n\n\n\\subsection{Single-valued homomorphisms}\n\nGiven $\\Sigma$-algebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$, a function $\\varphi:A\\rightarrow B$ is said to be a homomorphism from $\\mathcal{A}$ to $\\mathcal{B}$ if, for every $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$ we have that\n\\[\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))=\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})).\\]\n\nGiven $\\Sigma$-multialgebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$, we would like to define once again a morphism between the two of them. But in the case of multialgebras, one can come up with many possible straightforward definitions of homomorphisms, all of them starting with a function $\\varphi:A\\rightarrow B$. The following two definitions are the most useful to our purposes:\n\\begin{enumerate}\n\\item if, for all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$, \n\\[\\{\\varphi(a)\\ :\\ a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq \\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})),\\]\nwe call $\\varphi$ a homomorphism\\index{Homomorphism}, or $\\Sigma$-homomorphism;\n\\item we call $\\varphi$ a full homomorphism\\index{Homomorphism, Full} if the above condition is replaced by \n\\[\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})).\\]\n\\end{enumerate}\nIf the function $\\varphi:A\\rightarrow B$ is a homomorphism from $\\mathcal{A}$ to $\\mathcal{B}$, we will simply write $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$\\label{homomorphism}.\n\n\\begin{theorem}\nThe class of all $\\Sigma$-multialgebras becomes a category $\\textbf{MAlg}(\\Sigma)$\\label{MAlg} or $\\textbf{MAlg}_{=}(\\Sigma)$\\label{MAlg=} when the set of morphisms between two $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ is, respectively:\n\\begin{enumerate}\n\\item the set of all homomorphisms between $\\mathcal{A}$ and $\\mathcal{B}$;\n\\item the set of all full homomorphisms between $\\mathcal{A}$ and $\\mathcal{B}$.\n\\end{enumerate}\nIn both cases, the composition of morphisms is the usual composition of functions.\n\\end{theorem}\n\n\\begin{proof}\nWe must show that all two of those alleged categories have identity morphisms and that their compositions are well-defined, meaning that composing two morphisms returns again a morphism; clearly there is no need to show the associativity of composition, since it is know that the composition of functions is indeed associative.\n\nFor every multialgebra $\\mathcal{A}$ we consider the morphism $Id_{\\mathcal{A}}:\\mathcal{A}\\rightarrow \\mathcal{A}$ given by, for every $a\\in A$, $Id_{\\mathcal{A}}(a)=a$. This morphism is the desired identity morphism for $\\mathcal{A}$ in both categories, since: it is, in fact, a morphism in the two of them, given that it is a full homomorphism and therefore also a homomorphism; it is the identity for the composition of functions and therefore, for any morphisms $\\varphi:\\mathcal{A}\\rightarrow \\mathcal{B}$ and $\\psi:\\mathcal{C}\\rightarrow\\mathcal{A}$, \n\\[\\varphi\\circ Id_{\\mathcal{A}}=\\varphi\\quad\\text{and}\\quad Id_{\\mathcal{A}}\\circ\\psi=\\psi.\\]\n\nNow to prove that the composition is well-defined, fix $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$.\n\\begin{enumerate}\n\\item If $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ are simply homomorphisms,\n\\[\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\] \nand therefore \n\\[\\{\\psi\\circ\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq\\{\\psi(b)\\ :\\  b\\in \\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\};\\]\nsince $\\psi$ is a homomorphism, we have that\n\\[\\{\\psi(b)\\ :\\  b\\in \\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\}\\subseteq \\sigma_{\\mathcal{C}}(\\psi\\circ\\varphi(a_{1}), \\dotsc  , \\psi\\circ\\varphi(a_{n})),\\]\nand from that $\\psi\\circ\\varphi$ is also a homomorphism.\n\n\\item If $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ are full homomorphisms, is enough to replace all \"$\\subseteq$\" on the proof above by equalities to obtain a proof that $\\psi\\circ\\varphi$ is also a full homomorphism.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{definition}\nA full homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ is said to be an isomorphism\\index{Isomorphism} if $\\varphi:A\\rightarrow B$ is a bijection.\\footnote{We do not attempt to define isomorphisms for homomorphisms that are not full since the latter class is not closed under inverses: the inverse of a non-full homomorphism is an antihomomorphism, which satisfies $\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc, \\varphi(a_{n}))\\subseteq \\{\\varphi(a)\\ :\\ a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc a_{n})\\}$; considering antihomomorphisms does not make the underlying theory uninteresting, but it does make the theory much harder.}\n\\end{definition}\n\n\\begin{proposition}\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be $\\Sigma$-multialgebras and $\\varphi:A\\rightarrow B$ be a bijective function with inverse $\\psi:B\\rightarrow A$: if $\\varphi$ is a full homomorphism, so is $\\psi$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\sigma\\in\\Sigma_{n}$, $b_{1}, \\dotsc  , b_{n}\\in B$ and $a_{1}=\\psi(b_{1}), \\dotsc  , a_{n}=\\psi(b_{n})$, so that $b_{1}=\\varphi(a_{1}), \\dotsc  , b_{n}=\\varphi(a_{n})$. We have that\n\\[ \\{\\psi(b)\\ :\\  b\\in \\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})\\}=\\{\\psi(b)\\ :\\  b\\in\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\}=\\]\n\\[\\{\\psi(b)\\ :\\  b\\in\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\}=\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sigma_{\\mathcal{A}}(\\psi(b_{1}), \\dotsc  , \\psi(b_{n})),\\]\nand therefore $\\psi$ is indeed a full homomorphism.\n\\end{proof}\n\n\n\n\n\n\\subsection{Multi-valued homomorphisms}\\label{Multi-valued homomorphisms}\n\nOne natural consideration when defining morphisms between multialgebras is that, if the operations are of a non-deterministic nature, perhaps the morphisms should be as well.\n\nAlthough there exist a plethora of possible definitions in this case, we will mention only two of them, even avoiding considerations about isomorphisms for they are out of scope for this text. Given a signature $\\Sigma$ and $\\Sigma$-multialgebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$, a function $\\varphi:A\\rightarrow\\mathcal{P}(B)\\setminus\\{\\emptyset\\}$ is said to be a multihomomorphism\\index{Multihomomorphism} if, for all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\\[\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\varphi(a)\\subseteq\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\varphi(a_{1})\\times\\cdots\\times\\varphi(a_{n})}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n});\\]\nthe same function is said to be a full multihomomorphism\\index{Multihomomorphism, Full} if this conditions is replaced by \n\\[\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\varphi(a)=\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\varphi(a_{1})\\times\\cdots\\times\\varphi(a_{n})}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n}).\\]\nWe will denote a multihomomorphism $\\varphi$ between $\\mathcal{A}$ and $\\mathcal{B}$ simply by $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$.\n\n\\begin{lemma}\\label{lemma about unions}\nFor an $n\\in\\mathbb{N}$, take sets $X_{1}, \\dotsc  , X_{n}$; for $X=\\bigcup_{i=1}^{n}X_{i}$, let $\\{Y_{x}\\}_{x\\in X}$ be a family indexed by $X$; then\n\\[\\bigcup_{(x_{1}, \\dotsc  , x_{n})\\in X_{1}\\times\\cdots\\times X_{n}} Y_{x_{1}}\\times\\cdots\\times Y_{x_{n}}\\subseteq\\bigcup_{x_{1}\\in X_{1}}Y_{x_{1}}\\times\\cdots\\times\\bigcup_{x_{n}\\in X_{n}}Y_{x_{n}}.\\]\nIf $n=1$, we have instead an equality.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $(y_{1}, \\dotsc  , y_{n})$ belongs to the left side of the inequality, and there must exist\\\\ $(x_{1}, \\dotsc  , x_{n})$ in $X_{1}\\times\\cdots\\times X_{n}$ such that $(y_{1}, \\dotsc  , y_{n})\\in Y_{x_{1}}\\times\\cdots\\times Y_{x_{n}}$.\n\nNow, for any $i\\in\\{1, \\dotsc  , n\\}$, since $y_{i}\\in Y_{x_{i}}$ for an $x_{i}\\in X_{i}$, $y_{i}\\in \\bigcup_{x\\in X_{i}}Y_{x}$. It follows that $(y_{1}, \\dotsc  , y_{n})\\in\\bigcup_{x_{1}\\in X_{1}}Y_{x_{1}}\\times\\cdots\\times\\bigcup_{x_{n}\\in X_{n}}Y_{x_{n}}$.\n\nThe equality if $n=1$ is trivial.\n\\end{proof}\n\n\n\n\\begin{theorem}\nThe class of all $\\Sigma$-multialgebras becomes a category $\\textbf{MMAlg}(\\Sigma)$\\label{MMAlg} or $\\textbf{MMAlg}_{=}(\\Sigma)$\\label{MMAlg=} when the set of morphisms between two $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ is, respectively:\n\\begin{enumerate}\n\\item the set of all multihomomorphisms between $\\mathcal{A}$ and $\\mathcal{B}$;\n\\item the set of all full multihomomorphisms between $\\mathcal{A}$ and $\\mathcal{B}$.\n\\end{enumerate}\nIn both cases, the composition $\\psi\\circ\\varphi$ of multihomomorphisms $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ is given by, on an element $a\\in A$, $\\psi\\circ\\varphi(a)=\\bigcup_{b\\in\\varphi(a)}\\psi(b)$.\n\\end{theorem}\n\n\\begin{proof}\nWe must show the existence of identity morphisms and that the composition of morphisms is well-defined and associative.\n\nFor every $\\Sigma$-multialgebra $\\mathcal{A}$ we consider the morphism $Id_{\\mathcal{A}}:\\mathcal{A}\\rightarrow\\mathcal{A}$ given by, for every $a\\in A$, $Id_{\\mathcal{A}}(a)=\\{a\\}$. It is a full multihomomorphism, and therefore also a multihomomorphism, given that, for $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\\[\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}Id_{\\mathcal{A}}(a)=\\bigcup_{a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\{a\\}=\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\{a_{1}\\}\\times\\cdots\\times\\{a_{n}\\}}\\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n})=\\]\n\\[\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in Id_{\\mathcal{A}}(a_{1})\\times\\cdots\\times Id_{\\mathcal{A}}(a_{n})}\\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n}).\\]\nFor any multihomomorphisms $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{C}\\rightarrow\\mathcal{A}$ and elements $a\\in A$ and $c\\in C$, we have that\n\\[\\varphi\\circ Id_{\\mathcal{A}}(a)=\\bigcup_{d\\in Id_{\\mathcal{A}}(a)}\\varphi(d)=\\bigcup_{d\\in \\{a\\}}\\varphi(d)=\\varphi(a)\\]\nand\n\\[Id_{\\mathcal{A}}\\circ\\psi(c)=\\bigcup_{e\\in\\psi(c)}Id_{\\mathcal{A}}(e)=\\bigcup_{e\\in\\psi(c)}\\{e\\}=\\psi(c),\\]\nso that $\\varphi\\circ Id_{\\mathcal{A}}=\\varphi$ and $Id_{\\mathcal{A}}\\circ\\psi=\\psi$, and $Id_{\\mathcal{A}}$ is indeed an identity.\n\nTo see that the composition is well-defined, fix $\\sigma\\in\\Sigma$ and $a_{1}, \\dotsc  , a_{n}\\in A$.\n\n\\begin{enumerate}\n\\item If $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ are multihomomorphisms, \n\\[\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\psi\\circ\\varphi(a)=\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\bigcup_{b\\in \\varphi(a)}\\psi(b);\\]\nsince \n\\[\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\varphi(a)\\subseteq \\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in\\varphi(a_{1})\\times\\cdots\\times\\varphi(a_{n})}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n}),\\]\nwe have that the last set on the equality is contained in \n\\[\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\varphi(a_{1})\\times\\cdots\\times\\varphi(a_{n})}\\bigcup_{b\\in \\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})}\\psi(b)\\subseteq\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\varphi(a_{1})\\times\\cdots\\times\\varphi(a_{n})}\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in \\psi(b_{1})\\times\\cdots\\times\\psi(b_{n})}\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n});\\]\nby Lemma \\ref{lemma about unions}, for $X_{i}=\\varphi(a_{i})$ and $Y_{x}=\\psi(x)$, this last set is contained in\n\\[\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in \\bigcup_{b_{1}\\in \\varphi(a_{1})}\\psi(b_{1})\\times\\cdots\\times\\bigcup_{b_{n}\\in \\varphi(a_{n})}\\psi(b_{n})}\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n})=\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in \\psi\\circ\\varphi(a_{1})\\times\\cdots\\times\\psi\\circ\\varphi(a_{n})}\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n}),\\]\nand from that $\\psi\\circ\\varphi$ is also a multihomomorphism.\n\n\\item If $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ are full multihomomorphisms, is enough to replace all occurrences of \"$\\subseteq$\" in the proof above by \"$=$\" to obtain a proof that $\\psi\\circ\\varphi$ is also a full multihomomorphism.\n\\end{enumerate}\n\nFinally, it remains to be proved that such a notion of composition is associative: let $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$, $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ and $\\theta:\\mathcal{C}\\rightarrow\\mathcal{D}$ be multihomomorphisms and $a\\in A$; then, using Lemma \\ref{lemma about unions} again, we have that\n\\[[\\theta\\circ(\\psi\\circ\\varphi)](a)=\\bigcup_{c\\in \\psi\\circ\\varphi(a)}\\theta(c)=\\bigcup_{c\\in \\bigcup_{b\\in\\varphi(a)}\\psi(b)}\\theta(c)=\\bigcup_{b\\in\\varphi(a)}\\bigcup_{c\\in\\psi(b)}\\theta(c)=\\]\n\\[\\bigcup_{b\\in\\varphi(a)}\\theta\\circ\\psi(b)=[(\\theta\\circ\\psi)\\circ\\varphi](a)\\]\n\\end{proof}\n\nWe can arrange the categories that have so far appeared in our study of multialgebras in the following helpful diagram.\n\n\\[ \\begin{tikzcd}\n\\textbf{MAlg}_{=}(\\Sigma)\\arrow[hook]{dd}\\arrow{rr}{J_{=}}  & & \\textbf{MMAlg}_{=}(\\Sigma)\\arrow[hook]{dd}\\\\\n&&\\\\\n\\textbf{MAlg}(\\Sigma) \\arrow{rr}{J} & & \\textbf{MMAlg}(\\Sigma)\n\\end{tikzcd}\n\\]\n\nIt is clear that $\\textbf{MAlg}_{=}(\\Sigma)$ is a subcategory of $\\textbf{MAlg}(\\Sigma)$, and that $\\textbf{MMAlg}_{=}(\\Sigma)$ is a subcategory of $\\textbf{MMAlg}(\\Sigma)$, all of these four categories having as objects the class of all $\\Sigma$-multialgebras. And, although $\\textbf{MAlg}_{=}(\\Sigma)$ is not a subcategory of $\\textbf{MMAlg}_{=}(\\Sigma)$, nor is $\\textbf{MAlg}(\\Sigma)$ a subcategory of $\\textbf{MMAlg}(\\Sigma)$, we can easily define functors \n\\[J_{=}:\\textbf{MAlg}_{=}(\\Sigma)\\rightarrow \\textbf{MMAlg}_{=}(\\Sigma)\\quad\\text{and}\\quad J:\\textbf{MAlg}(\\Sigma)\\rightarrow \\textbf{MMAlg}(\\Sigma)\\]\nsuch that the image of $J$ (respectively $J_{=}$) is a subcategory of $\\textbf{MMAlg}(\\Sigma)$ ($\\textbf{MMAlg}_{=}(\\Sigma)$) isomorphic to $\\textbf{MAlg}(\\Sigma)$ ($\\textbf{MAlg}_{=}(\\Sigma)$); we define $J$ ($J_{=}$) as the identity on objects, and for a (full) homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$, the (full) multihomomorphism $J\\varphi$ (respectively $J_{=}\\varphi$) from $\\mathcal{A}$ to $\\mathcal{B}$, on an element $a$ of $\\mathcal{A}$, is just $\\{\\varphi(a)\\}$.\n\n\n\n\\section{Formulas and how to interpret them}\\label{Formulas and how to interpret them}\n\n\\begin{definition}\nGiven a set $\\mathcal{V}$, whose elements we will call propositional variables, we define the formulas\\index{Formula} over the signature $\\Sigma$ on the variables $\\mathcal{V}$ by recursion, and only by the following rules:\n\\begin{enumerate}\n\\item all elements of $\\mathcal{V}$ are formulas;\n\\item all elements of $\\Sigma_{0}$ are formulas;\n\\item if $\\sigma\\in\\Sigma_{n}$ and $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ are formulas, $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$ is a formula.\n\\end{enumerate}\n\\end{definition}\n\nHowever, the expression $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$ is not entirely formally defined: in what would be the correct, formal definition, yet not as clear, a formula over the signature $\\Sigma$ on the variables $\\mathcal{V}$ is any function $f:I_{k}\\rightarrow\\mathcal{V}\\cup\\bigcup_{n\\in\\mathbb{N}}\\Sigma_{n}$, where $I_{k}=\\{0, \\dotsc  , k\\}$ is a initial segment of $\\mathbb{N}$, such that:\n\\begin{enumerate}\n\\item either $k=0$ and $f(0)\\in\\mathcal{V}\\cup\\Sigma_{0}$;\n\\item or there exist $m\\in\\mathbb{N}\\setminus\\{0\\}$, $\\sigma\\in\\Sigma_{m}$ and formulas \n\\[f_{1}:I_{k_{1}}\\rightarrow \\mathcal{V}\\cup\\bigcup_{n\\in\\mathbb{N}}\\Sigma_{n}\\quad\\text{through}\\quad f_{m}:I_{k_{m}}\\rightarrow\\mathcal{V}\\cup\\bigcup_{n\\in\\mathbb{N}}\\Sigma_{n}\\]\nsuch that $k=1+\\sum_{j=1}^{m}k_{j}$, $f(0)=\\sigma$ and\n\\[\\text{$f(i-k_{p}+\\sum_{j=1}^{p}k_{j})=f_{p}(i)$, for $p\\in\\{1, \\dotsc  , m\\}$ and $i\\in I_{k_{p}}$.}\\]\n\\end{enumerate}\n\nThe set of all formulas over the signature $\\Sigma$ on the variables $\\mathcal{V}$ is denoted by $F(\\Sigma, \\mathcal{V})$\\label{FSigmaV}. If we define, for a $\\sigma\\in\\Sigma_{n}$, the function $\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}:F(\\Sigma, \\mathcal{V})^{n}\\rightarrow F(\\Sigma, \\mathcal{V})$ to be, for formulas $\\alpha_{1}, \\dotsc  , \\alpha_{n}$,\n\\[\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}(\\alpha_{1}, \\dotsc  , \\alpha_{n})=\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}),\\]\nwe obtain a $\\Sigma$-algebra \n\\[\\textbf{F}(\\Sigma, \\mathcal{V})=(F(\\Sigma, \\mathcal{V}), \\{\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}\\}_{\\sigma\\in\\Sigma}),\\]\n\\label{tFSigmaV}said to be the free $\\Sigma$-algebra on the variables $\\mathcal{V}$, or the $\\Sigma$-algebra of formulas on the variables $\\mathcal{V}$.\n\nWe will also use $\\textbf{F}(\\Sigma, \\mathcal{V})$ to denote the corresponding $\\Sigma$-multialgebra.\n\n\\begin{definition}\nWe define the order\\index{Order of a formula}\\label{order}, or complexity\\index{Complexity of a formula}, $|\\alpha|$ of a formula $\\alpha$ in $F(\\Sigma, \\mathcal{V})$ as:\n\\begin{enumerate}\n\\item $0$, if $\\alpha$ is an element of $\\mathcal{V}$;\n\\item $0$, if $\\alpha$ is an element of $\\Sigma_{0}$;\n\\item if $\\alpha$ is of the form $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$, \n\\[|\\alpha|=1+\\max_{1\\leq i\\leq n}|\\alpha_{i}|.\\]\n\\end{enumerate}\n\\end{definition}\n\nNow, we have given the simplest definition of formula: when we approach choice-dependent freely generated multialgebras, we will find what one can identify as a broader definition of formula, but with which we shall not deal with until later. \n\n\n\\subsection{Different notions of interpretation}\n\nPerhaps more important than defining what is a formula is interpreting this formula, and here the non-determinism of multialgebras once again gives us an array of different notions.\n\n\\begin{definition}\nGiven a $\\Sigma$-multialgebra $\\mathcal{A}$, a homomorphism $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow \\mathcal{A}$ is called a legal valuation.\\index{Valuation, Legal}\n\\end{definition}\n\nThe notion of legal valuation is one attempt to give an interpretation of a formula $\\alpha$: in this case, the formula $\\alpha$, under the legal valuation $\\nu$, takes the value $\\nu(\\alpha)$.\n\nA serious problem that arises from the notion of legal valuation is that one interpretation of the variables can lead to several legal valuations, and therefore different interpretations of a formula. This does not occur to $\\Sigma$-algebras, where one interpretation of the variables implies an unique interpretation of the formulas.\n\n\\begin{example}\nTake the signature $\\Sigma$ with $\\Sigma_{1}=\\{s\\}$ and $\\Sigma_{n}=\\emptyset$ for $n\\neq 1$; take the $\\Sigma$-multialgebras $\\textbf{F}(\\Sigma,  \\mathcal{V})$, for $\\mathcal{V}=\\{x\\}$, and $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ such that $A=\\{0,1\\}$ and $s_{\\mathcal{A}}(0)=s_{\\mathcal{A}}(1)=\\{0,1\\}$.\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tikzcd}[row sep=1cm]\n\\vdots\\\\\ns^{2}(x)\\arrow{u}{s_{\\textbf{F}(\\Sigma, \\mathcal{V})}}\\\\\ns(x)\\arrow{u}{s_{\\textbf{F}(\\Sigma, \\mathcal{V})}}\\\\\nx\\arrow{u}{s_{\\textbf{F}(\\Sigma, \\mathcal{V})}}\n\\end{tikzcd}\n\\caption*{$\\textbf{F}(\\Sigma, \\mathcal{V})$}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tikzcd}[column sep=1cm]\n0\\arrow[loop left, out=150, in=-150, distance=3em, swap]{}{s_{\\mathcal{A}}}\\arrow[shift right, swap]{r}{s_{\\mathcal{A}}} & 1\\arrow[loop right, out=30, in=-30, distance=3em]{}{s_{\\mathcal{A}}}\\arrow[shift right, swap]{l}{s_{\\mathcal{A}}}\n\\end{tikzcd}\n\\caption*{$\\mathcal{A}$}\n\\end{minipage}\n\\end{figure}\nWe state that $\\nu_{1}:F(\\Sigma, \\mathcal{V})\\rightarrow A$, given by $\\nu_{1}(\\alpha)=0$ for every $\\alpha\\in F(\\Sigma, \\mathcal{V})$, and $\\nu_{2}:F(\\Sigma, \\mathcal{V})\\rightarrow A$, given by $\\nu_{2}(\\alpha)=1$ for every $\\alpha\\in F(\\Sigma, \\mathcal{V})\\setminus\\{x\\}$ and $\\nu_{2}(x)=0$, are homomorphisms. In fact, for every $\\alpha\\in F(\\Sigma, \\mathcal{V})$, we have that\n\\[\\{\\nu_{i}(\\beta)\\ :\\  \\beta\\in s_{\\textbf{F}(\\Sigma, \\mathcal{V})}(\\alpha)\\}=\\{\\nu_{i}(s(\\alpha))\\}\\subseteq \\{0,1\\}=s_{\\mathcal{A}}(\\nu_{i}(\\alpha)),\\]\nwhere $i\\in \\{1,2\\}$. So $\\nu_{1}$ and $\\nu_{2}$ are different legal valuations, despite being the same over the variables.\n\\end{example}\n\nWe will say that a legal valuation $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ is associated\\index{Valuation, Associated legal} to the function $\\chi:\\mathcal{V}\\rightarrow A$ given by $\\chi=\\nu|_{\\mathcal{V}}$,\\footnote{Here, it is important to clarify the notation: for a function $f:X\\rightarrow Y$ and a set $Z\\subset X$, we will denote by $f|_{Z}$ the restriction of $f$ to $Z$.} which we shall call a valuation or interpretation of the variables; more generally, we say $\\nu$ is associated to the function $\\chi:\\mathcal{V}\\rightarrow\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ if $\\nu(x)\\in \\chi(x)$ for every $x\\in\\mathcal{V}$. What we have shown in the previous example is that two distinct legal valuations may be associated to the same interpretation of the variables.\n\nSo, if legal valuations are problematic, what other definition should we take? Many attempts to correctly interpret a formula in a multialgebra have been made, each with its own advantages and drawbacks. When studying choice-dependent freely generated multialgebras, we will show that maybe one needs to specify both an interpretation of the variables and what we shall call a collection of choices. For now, we shall offer a few other possibilities.\n\n\\begin{definition}\nGiven a $\\Sigma$-multialgebra $\\mathcal{A}$, a full multihomomorphism $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ is called a full valuation\\index{Valuation, Full}.\n\\end{definition}\n\n\\begin{proposition}\nTwo full valuations $\\nu_{1}, \\nu_{2}:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ such that $\\nu_{1}|_{\\mathcal{V}}=\\nu_{2}|_{\\mathcal{V}}$ are the same.\n\\end{proposition}\n\n\\begin{proof}\nWe will prove that, for any formula $\\alpha$, $\\nu_{1}(\\alpha)=\\nu_{2}(\\alpha)$ by induction on the order of $\\alpha$. If $|\\alpha|=0$, either $\\alpha$ is an element $x\\in\\mathcal{V}$ and then $\\nu_{1}(x)=\\nu_{2}(x)$ since $\\nu_{1}|_{\\mathcal{V}}=\\nu_{2}|_{\\mathcal{V}}$, or $\\alpha$ is a $\\sigma\\in\\Sigma_{0}$, and then \n\\[\\nu_{1}(\\alpha)=\\sigma_{\\mathcal{A}}=\\nu_{2}(\\alpha).\\]\n\nAssuming the proposition is true for formulas of order at most $m$, if $\\alpha=\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$ is a formula of order $m+1$ then $\\alpha_{1}$ through $\\alpha_{n}$ are formulas of order at most $m$, and therefore $\\nu_{1}(\\alpha_{1})=\\nu_{2}(\\alpha_{2}), \\dotsc  , \\nu_{1}(\\alpha_{n})=\\nu_{2}(\\alpha_{n})$. It follows that \n\\[\\nu_{1}(\\alpha)=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\nu_{1}(\\alpha_{1})\\times\\cdots\\nu_{1}(\\alpha_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\nu_{2}(\\alpha_{1})\\times\\cdots\\nu_{2}(\\alpha_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\nu_{2}(\\alpha).\\]\n\\end{proof}\n\nSo, to every function $\\chi:\\mathcal{V}\\rightarrow \\mathcal{P}(A)\\setminus\\{\\emptyset\\}$, there corresponds at most one full valuation: but we can also show that there is at least one full valuation associated to $\\chi$, which we denote by $\\overline{\\chi}$. We define it by induction on the order of a formula $\\alpha$:\n\\begin{enumerate}\n\\item if $|\\alpha|=0$, either $\\alpha$ is a $x\\in\\mathcal{V}$, when we define $\\overline{\\chi}(\\alpha)=\\chi(x)$, or $\\alpha$ is a $\\sigma\\in\\Sigma_{0}$, when $\\overline{\\chi}(\\alpha)=\\sigma_{\\mathcal{A}}$;\n\\item supposing $\\overline{\\chi}$ is defined for formulas of degree at most $m$, if $\\alpha=\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$ is of degree $m+1$, and so $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ are of degree at most $m$, we define\n\\[\\overline{\\chi}(\\alpha)=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in\\overline{\\chi}(\\alpha_{1})\\times\\cdots\\times\\overline{\\chi}(\\alpha_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}).\\]\n\\end{enumerate}\n\nWe can generalize both legal and full valuations through what we will call simply valuations: any multihomomorphism $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow \\mathcal{A}$ will be said to be a valuation on $\\mathcal{A}$, associated to any of the functions $\\chi:\\mathcal{V}\\rightarrow \\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ such that $\\nu(x)\\subseteq\\chi(x)$ for every $x\\in\\mathcal{V}$.\n\nIt is quite clear how this is a generalization of a full valuation, but one can see this is not a direct generalization of a legal valuation, since those are function from $F(\\Sigma, \\mathcal{V})$ to $A$, instead of $\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$; however, when one considers for the legal valuation $\\nu$ the function $\\nu^{*}:F(\\Sigma, \\mathcal{V})\\rightarrow \\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ such that $\\nu^{*}(\\alpha)=\\{\\nu(\\alpha)\\}$ for every $\\alpha\\in F(\\Sigma, \\mathcal{V})$, one sees that $\\nu^{*}$ is a valuation.\n\nWithout much further ado, we will identify $\\nu^{*}$ with $\\nu$, and characterize some valuations as legal ones.\n\n\\begin{proposition}\nFixed $\\textbf{F}(\\Sigma,\\mathcal{V})$, the set $V(\\mathcal{A})$ of all valuations from it to the $\\Sigma$-multialgebra $\\mathcal{A}$ is a partially ordered set, when for valuations $\\nu_{1}, \\nu_{2}\\in V(\\mathcal{A})$ one considers $\\nu_{1}\\leq \\nu_{2}$ if and only if \n\\[\\nu_{1}(\\alpha)\\subseteq \\nu_{2}(\\alpha),\\quad \\forall \\alpha\\in F(\\Sigma, \\mathcal{V}),\\]\nclosed under suprema of non-empty sets, with as minimal elements the set of all legal valuations $LV(\\mathcal{A})$  on $\\mathcal{A}$.\n\\end{proposition}\n\n\\begin{proof}\nFirst, we prove that the mentioned order is in fact an order.\n\\begin{enumerate}\n\\item For any $\\nu\\in V(\\mathcal{A})$ and all $\\alpha\\in F(\\Sigma, \\mathcal{V})$, it is true that $\\nu(\\alpha)\\subseteq\\nu(\\alpha)$, and therefore $\\nu\\leq \\nu$.\n\\item For any two $\\nu_{1}, \\nu_{2}\\in V(\\mathcal{A})$, if $\\nu_{1}\\leq\\nu_{2}$ and $\\nu_{2}\\leq \\nu_{1}$, then for every $\\alpha\\in F(\\Sigma, \\mathcal{V})$ we have that $\\nu_{1}(\\alpha)\\subseteq\\nu_{2}(\\alpha)$ and $\\nu_{2}(\\alpha)\\subseteq\\nu_{1}(\\alpha)$, and therefore $\\nu_{1}(\\alpha)=\\nu_{2}(\\alpha)$; it follows that $\\nu_{1}=\\nu_{2}$.\n\\item For any three $\\nu_{1}, \\nu_{2}, \\nu_{3}\\in V(\\mathcal{A})$, if $\\nu_{1}\\leq\\nu_{2}$ and $\\nu_{2}\\leq\\nu_{3}$, then for every $\\alpha\\in F(\\Sigma, \\mathcal{V})$ we see that $\\nu_{1}(\\alpha)\\subseteq\\nu_{2}(\\alpha)$ and $\\nu_{2}(\\alpha)\\subseteq\\nu_{3}(\\alpha)$, and therefore $\\nu_{1}(\\alpha)\\subseteq\\nu_{3}(\\alpha)$; it follows that $\\nu_{1}\\leq\\nu_{3}$.\n\\end{enumerate}\n\nNow, we state that the supremum of a non-empty set $\\Lambda\\subseteq V(\\mathcal{A})$ is the valuation such that, for every $\\alpha\\in F(\\Sigma, \\mathcal{V})$, \n\\[\\nu(\\alpha)=\\bigcup_{\\lambda\\in \\Lambda}\\lambda(\\alpha);\\]\nit is, in fact, a valuation since it is a multihomomorphism: if $\\sigma\\in\\Sigma_{n}$ and $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ are formulas in $F(\\Sigma, \\mathcal{V})$, we have that\n\\[\\nu(\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}))=\\bigcup_{\\lambda\\in\\Lambda}\\lambda(\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}))\\subseteq\\bigcup_{\\lambda\\in\\Lambda}\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\lambda(\\alpha_{1})\\times\\cdots\\times\\lambda(\\alpha_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\subseteq\\]\n\\[\\bigcup_{\\lambda\\in\\Lambda}\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\nu(\\alpha_{1})\\times\\cdots\\times\\nu(\\alpha_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\nu(\\alpha_{1})\\times\\cdots\\times\\nu(\\alpha_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}).\\]\n\nClearly $\\nu$ is an upper bound of $\\Lambda$, so suppose $\\nu^{*}$ is another upper bound: for every formula $\\alpha$ and $\\lambda\\in \\Lambda$ we have that $\\lambda(\\alpha)\\subseteq \\nu^{*}(\\alpha)$, and from that $\\nu(\\alpha)=\\bigcup_{\\lambda\\in\\Lambda}\\lambda(\\alpha)\\subseteq \\nu^{*}(\\alpha)$, that is, $\\nu\\leq\\nu^{*}$ and so $\\nu=\\sup\\Lambda$.\n\nSince for any legal valuation $\\nu\\in LV(\\mathcal{A})$ and formula $\\alpha$ the set $\\nu(\\alpha)$ is always of cardinality $1$, legal valuations are clearly minimal elements.\n\\end{proof}\n\nFor a function $\\chi:\\mathcal{V}\\rightarrow\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$, we would like to consider the set \n\\[V(\\chi)=\\{\\nu\\in V(\\mathcal{A})\\ :\\  \\nu\\leq\\overline{\\chi}\\}.\\]\nIt is clearly a partially ordered set, containing $\\overline{\\chi}$ and all valuations, legal or not, associated to $\\chi$; it has as minimal elements $LV(\\chi)=LV(\\mathcal{A})\\cap V(\\chi)$ and maximum $\\overline{\\chi}$; it is closed under suprema since for any $\\Lambda\\subseteq V(\\chi)$ we have that $\\overline{\\chi}$ is an upper bound for $\\Lambda$, and therefore $\\sup\\Lambda\\leq \\overline{\\chi}$, meaning that $\\sup\\Lambda\\in V(\\chi)$.\n\nNotice that, alternatively, $\\nu\\in V(\\chi)$ if, and only if, $\\nu$ is a valuation for which, for every $x\\in\\mathcal{V}$, $\\nu(x)\\subseteq \\chi(x)$.\n\nThe join-semilattice $V(\\chi)$ is then a very interesting object: if one is in doubt of how to interpret a formula given the interpretation of variables $\\chi$, using valuations, legal valuations or full valuations, $V(\\chi)$ captures the information offered by all of these attempts, despite being considerably more complex.\n\n \n\n\\subsection{Full valuations and adjointedness}\n\nTake the class of all sets and, as morphisms between any two sets $X$ and $Y$, take the functions $f:X\\rightarrow\\mathcal{P}(Y)\\setminus\\{\\emptyset\\}$, with the composition of two such functions $f:X\\rightarrow\\mathcal{P}(Y)\\setminus\\{\\emptyset\\}$ and $g:Y\\rightarrow\\mathcal{P}(Z)\\setminus\\{\\emptyset\\}$ being defined as the function $g\\circ f:X\\rightarrow\\mathcal{P}(Z)\\setminus\\{\\emptyset\\}$ such that, for any $x\\in X$, \n\\[g\\circ f(x)=\\bigcup_{y\\in f(x)}g(y).\\]\n\nAs we saw before, such a composition is associative, and has as identity, for a given set $X$, the function $Id_{X}:X\\rightarrow\\mathcal{P}(X)\\setminus\\{\\emptyset\\}$ such that, for every $x\\in X$, $Id_{X}(x)=\\{x\\}$. Therefore, this object is a category, which we will call the category of sets with multifunctions, and denote by $\\textbf{MSet}$\\label{MSet}; its morphisms will be called multifunctions.\n\nNow, fixed a signature $\\Sigma$, take for every set $X$ the $\\Sigma$-multialgebra $FX=\\textbf{F}(\\Sigma, X)$ and, for every morphism $f$ from $X$ to $Y$, that is, function $f:X\\rightarrow \\mathcal{P}(Y)\\setminus\\{\\emptyset\\}$, take the full multihomomorphism $Ff=\\overline{f}:\\textbf{F}(\\Sigma, X)\\rightarrow \\textbf{F}(\\Sigma, Y)$.\n\nGiven the identity $Id_{X}:X\\rightarrow X$ on $\\textbf{MSet}$, we state that $\\overline{Id_{X}}:\\textbf{F}(\\Sigma, X)\\rightarrow \\textbf{F}(\\Sigma, X)$ is exactly the identity $Id_{\\textbf{F}(\\Sigma, X)}$ of this multialgebra in $\\textbf{MMAlg}_{=}(\\Sigma)$: one can see this by induction over the order of a formula $\\alpha$.\n\\begin{enumerate}\n\\item If $\\alpha$ is of order $0$, either $\\alpha$ is an $x\\in X$ and then $\\overline{Id_{X}}(x)=Id_{X}(x)=\\{x\\}$, or $\\alpha$ is a $\\sigma\\in\\Sigma_{0}$, when $\\overline{Id_{X}}(\\sigma)=\\sigma_{\\textbf{F}(\\Sigma, X)}=\\{\\sigma\\}$.\n\\item If $\\alpha=\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$, for $\\sigma\\in\\Sigma_{n}$ and $\\alpha_{1}$ through $\\alpha_{n}$ of order less than that of $\\alpha$, we have that\n\\[\\overline{Id_{X}}(\\alpha)=\\bigcup_{(\\beta_{1}, \\dotsc  , \\beta_{n})\\in \\overline{Id_{X}}(\\alpha_{1})\\times\\cdots\\times\\overline{Id_{X}}(\\alpha_{n})}\\sigma_{\\textbf{F}(\\Sigma, X)}(\\beta_{1}, \\dotsc  , \\beta_{n})=\\bigcup_{(\\beta_{1}, \\dotsc  , \\beta_{n})\\in \\{\\alpha_{1}\\}\\times\\cdots\\times\\{\\alpha_{n}\\}}\\{\\sigma(\\beta_{1}, \\dotsc  , \\beta_{n})\\}=\\]\n\\[\\{\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})\\}=\\{\\alpha\\}.\\]\n\\end{enumerate}\n\nFurthermore, given morphisms $f$ from $X$ to $Y$ and $g$ from $Y$ to $Z$ in $\\textbf{MSet}$, we see that for every $x\\in X$ \n\\[\\overline{g}\\circ\\overline{f}(x)=\\bigcup_{y\\in \\overline{f}(x)}\\overline{g}(y)=\\bigcup_{y\\in f(x)}\\overline{g}(y),\\]\nand since $f(x)\\subseteq Y$, for any $y\\in f(x)$ we have that $\\overline{g}(y)=g(y)$ and therefore\n\\[\\bigcup_{y\\in f(x)}\\overline{g}(y)=\\bigcup_{y\\in f(x)}g(y)=g\\circ f(x)=\\overline{g\\circ f}(x),\\]\nsince $g\\circ f$ goes from $X$ to $Z$; therefore both $\\overline{g}\\circ\\overline{f}$ and $\\overline{g\\circ f}$ are full multihomomorphisms equal over the variables, and are therefore equal. \n\nSince $FId_{X}=Id_{FX}$ and $Fg\\circ Ff=F(g\\circ f)$, we have proved\n\\[F:\\textbf{MSet}\\rightarrow\\textbf{MMAlg}_{=}(\\Sigma)\\]\nis a functor. Now, we can also consider the forgetful functor\n\\[\\mathcal{U}:\\textbf{MMAlg}_{=}(\\Sigma)\\rightarrow \\textbf{MSet},\\]\nsince full multihomomorphisms are multifunctions; what we shall prove now is that $F$ and $\\mathcal{U}$ are actually adjoint. So, we consider the functions, for a set $X$ and a $\\Sigma$-multialgebra $\\mathcal{A}$,\n\\[\\Phi_{\\mathcal{A}, X}:Hom_{\\textbf{MSet}}(X, \\mathcal{U}\\mathcal{A})\\rightarrow Hom_{\\textbf{MMAlg}_{=}(\\Sigma)}(FX, \\mathcal{A})\\]\nassociating a map $f:X\\rightarrow\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ to the full multihomomorphism $\\overline{f}:\\textbf{F}(\\Sigma, X)\\rightarrow\\mathcal{A}$ extending $f$. Since, for every $f$, there corresponds one and only one $\\overline{f}$, the $\\Phi_{\\mathcal{A}, X}$ are bijections.\n\nNow, given sets $X$ and $Y$, $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, a morphism $f$ from $Y$ to $X$ in $\\textbf{MSet}$ and a full multihomomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$, we must only prove that the following diagram commutes.\n\n\\[ \\begin{tikzcd}[row sep=5em, column sep=3em]\nHom_{\\textbf{MSet}}(X, \\mathcal{U}\\mathcal{A}) \\arrow{r}{\\Phi_{\\mathcal{A}, X}} \\arrow{d}{Hom(f, \\mathcal{U}\\varphi)}& Hom_{\\textbf{MMAlg}_{=}(\\Sigma)}(FX, \\mathcal{A}) \\arrow{d}{Hom(Ff, \\varphi)} \\\\%\nHom_{\\textbf{MSet}}(Y, \\mathcal{U}\\mathcal{B}) \\arrow{r}{\\Phi_{\\mathcal{B}, Y}}& Hom_{\\textbf{MMAlg}_{=}(\\Sigma)}(FY, \\mathcal{B})\n\\end{tikzcd}\n\\]\n\nSo, denoting the universe of $\\mathcal{A}$ by $A$ as usual, we take a function $g:X\\rightarrow\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ in $Hom_{\\textbf{MSet}}(X, \\mathcal{U}\\mathcal{A})$: on the top edge of the diagram we obtain the full multihomomorphism $\\Phi_{\\mathcal{A}, X}(g)=\\overline{g}$ from $\\textbf{F}(\\Sigma, X)$ to $\\mathcal{A}$; and on the right edge we obtain the once again full multihomomorphism $\\varphi\\circ \\overline{g}\\circ\\overline{f}$, since $Ff=\\overline{f}$, from $\\textbf{F}(\\Sigma, Y)$ to $\\mathcal{B}$.\n\nOn the left edge, we obtain the multifunction $\\varphi\\circ g\\circ f$ from $Y$ to the universe $B$ of $\\mathcal{B}$, since $\\mathcal{U}\\varphi=\\varphi$; applying $\\Phi_{\\mathcal{B}, Y}$ on the bottom edge of the diagram we obtain $\\overline{\\varphi\\circ g\\circ f}$, also a full multihomomorphism from $\\textbf{F}(\\Sigma, Y)$ to $\\mathcal{B}$.\n\nNow, to prove that the diagram commutes or, what is equivalent, that $\\varphi\\circ\\overline{g}\\circ\\overline{f}=\\overline{\\varphi\\circ g\\circ f}$, since both are full valuations on $\\textbf{F}(\\Sigma, Y)$ is enough to prove that, when restricted to $Y$, both are the same. So, for $y\\in Y$,\n\\[\\varphi\\circ\\overline{g}\\circ\\overline{f}(y)=\\bigcup_{x\\in \\overline{f}(x)}\\varphi\\circ\\overline{g}(x)=\\bigcup_{x\\in\\overline{f}(x)}\\bigcup_{a\\in \\overline{g}(x)}\\varphi(a)=\\bigcup_{x\\in f(x)}\\bigcup_{a\\in\\overline{g}(x)}\\varphi(a)=\\bigcup_{x\\in f(y)}\\bigcup_{a\\in g(x)}\\varphi(a)=\\]\n\\[\\bigcup_{x\\in f(y)}\\varphi\\circ g(x)=\\varphi\\circ g\\circ f(y)=\\overline{\\varphi\\circ g\\circ f}(y),\\]\nwhat ends the proof that $F$ and $\\mathcal{U}$ are adjoint.\n\n\n\\begin{proposition}\nLet $\\chi:\\mathcal{V}\\rightarrow\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ be a function and $\\varphi:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ a multihomomorphism such that, for every $x\\in\\mathcal{V}$, $\\varphi(x)\\subseteq\\chi(x)$: then $\\varphi\\leq \\overline{\\chi}$.\n\\end{proposition}\n\n\\begin{proof}\nIt is enough to proceed by induction over the order of a formula $\\alpha$.\n\nIf it is $0$, either $\\alpha$ is an element $x\\in\\mathcal{V}$, when the result is true by hypothesis; or $\\alpha$ is a $\\sigma\\in\\Sigma_{0}$, and since $\\varphi$ is a multihomomorphism we have that $\\varphi(\\sigma)\\subseteq \\sigma_{\\mathcal{A}}=\\overline{\\chi}(\\sigma)$.\n\nNow, suppose the result is true for formulas of order smaller than that of $\\alpha$, and suppose $\\alpha$ is of the form $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$: then \n\\[\\varphi(\\alpha)=\\bigcup_{\\beta\\in\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}(\\alpha_{1}, \\dotsc  , \\alpha_{n})}\\varphi(\\beta)\\subseteq\\bigcup_{(\\beta_{1}, \\dotsc  , \\beta_{n})\\in \\varphi(\\alpha_{1})\\times\\cdots\\times\\varphi(\\alpha_{n})}\\sigma_{\\mathcal{A}}(\\beta_{1}, \\dotsc  , \\beta_{n})\\subseteq\\]\n\\[\\bigcup_{(\\beta_{1}, \\dotsc  , \\beta_{n})\\in\\overline{\\chi}(\\alpha_{1})\\times\\cdots\\times\\overline{\\chi}(\\alpha_{n})}\\sigma_{\\mathcal{A}}(\\beta_{1}, \\dotsc  , \\beta_{n})=\\bigcup_{\\beta\\in\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}(\\alpha_{1}, \\dotsc  , \\alpha_{n})}\\overline{\\chi}(\\beta)=\\overline{\\chi}(\\alpha).\\]\n\\end{proof}\n\nThis way, we see that full valuations have many important properties: they allow, together with the algebras of formulas, to build a functor adjoint to a forgetful functor on a rather natural category ($\\textbf{MSet}$); furthermore, they are maximal among valuations associated to a given evaluation of the variables.\n\nAnd, at the same time, they are not very useful to most applications in logic, one reason being that, from the point of view of full valuations, the free objects of $\\textbf{MMAlg}_{=}(\\Sigma)$ are the algebras of formulas $\\textbf{F}(\\Sigma, \\mathcal{V})$, meaning we gain no new structures to analyze. A second reason for or disinterest on full valuations is that they are not at all very precise: legal valuations, that are only functions instead of multifunctions, are much more useful and \"precise\". Unfortunately, to an evaluation of the variables there are many legal valuations associated: one approach in the direction of fixing this problem is to consider collections of choices (Definition \\ref{Collection of Choices}), which will allow us at the same time to consider generalizations of the algebras of formulas (Definition \\ref{Multialgebra of formulas}).\n\nWe will work those objects in Chapter \\ref{Chapter2}. Their algebraic structure is rather rich, regardless of how difficult their treatment is trough category theory. And, speaking of category theory, the categories shown up until now can be arranged as in the diagram below. The functor $J_{\\textbf{Set}}$ is the identity on sets and takes functions to multifunctions in the obvious way, like $J$ and $J_{=}$; the question of whether the forgetful functor from $\\textbf{MAlg}_{=}(\\Sigma)$ to $\\textbf{Set}$, here represented by a dashed arrow, has a left adjoint is answered negatively at the end of Section \\ref{Freely generated multialgebra}.\n\n\\[ \\begin{tikzcd}\n\\textbf{MAlg}(\\Sigma) \\arrow{dd}{J} && \\textbf{MAlg}_{=}(\\Sigma)\\arrow[hook]{ll}\\arrow[dashed]{rr} \\arrow{dd}{J_{=}} &&\\textbf{Set}\\arrow{dd}{J_{\\textbf{Set}}}  \\\\\n&&&&\\\\\n \\textbf{MMAlg}(\\Sigma) && \\textbf{MMAlg}_{=}(\\Sigma)\\arrow[hook]{ll}\\arrow[transform canvas={yshift=.7ex}]{rr}{\\mathcal{U}} && \\textbf{MSet}\\arrow[transform canvas={yshift=-.7ex}]{ll}{F}\n\\end{tikzcd}\n\\]\n\n\n\n\n\\section{Lattices, and Boolean and Heyting algebras}\\label{Lattices, and Boolean... }\n\nConsider the signature $\\Sigma_{\\textbf{Lat}}$\\label{SigmaLat} with two binary connectives, $\\vee$ and $\\wedge$: that is $(\\Sigma_{\\textbf{Lat}})_{2}=\\{\\vee, \\wedge\\}$ and $(\\Sigma_{\\textbf{Lat}})_{n}=\\emptyset$, for $n\\neq 2$. \n\n\\begin{definition}\nA lattice\\index{Lattice} is a $\\Sigma_{\\textbf{Lat}}$-algebra $\\mathcal{L}=(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma})$ satisfying, for every $x,y,z\\in L$,\n\\begin{enumerate}\n\\item $x\\vee y=y\\vee x$ and $x\\wedge y=y\\wedge x$ (commutative laws);\n\\item $x\\vee(y\\vee z)=(x\\vee y)\\vee z$ and $x\\wedge(y\\wedge z)=(x\\wedge y)\\wedge z$ (associative laws);\n\\item $x\\vee x=x$ and $x\\wedge x=x$ (idempotent laws);\n\\item $x\\vee(x\\wedge y)=x$ and $x\\wedge(x\\vee y)=x$ (absorption laws).\n\\end{enumerate}\n\\end{definition}\nIn this definition, for simplicity, we have denoted $\\vee_{\\mathcal{L}}(x,y)$ and $\\wedge_{\\mathcal{L}}(x,y)$ by, respectively, $x\\vee y$ and $x\\wedge y$.\n\nThis is often regarded as the algebraic definition of a lattice, an order-theoretic one existing as well. A partial order in a set $X$ is a binary relation $\\leq$ on $X$, meaning a subset of $X\\times X$, such that, for all $x, y, z\\in X$,\n\\begin{enumerate}\n\\item $x\\leq x$ (reflexivity);\n\\item $x\\leq y$ and $y\\leq x$ imply $x=y$ (antisymmetry);\n\\item $x\\leq y$ and $y\\leq z$ imply $x\\leq z$ (transitivity),\n\\end{enumerate}\nwhere we will denote the fact that $(x,y)\\in\\leq$ by $x\\leq y$. A set $X$, together with a partial order $\\leq$ in it, will be named a partially ordered set, or poset. \n\nIn a poset $(X, \\leq)$, we say $x$ is an upper bound of $S\\subseteq X$ if $y\\leq x$ for every $y\\in S$; $x$ is a lower bound for $S$ if $x\\leq y$ for every $y\\in S$. Then, the supremum of $S\\subseteq X$, if it exists, is its least upper bound, meaning an upper bound $\\sup S\\in X$ for $S$ such that, for any other upper bound $x$ for $S$, $\\sup S\\leq x$; the infimum of $S$, if it exists, is its greatest lower bound, meaning a lower bound $\\inf S$ for $S$ such that, for any other lower bound $x$ for $S$, $x\\leq \\inf S$.\n\n\\begin{definition}\nA lattice is a poset $\\mathcal{L}=(L,\\leq)$ such that, for any elements $x, y\\in L$, $\\sup\\{x,y\\}$ and $\\inf\\{x,y\\}$ both exist.\n\\end{definition}\n\nOne can translate between the two definitions: given a lattice $\\mathcal{L}$ presented as a $\\Sigma_{\\textbf{Lat}}$-algebra $(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Lat}}})$, we may define an order $\\leq$ in $L$ by \n\\[x\\leq y\\quad\\text{if and only if}\\quad x=x\\wedge y,\\]\nand it is easy to prove that in this case $(L,\\leq)$ is a lattice, where $x\\wedge y$ and $x\\vee y$ are, respectively, the infimum and supremum of $x$ and $y$. Reciprocally, given a lattice $\\mathcal{L}$ presented as a poset $(L,\\leq)$, we may define operations $\\vee$ and $\\wedge$ on $L$ by\n\\[x\\wedge y=\\inf\\{x,y\\}\\quad\\text{and}\\quad x\\vee y=\\sup\\{x,y\\},\\]\nfrom what we obtain a $\\Sigma_{\\textbf{Lat}}$-algebra on which $x\\leq y$ if, and only if, $x=x\\wedge y$.\n\nAn element $0$ of a lattice $\\mathcal{L}$ is a minimum, or bottom element, when either: $x\\vee 0=x$, for every $x\\in L$; $x\\wedge 0=0$, for every $x\\in L$; or $0\\leq x$, for every $x\\in X$. Notice that all three conditions are equivalent. An element $1$ of $\\mathcal{L}$ is a maximum, or top element, when either: $x\\vee 1=1$, for every $x\\in L$; $x\\wedge 1=x$, for every $x\\in L$; or $x\\leq 1$, for every $x\\in L$. Again, all three conditions are equivalent. \n\nTo accommodate bottom and top elements, we extend the signature $\\Sigma_{\\textbf{Lat}}$ to $\\Sigma_{\\textbf{Lat}}^{0}$\\label{SigmaLat0}, $\\Sigma_{\\textbf{Lat}}^{1}$\\label{SigmaLat1} and $\\Sigma_{\\textbf{Lat}}^{0,1}$\\label{SigmaLat01} by adding, respectively, a $0$-ary symbol $0$, a $0$-ary symbol $1$ and $0$-ary symbols $0$ and $1$. For simplicity, we will drop the indexes $\\mathcal{L}$ from $0_{\\mathcal{L}}$ and $1_{\\mathcal{L}}$.\n\n\\begin{definition}\n\\begin{enumerate}\n\\item A lattice with bottom (respectively top) element is a $\\Sigma_{\\textbf{Lat}}^{0}$-algebra ($\\Sigma_{\\textbf{Lat}}^{1}$-algebra) $\\mathcal{L}=(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Lat}}^{0}})$ such that $(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Lat}}})$ is a lattice and $0$ is a bottom element ($1$ is a top element).\n\\item A bounded lattice\\index{Lattice, Bounded} is a $\\Sigma_{\\textbf{Lat}}^{0,1}$-algebra $\\mathcal{L}=(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Lat}}^{0,1}})$ such that $(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Lat}}})$ is a lattice, $0$ is a bottom element and $1$ is a top element.\n\\end{enumerate}\n\\end{definition}\n\nIn a lattice, \"$x$ implies $y$\" is defined as the element\n\\[\\sup\\{z\\in L: x\\wedge z\\leq y\\},\\]\nif it exists, and denoted by $x\\rightarrow y$. We add to the signatures $\\Sigma_{\\textbf{Lat}}^{1}$ and $\\Sigma_{\\textbf{Lat}}^{0,1}$ the binary symbol $\\rightarrow$, obtaining respectively the signatures $\\Sigma_{\\textbf{Imp}}$\\label{SigmaImp} and $\\Sigma_{\\textbf{Hey}}$\\label{SigmaHey}.\n\n\\begin{definition}\n\\begin{enumerate}\n\\item An implicative lattice\\index{Lattice, Implicative} is a $\\Sigma_{\\textbf{Imp}}$-algebra $\\mathcal{L}=(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Imp}}})$ such that $(L, \\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Lat}}^{1}})$ is a lattice with top element, $\\sup\\{z\\in L: x\\wedge z\\leq y\\}$ exists for any two $x,y\\in L$ and \n\\[x\\rightarrow y=\\sup\\{z\\in L: x\\wedge z\\leq y\\}.\\]\n\n\\item A Heyting algebra\\index{Heyting algebra} is a $\\Sigma_{\\textbf{Hey}}$-algebra such that $(L,\\{\\sigma_{\\mathcal{L}}\\}_{\\sigma\\in\\Sigma_{\\textbf{Imp}}})$ is an implicative lattice and $0$ is a bottom element.\n\\end{enumerate}\n\\end{definition}\n\nIt is important here to point out that various different definitions of implicative lattices and Heyting algebras exist in the literature, most, if not all, equivalent to each other; we chose definitions most befitting of our purposes, taken from \\cite{ParLog}.\n\nAnother relevant point to make is that Heyting algebras are no strange beasts: classical propositional logic is to Boolean algebras, which we shall soon formally define, as intuitionistic logic is to Heyting algebras, meaning that not only are Heyting algebras models of intuitionistic logic, but their class characterizes intuitionistic logic (unfortunately, the parallel ends there: although the two-valued Boolean algebra is, by itself, capable of characterizing $\\textbf{CPL}$, no finite algebra, Heyting or not, can do the same to intuitionistic logic, see \\cite{Godel}). One could argue that while intuitionistic logic possesses a negation, Heyting algebras do not, but this can be easily dealt with: it is enough to define\n\\[\\neg x=x\\rightarrow 0.\\]\n\nWith this, we can define Heyting algebras over the signature $\\Sigma_{\\textbf{Boo}}$\\label{SigmaBoo}, obtained from $\\Sigma_{\\textbf{Hey}}$ by addition of an unary symbol \"$\\neg$\", by merely requesting that $\\neg x=x\\rightarrow 0$ for all elements $x$; we say $\\neg x$ is the partial complement, or (intuitionistic) negation of $x$.\n\n\\begin{definition}\nA Boolean algebra\\index{Boolean algebra} is a Heyting algebra over $\\Sigma_{\\textbf{Boo}}$ such that, for every $x$ in its universe, we have the law of excluded middle, or \\textit{tertium non datur}:\n\\[x\\vee\\neg x=1.\\]\n\\end{definition}\n\nThe following lemma will need an observation about the definition of implication on a Heyting algebra: notice that, if $x\\wedge z\\leq y$, then $z\\leq x\\rightarrow y$, by the very definition of $x\\rightarrow y$ as the supremum of $\\{z\\in L: x\\wedge z\\leq y\\}$. Reciprocally, if $z\\leq x\\rightarrow y$, $x\\wedge z\\leq x\\wedge(x\\rightarrow y)\\leq y$,\\footnote{Here we are using that $x\\leq y$ implies $x\\wedge z\\leq y\\wedge z$, but this is trivial to prove: if $x\\leq y$, $x=x\\wedge y$, and so $(x\\wedge z)\\wedge(y\\wedge z)=(x\\wedge y)\\wedge z=x\\wedge z$.} meaning that $z\\leq x\\rightarrow y$ if, and only if, $x\\wedge z\\leq y$. The following proof, although quite standard, follows closely the one found in \\cite{HeytingAlgebras}.\n\n\\begin{lemma}\nIn a Heyting algebra, $\\vee$ is distributive over $\\wedge$ and vice-versa, meaning that\n\\[x\\vee(y\\wedge z)=(x\\vee y)\\wedge(x\\vee z)\\quad\\text{and}\\quad x\\wedge(y\\vee z)=(x\\wedge y)\\vee(x\\wedge z),\\]\nfor all $x$, $y$ and $z$ in the universe.\n\\end{lemma}\n\n\\begin{proof}\nWe will prove that $x\\wedge(y\\vee z)=(x\\wedge y)\\vee(x\\wedge z)$, being the other equality proved analogously. Even more, we need only to prove that $x\\wedge(y\\vee z)\\leq (x\\wedge y)\\vee(x\\wedge z)$, since in any lattice one finds that $y\\leq y\\vee z$ and $z\\leq y\\vee z$, meaning that $x\\wedge y\\leq x\\wedge (y\\vee z)$ and $x\\wedge z\\leq x\\wedge(y\\vee z)$, and therefore $(x\\wedge y)\\vee(x\\wedge z)\\leq x\\wedge (y\\vee z)$.\n\nLet $w$ denote $(x\\wedge y)\\vee(x\\wedge z)$, and since $x\\wedge y\\leq w$ and $x\\wedge z\\leq w$, we obtain $y\\leq x\\rightarrow w$ and $z\\leq x\\rightarrow w$. So $y\\vee z\\leq x\\rightarrow w$, and we get, as desired,\n\\[x\\wedge(y\\vee z)\\leq x\\wedge(x\\rightarrow w)\\leq w=(x\\wedge y)\\vee(x\\wedge z).\\]\n\\end{proof}\n\nIn the following proposition, we omit several more trivial steps, such as proving that $\\neg x\\wedge x=0$. The reader is invited to fill them in as they appear.\n\n\\begin{proposition}\nA Heyting algebra $\\mathcal{H}$, with universe $H$, is a Boolean algebra if, and only if, $x\\vee(x\\rightarrow y)=1$ for all $x, y\\in H$.\n\\end{proposition}\n\n\\begin{proof}\nStart by assuming that we have $x\\vee(x\\rightarrow y)=1$ for all $x, y\\in H$: then, since in particular $0\\in H$, $x\\vee\\neg x=x\\vee(x\\rightarrow 0)=1$ for all $x\\in H$, meaning $\\mathcal{H}$ is a Boolean algebra.\n\nReciprocally, suppose $\\mathcal{H}$ is a Boolean algebra. We state that $x\\rightarrow y=\\neg x\\vee y$, being then necessary to prove that $\\neg x\\vee y$ is the supremum of $\\{z\\in H: x\\wedge z\\leq y\\}$: since \n\\[x\\wedge(\\neg x\\vee y)=(x\\wedge\\neg x)\\vee(x\\wedge y)=0\\vee (x\\wedge y)=x\\wedge y,\\]\nwhich is smaller or equal to $y$, $\\neg x\\vee y$ is in fact an element of this set; furthermore, if $x\\wedge z\\leq y$, then \n\\[z\\leq \\neg x\\vee z=1\\wedge (\\neg x\\vee z)=(\\neg x\\vee x)\\wedge (\\neg x\\vee z)=\\neg x\\vee(x\\wedge z)\\leq \\neg x\\vee y.\\]\n\nThen, since $x\\rightarrow y=\\neg x\\vee y$,\n\\[x\\vee (x\\rightarrow y)=x\\vee(\\neg x\\vee y)=(x\\vee \\neg x)\\vee y=1\\vee y=1,\\]\nas we wished to prove.\n\\end{proof}\n\nOf course, this means we could avoid adding a negation to the signature of Heyting algebras to be able to express Boolean algebras: a Boolean algebra is a $\\Sigma_{\\textbf{Hey}}$-algebra which is a Heyting algebra and satisfies, for all $x$ and $y$ in its universe, $x\\vee(x\\rightarrow y)=1$. But we prefer to have a negation at hand when dealing with Boolean algebras, it is just more convenient that way.\n\nNow, it is important to point out that some symbols on the signature $\\Sigma_{\\textbf{Boo}}$ can be changed, usually because we are inserting Boolean algebras in a context where they are already in use. So $0$ may be replaced with \"$\\bot$\", $1$ with \"$\\top$\" and $\\neg$ with \"$\\sim$\". We will make such changes more or less freely, without much ado.\n\nAs a final remark, we would like to make clear that lattices, implicative or not, and Heyting algebras will play a very minor role in what is to come: they are, more importantly, milestones in defining Boolean algebras that also appear when dealing with some swap structures, specifically for paraconsistent logics (\\cite{ParLog}). Boolean algebras, however, will be used time and time again in our study: in Section \\ref{bottomless Boolean algebras} we will use them to search for a category equivalent to that of multialgebras, when we will offer a more order-theoretic approach to the subject; Section \\ref{B-valuations} shows a new semantics of valuations for da Costa's logics $C_{n}$ based on Boolean algebras, which is used in Section \\ref{RNmatrices RMBCn} to offer yet another semantics for those logics, proven in Section \\ref{BA and RSwap} to generate a category of models for $C_{n}$ isomorphic to the category of Boolean algebras itself.\n\n\n\n\n\n\n\n\n\\newpage\n\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{1}\n\\chapter{Weakly free multialgebras}\\label{Chapter2}\\label{Chapter 2}\n\n\nIn the study of universal algebra, it is well known (\\cite{Burris}) that there exist $\\Sigma$-algebras $\\mathcal{A}$ freely generated\\index{Freely generated} (or absolutely freely generated) by subsets $X$ of their universes $A$, meaning that for any $\\Sigma$-algebra $\\mathcal{B}$, with universe $B$, and function $f:X\\rightarrow B$, there exists precisely one homomorphism $\\overline{f}: \\mathcal{A}\\rightarrow\\mathcal{B}$ extending $f$, and therefore commuting the following diagram in $\\textbf{Set}$, for $j:X\\rightarrow A$ the inclusion.\n\\[\n\\begin{tikzcd}\n A \\arrow{ddrr}{\\overline{f}} & & \\\\\n & & \\\\\nX \\arrow{uu}{j} \\arrow{rr}{f} & & B\n\\end{tikzcd}\n\\]\nMoreover, the $\\Sigma$-algebra freely generated by $X$ is isomorphic to the $\\Sigma$-algebra of formulas over $X$, that is $\\textbf{F}(\\Sigma, X)$, and therefore unique up to isomorphisms; even more, given $\\textbf{F}(\\Sigma, X)$ and $\\textbf{F}(\\Sigma, Y)$ are isomorphic whenever $X$ and $Y$ are of the same cardinality, we discover there is precisely one freely generated $\\Sigma$-algebra, up to isomorphisms, for each cardinality. Equivalently, in the language of categories, the forgetful functor \n\\[U:\\textbf{Alg}(\\Sigma)\\rightarrow\\textbf{Set},\\]\nfrom the category of $\\Sigma$-algebras (with homomorphisms between them) to the category of sets, which takes a $\\Sigma$-algebra and returns its underlying universe, has a left adjoint $F$: it associates to a set $X$ any $\\Sigma$-algebra freely generated by $X$, and to a function the only homomorphism extending it.\n\nHowever, an algebraic structure being absolutely free is a concept that does not extend well to the context of multialgebras (\\cite{Marty}), specially when one restricts oneself to the non-partial multialgebras (whose multioperations do not return the empty-set), as we often do here given our interest on non-deterministic semantics, specifically those designed for paraconsistency. It is easy to prove that, first of all, freely generated multialgebras, which generalize freely generated algebras in the most obvious way, do not exist, and second, that the forgetful functor \n\\[\\mathcal{U}:\\textbf{MAlg}(\\Sigma)\\rightarrow\\textbf{Set},\\]\nfrom the category of multialgebras over the signature $\\Sigma$ to the category of sets, does not have a left adjoint. These results are deeply folkloric, being difficult to pinpoint a proof of them in standard literature, although it seems no one is not aware of their validity.\n\nWe here expose our reasons to believe that understanding as formulas, in $\\textbf{MAlg}(\\Sigma)$, only those elements found in the universe of $\\textbf{F}(\\Sigma, \\mathcal{V})$ disregards other multialgebras with an astoundingly similar behavior, so that we generalize the algebras of formulas to multialgebras of formulas. These structures indeed share many of the properties one expects of the algebras of formulas (or, equivalently, freely generated algebras):\n\n\\begin{enumerate}\n\\item they possess the unique extension property not for functions, but rather pairs of functions and what we will call collections of choices, that ``select'' how a homomorphism will approach indeterminacies; \n\nin fact, given multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ and an $n$-ary $\\sigma$, for all $n$-tuples $(a_{1},\\dotsc , a_{n})\\in A^{n}$ and $(b_{1}, \\dotsc , b_{n})\\in B^{n}$ a collection of choices determines how to map those elements of $\\sigma_{\\mathcal{A}}(a_{1},\\dotsc , a_{n})$ into those of $\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc , b_{n})$;\n\n\\item they are somewhat ``free'' of identities, an intuition we formalize through disconnected multialgebras, and they are generated by a set of ``indecomposable'' elements we shall call the ground of the multialgebra, much like variables (which are formulas without proper subformulas and therefore indecomposable in some sense); \n\nmore formally, a multialgebra is disconnected whenever different operations, or the same operation performed on different elements, always return disjoint results, while the ground $G(\\mathcal{A})$ of a multialgebra $\\mathcal{A}$ is the subset of its universe of all elements $a$ for which there do not exist $\\sigma$ (of arity $n$) and elements $a_{1}, \\dotsc  , a_{n}$ of $\\mathcal{A}$ such that $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$;\n\n\\item strengthening the previous point, they are disconnected and have a minimum generating set that behaves quite similarly to a basis, of e.g. a vector space;\n\nnotice, however, that while a basis of a vector space is a minimal generating set, we are looking here at minimum generating sets, so we use the terminology of ``strong basis'', which end up being precisely the grounds we have mentioned earlier;\n\n\\item a final pair of properties we present is that they are simultaneously disconnected and satisfy that every sequence of further simpler and simpler elements eventually ends on an indecomposable element, condition we call being ``chainless'' and that implies having a strong basis;\n\nessentially, by $a$ being simpler than $b$ we mean that there exist $\\sigma$ (of arity $n$) and elements $a_{1}, \\dotsc  , a_{n}$ such that $a_{i}=a$, for some $i\\in\\{1, \\dotsc  , n\\}$, and $b\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$.\n\\end{enumerate}\n\nFurthermore, we prove all these four listed items are equivalent, characterizing exactly the same multialgebras; and with these weakly free multialgebras we here define at hand, we can offer simple proofs that freely generated multialgebras (now in the naive generalization) do not exist, and that $\\mathcal{U}$ does not have a left adjoint.\n\nMost of the research presented in this chapter was submitted as a preprint in \\cite{AbsFreeHyp} and finally published in \\cite{WeaklyFreeMultialgebras}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Formulas}\n\nHere we briefly recall the basic notions involving formulas found in Section \\ref{Formulas and how to interpret them}. Given a set $\\mathcal{V}$ of propositional variables and a signature $\\Sigma=\\{\\Sigma_{n}\\}_{n\\in\\mathbb{N}}$, the algebra of formulas freely generated by $\\mathcal{V}$ over $\\Sigma$ will be denoted $\\textbf{F}(\\Sigma, \\mathcal{V})$, and its universe will be denoted $F(\\Sigma, \\mathcal{V})$. Intuitively, the set of formulas $F(\\Sigma, \\mathcal{V})$ is the smallest set containing:\n\\begin{enumerate}\n\\item the variables $\\mathcal{V}$;\n\\item the formula $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$, given a $\\sigma\\in\\Sigma_{n}$ and already defined formulas $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ in $F(\\Sigma, \\mathcal{V})$.\n\\end{enumerate}\n\nMore formally, $F(\\Sigma, \\mathcal{V})$ should be smallest set containing:\n\\begin{enumerate}\n\\item for every $x\\in\\mathcal{V}$, the function $f_{x}:\\{0\\}\\rightarrow \\mathcal{V}\\cup \\Sigma$ defined by $f_{x}(0)=x$;\n\\item the function \n\\[f_{\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})}:\\{0, \\dotsc  , 1+\\sum_{i=1}^{n}m_{n}\\}\\rightarrow \\mathcal{V}\\cup\\Sigma,\\]\ngiven a $\\sigma\\in\\Sigma_{n}$ and already defined formulas $f_{\\alpha_{1}}:\\{0, \\dotsc  , m_{1}\\}\\rightarrow\\mathcal{V}\\cup\\Sigma, \\dotsc  , f_{\\alpha_{n}}:\\{0, \\dotsc  , m_{n}\\}\\rightarrow\\mathcal{V}\\cup\\Sigma$, such that $f_{\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})}(0)=\\sigma$ and, for every $j\\in\\{1, \\dotsc  , n\\}$ and $k\\in\\{0, \\dotsc  , m_{j}\\}$,\n\\[f_{\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})}(k+\\sum_{i=0}^{j-1}m_{i})=f_{\\alpha_{j}}(k),\\]\nwhere for simplicity we define $m_{0}=1$.\n\\end{enumerate}\nOne should notice that $f_{\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})}$ is simply the formalization of the polish notation of $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$.\n\nThe set $F(\\Sigma, \\mathcal{V})$ becomes the $\\Sigma-$algebra $\\textbf{F}(\\Sigma, \\mathcal{V})$ when we define, for a $\\sigma\\in\\Sigma_{n}$ and formulas $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ in $F(\\Sigma, \\mathcal{V})$, \n\\[\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}(\\alpha_{1}, \\dotsc  , \\alpha_{n})=\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}).\\]\n\nWe define the order, or complexity, of an element of $F(\\Sigma, \\mathcal{V})$ as: $0$ if the formula is a variable or a constant, that is, a $\\sigma\\in \\Sigma_{0}$; as $1+\\max\\{p_{1}, \\dotsc  , p_{n}\\}$ if the formula is of the form $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$, with $\\alpha_{j}$ being of order $p_{j}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Multialgebra of non-deterministic formulas}\n\n\\begin{definition}\nGiven a signature $\\Sigma$ and a cardinal $\\kappa>0$, the expanded signature\\index{Signature, Expanded}\\label{expandedsignature} $\\Sigma^{\\kappa}=\\{\\Sigma_{n}^{\\kappa}\\}_{n\\in\\mathbb{N}}$ is the signature such that $\\Sigma_{n}^{\\kappa}=\\Sigma_{n}\\times \\kappa$, where we will denote the pair $(\\sigma, \\beta)$ by $\\sigma^{\\beta}$, for $\\sigma\\in\\Sigma$ and $\\beta\\in\\kappa$. \n\\end{definition}\n\nWe demand that $\\kappa$ is greater than zero: hence, if $\\Sigma$ is non-empty, so is $\\Sigma^{\\kappa}$.\n\n\\begin{definition}\\label{Multialgebra of formulas}\nGiven a set of  variables $\\mathcal{V}$, a signature $\\Sigma$ and a cardinal $\\kappa>0$, we define the $\\Sigma$-multialgebra of non-deterministic formulas, or simply multialgebra of formulas\\index{Formulas, Non-deterministic}\\index{Multialgebra of non-deterministic formulas}\\label{mFSigmakappaV}, as\n\\[\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)=(F(\\Sigma^{\\kappa}, \\mathcal{V}), \\{\\sigma_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}\\}_{\\sigma\\in\\Sigma})\\]\nwith universe $F(\\Sigma^{\\kappa}, \\mathcal{V})$ and such that, for $\\sigma\\in \\Sigma_{n}$ and $\\alpha_{1}, \\dotsc   , \\alpha_{n}\\in {T}(\\Sigma^{\\kappa}, \\mathcal{V})$,\n\\[\\sigma_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}(\\alpha_{1}, \\dotsc   , \\alpha_{n})=\\{\\sigma^{\\beta}(\\alpha_{1}, \\dotsc  , \\alpha_{n}) \\ : \\  \\beta\\in\\kappa\\}.\\]\n\\end{definition}\n\nThe intuition behind this definition is that, connecting given formulas $\\alpha_{1}$ through $\\alpha_{n}$ with a connective $\\sigma$ can, in a broader interpretation taking into account non-determinism, return many formulas with the same general shape, that is $\\sigma(\\alpha_{1}, \\dotsc  ,\\alpha_{n})$, over which we maintain certain degree of control by counting them, what we achieve by using an index to our connective, $\\sigma^{\\beta}$.\n\nOne can ask why all connectives must return the exact same number of generalized formulas, that is $\\kappa$, but this will not be the case: more useful to us shall be the submultialgebras of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$, where the cardinality will vary as long as it is bounded by $\\kappa$; we have defined the multialgebras of formulas as above since defining its submultialgebras directly is substantially more difficult.\n\n Here, we will restrict ourselves to the cases where $\\Sigma_{0}\\neq\\emptyset$ or $\\mathcal{V}\\neq\\emptyset$, so that $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ is always well defined. \n\nWe will understand as the order of an element $\\alpha$ of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ simply its order as an element of $F(\\Sigma^{\\kappa}, \\mathcal{V})$. Notice that, if \n\\[\\sigma_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}(\\alpha_{1}, \\dotsc   , \\alpha_{n})\\cap \\theta_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}(\\beta_{1}, \\dotsc    ,\\beta_{m})\\neq\\emptyset,\\]\nthen $\\sigma=\\theta$, $n=m$ and $\\alpha_{1}=\\beta_{1}, \\dotsc , \\alpha_{n}=\\beta_{m}$, since if the intersection is not empty there are $\\beta, \\gamma\\in\\kappa$ such that $\\sigma^{\\beta}(\\alpha_{1}, \\dotsc  , \\alpha_{n})= \\theta^{\\gamma}(\\beta_{1}, \\dotsc  ,\\beta_{m})$ and by the structure of $F(\\Sigma^{\\kappa}, \\mathcal{V})$ we find that $\\sigma^{\\beta}=\\theta^{\\gamma}$. \n\n\\begin{example}\nThe $\\Sigma$-algebras of formulas $\\textbf{F}(\\Sigma, \\mathcal{V})$, when considered as multialgebras such that $\\sigma_{\\textbf{F}(\\Sigma, \\mathcal{V})}(\\alpha_{1}, \\dotsc   , \\alpha_{n})=\\{\\sigma(\\alpha_{1}, \\dotsc  ,\\alpha_{n})\\}$, are multialgebras of formulas, with $\\kappa=1$; that is, $\\textbf{F}(\\Sigma, \\mathcal{V})$ and $\\textbf{mF}(\\Sigma, \\mathcal{V}, 1)$ are isomorphic.\n\\end{example}\n\nFrom now on, the cardinal of a set $X$ will be denoted by $|X|$.\\label{|X|}\n\n\\begin{example}\\label{s}\nA directed graph is a pair $(V, A)$, with $V$ a non-empty set of elements called vertices and $A\\subseteq V^{2}$ a set of elements called arrows, where we say that there is an arrow from $u$ to $v$, both in $V$, if $(u,v)\\in A$; we say that the $n$-tuple $(v_{1}, \\dotsc   , v_{n})$ is a path between $u$ and $v$ if $u=v_{1}$, $v=v_{n}$ and $(v_{i}, v_{i+1})\\in A$ for every $i\\in\\{1, \\dotsc   , n-1\\}$; we say that a vertex $u\\in V$ has a successor if there exists $v\\in V$ such that $(u,v)\\in A$, and $u$ has a predecessor if there exists $v\\in V$ such that $(v,u)\\in A$.\n\nA directed graph $F=(V, A)$ is a forest if, for any two vertices $u, v\\in V$, there exists at most one path between $u$ and $v$, and a forest is said to have height $\\omega$ if every vertex has a successor. Then, we state that forests of height $\\omega$ are in bijection with the submultialgebras of the multialgebras of formulas over the signature $\\Sigma_{s}$\\label{Sigmas} with exactly one operator $s$ of arity $1$.\n\nEssentially, take as $\\mathcal{V}$ the set of elements of $F$ that have no predecessor and define, for $u\\in V$,\n\\[s_{\\mathcal{A}}(u)=\\{v\\in V \\ : \\ (u,v)\\in A\\},\\]\nand we have that the $\\Sigma_{s}$-multialgebra $\\mathcal{A}=(V, \\{s_{\\mathcal{A}}\\})$, submultialgebra of $\\textbf{mF}(\\Sigma_{s}, \\mathcal{V}, |V|)$, carries the same information that $F$.\n\\end{example}\n\n\\begin{example}\nMore generally, a directed multi-graph\\index{Multi-graph} \\cite{ConiglioSernadas}, or directed $m$-graph, is a pair $(V, A)$ with $V$ a non-empty set of vertices and $A$ a subset of $V^{+}\\times V$, where $V^{+}=\\bigcup_{n\\in\\mathbb{N}\\setminus\\{0\\}}V^{n}$ is the set of finite, non-empty, sequences over $V$. We will say that $(v_{1}, \\dotsc   , v_{n})$ is a path between $u$ and $v$ if $u=v_{1}$, $v=v_{n}$ and, for every $i\\in\\{1, \\dotsc   , n-1\\}$, there exist $v_{i_{1}}, \\dotsc   , v_{i_{m}}$ such that $((v_{i_{1}}, \\dotsc   , v_{i_{m}}), v_{i+1})$, with $v_{i}=v_{i_{j}}$ for some $j\\in\\{1, \\dotsc   , m\\}$.\n\nThen an $m$-forest is a directed $m$-graph such that any two elements are connected by at most one path; and an $m$-forest is said to have $n$-height $\\omega$, for $n\\in\\mathbb{N}\\setminus\\{0\\}$, if, for any $(u_{1}, \\dotsc   , u_{n})\\in V^{n}$, there exists $v\\in V$ such that $((u_{1}, \\dotsc   , u_{n}), v)\\in A$. Finally, we see that every $m$-forest $F=(V, A)$ with $n$-height $\\omega$, for every $n\\in S\\subseteq\\mathbb{N}\\setminus\\{\\emptyset\\}$, is essentially equivalent to the $\\Sigma_{S}$-multialgebra $\\mathcal{A}=(V, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{S}})$, with\n\\[\\sigma_{\\mathcal{A}}(u_{1}, \\dotsc   , u_{m})=\\{v\\in V \\ : \\ ((u_{1}, \\dotsc   , u_{m}), v)\\in A\\},\\]\nfor $\\sigma$ of arity $m$, and $\\Sigma_{S}$ the signature with exactly one operator of arity $n$, for every $n\\in S$. It is not hard to see that $\\mathcal{A}$ is a submultialgebra of $\\textbf{mF}(\\Sigma_{S}, \\mathcal{V}, |V|)$, with $\\mathcal{V}$ the set of elements $v$ of $V$ such that, for no $(u_{1}, \\dotsc   , u_{n})\\in V^{+}$, $((u_{1}, \\dotsc   , u_{n}), v)\\in A$.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Equivalences for being a submultialgebra of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$}\n\\subsection{Being $\\textbf{cdf}$-generated}\n \nNow, in universal algebra, the algebras of formulas $\\textbf{F}(\\Sigma, \\mathcal{V})$ are absolutely free, also said to be freely generated, also said to be freely generated in the variety of all $\\Sigma$-algebras: this means that there exists a set, in their case the set of variables $\\mathcal{V}$, such that, for every other $\\Sigma$-algebra $\\mathcal{B}$ with universe $B$ and function $f:\\mathcal{V}\\rightarrow B$, there exists an unique homomorphism $\\overline{f}:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{B}$ extending $f$, essentially defined as:\n\\begin{enumerate}\n\\item $\\overline{f}(p)=f(p)$, for every $p\\in \\mathcal{V}$;\n\\item $\\overline{f}(\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}))=\\sigma_{\\mathcal{B}}(\\overline{f}(\\alpha_{1}), \\dotsc  , \\overline{f}(\\alpha_{n}))$. \n\\end{enumerate}\nAs we mentioned before, this is no longer true when dealing with multialgebras, but we can define a closely related concept with the aid of what we will call collections of choices.\n\nCollections of choices are motivated by legal valuations, notion first defined in Avron and Lev's seminal paper \\cite{AvronLev} on non-deterministic logical matrices. A map $\\nu$ from $\\textbf{F}(\\Sigma, \\mathcal{V})$ to the universe of a $\\Sigma$-multialgebra $\\mathcal{A}$ is a legal valuation whenever \n\\[\\nu(\\sigma(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))\\in \\sigma_{\\mathcal{A}}(\\nu(\\alpha_{1}), \\dotsc   , \\nu(\\alpha_{n}));\\]\nessentially, at every formula $\\sigma(\\alpha_{1}, \\dotsc   ,\\alpha_{n})$, $\\nu$ ``chooses'' a value from all the possible values\\\\ $\\sigma_{\\mathcal{A}}(\\nu(\\alpha_{1}), \\dotsc   , \\nu(\\alpha_{n}))$, possible values which depend themselves on previous choices $\\nu(\\alpha_{1})$ through $\\nu(\\alpha_{n})$ performed by $\\nu$. What a collection of choices does is then to automatize these choices, what justifies its name. \n\n\\begin{definition}\\label{Collection of Choices}\nGiven multialgebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\sigma})$ over the signature $\\Sigma$, a collection of choices\\index{Collection of choices} from $\\mathcal{A}$ to $\\mathcal{B}$ is a collection $C=\\{C_{n}\\}_{n\\in\\mathbb{N}}$ of collections of functions\n\\[C_{n}=\\{C\\sigma_{a_{1}, \\dotsc  , a_{n}}^{b_{1}, \\dotsc  , b_{n}} : \\sigma\\in\\Sigma_{n}, a_{1}, \\dotsc  , a_{n}\\in A, b_{1}, \\dotsc   ,b_{n}\\in B\\}\\]\nsuch that, for $\\sigma\\in\\Sigma_{n}$, $a_{1}, \\dotsc  , a_{n}\\in A$ and $b_{1}, \\dotsc  , b_{n}\\in B$, $C\\sigma_{a_{1}, \\dotsc  , a_{n}}^{b_{1}, \\dotsc  , b_{n}}$ is a function of the form \n\\[C\\sigma_{a_{1}, \\dotsc  , a_{n}}^{b_{1}, \\dotsc  , b_{n}}:\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   ,a_{n})\\rightarrow\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n}).\\]\n\\end{definition}\n\n\\begin{example}\nIf $\\mathcal{B}$ is actually an algebra (meaning all its operations return singletons), there only exists one collection of choices from any $\\mathcal{A}$ to $\\mathcal{B}$ (since there exists only one function to a set with only one element); this means that in universal algebra, collections of choices are somewhat irrelevant.\n\\end{example}\n\n\\begin{example}\nA directed tree is a directed forest where there exists exactly one element without predecessor; we say that $v$ ramifies from $u$ if there exists an arrow from $u$ to $v$. Then, given two directed trees $T_{1}=(V_{1}, A_{1})$ and $T_{2}=(V_{2}, A_{2})$ of height $\\omega$, seem as $\\Sigma_{s}$-multialgebras, and a collection of choices $C$ from $T_{1}$ to $T_{2}$, for every $v\\in V_{1}$ and $u\\in V_{2}$ the function $Cs_{v}^{u}$ chooses, for each of the elements that ramify from $v$, one element that ramifies from $u$.\n\\end{example}\n\n\\begin{definition}\nGiven a signature $\\Sigma$, a $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is choice-dependent freely generated\\index{Choice-dependent freely generated} by $X$ if $X\\subseteq A$ and, for all $\\Sigma$-multialgebras $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$, all functions $f:X\\rightarrow B$ and all collections of choices $C$ from $\\mathcal{A}$ to $\\mathcal{B}$, there is a unique homomorphism $f_{C}:\\mathcal{A}\\rightarrow \\mathcal{B}$ such that:\n\n\\begin{enumerate} \\item $f_{C}|_{X}=f$;\n\n\\item for all $\\sigma\\in \\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}\\in A$, \n\\[f_{C}|_{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})}=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}.\\]\n\\end{enumerate}\n\\end{definition}\n\nFor simplicity, when $\\mathcal{A}$ is choice-dependent freely generated by $X$, we will write that $\\mathcal{A}$ is $\\textbf{cdf}$-generated\\label{cdfgenerated} by $X$, or merely that $\\mathcal{A}$ is $\\textbf{cdf}$-generated, when the set $X$ is not important.\n\nWe now introduce the concept of ground to indicate what elements of a multialgebra are not ``achieved'', ``reached'' by its multioperations; alternatively, while thinking of formulas and their respective algebras, the ground is the set of indecomposable formulas, that is, variables.\n\n\\begin{definition}\nGiven a $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$, we define its build\\index{Build}\\label{build} as\n\\[B(\\mathcal{A})=\\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in A \\big\\}.\\]\nWe define the ground\\index{Ground}\\label{ground} of $\\mathcal{A}$ as\n\\[G(\\mathcal{A})=A\\setminus B(\\mathcal{A}).\\]\n\\end{definition}\n\n\\begin{example}\\label{ground of formulas}\n$B(\\textbf{F}(\\Sigma, \\mathcal{V}))={T}(\\Sigma, \\mathcal{V})\\setminus\\mathcal{V}$ and $G(\\textbf{F}(\\Sigma, \\mathcal{V}))=\\mathcal{V}$.\n\\end{example}\n\n\\begin{example}\nIf $F=(V, A)$ is a directed forest of height $\\omega$, thought as a $\\Sigma_{s}$-multialgebra, its ground is the set of elements $v$ in $V$ without predecessors.\n\\end{example}\n\n\\begin{proposition}\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be $\\Sigma$-multialgebras.\n\\begin{enumerate} \n\\item If $f:\\mathcal{A}\\rightarrow\\mathcal{B}$ is a homomorphism between $\\Sigma$-multialgebras, then \n\\[B(\\mathcal{A})\\subseteq f^{-1}(B(\\mathcal{B}))\\quad\\text{and}\\quad f^{-1}(G(\\mathcal{B}))\\subseteq G(\\mathcal{A}).\\]\n\n\\item If $\\mathcal{B}$ is a submultialgebra of $\\mathcal{A}$, $B(\\mathcal{B})\\subseteq B(\\mathcal{A})$ and $G(\\mathcal{A})\\subseteq G(\\mathcal{B})$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate} \\item If $a\\in B(\\mathcal{A})$, there exist $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}\\in A$ such that $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$. Since $f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})) \\subseteq \\sigma_{\\mathcal{B}}(f(a_{1}), \\dotsc    ,f(a_{n}))$,\nwe find that $f(a)\\in \\sigma_{\\mathcal{B}}(f(a_{1}), \\dotsc   , f(a_{n}))$ and therefore $f(a)\\in B(\\mathcal{B})$, meaning that $a\\in f^{-1}(B(\\mathcal{B}))$. Using that $G(\\mathcal{A})=A\\setminus B(\\mathcal{A})$ and $G(\\mathcal{B})=B\\setminus B(\\mathcal{B})$ we obtain the second mentioned inclusion.\n\n\n\\item If $b\\in B(\\mathcal{B})$, there exist $\\sigma\\in\\Sigma_{n}$ and $b_{1}, \\dotsc   , b_{n}\\in B$ such that $b\\in \\sigma_{\\mathcal{B}}(b_{1}, \\dotsc   , b_{n})$, and given that $\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc   , b_{n})\\subseteq \\sigma_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{n})$ we obtain $b\\in B(\\mathcal{A})$. Using again that $G(\\mathcal{A})=A\\setminus B(\\mathcal{A})$ and $G(\\mathcal{B})=B\\setminus B(\\mathcal{B})$ we finish the proof.\n\\end{enumerate}\n\\end{proof}\n\nFrom this it also follows that if $f:\\mathcal{A}\\rightarrow\\mathcal{B}$ is a homomorphism, $G(\\mathcal{B})\\cap f(A)$ is contained in $\\{f(a)\\ : \\  a\\in G(\\mathcal{A})\\}$. Indeed,  if $b$ is in $G(\\mathcal{B})\\cap f(A)$, $a\\in A$ such that $f(a)=b$ is in $f^{-1}(G(\\mathcal{B}))$ and, by the previous proposition, it is in $G(\\mathcal{A})$, and therefore $b$ is in $\\{f(a) \\ : \\  a\\in G(\\mathcal{A})\\}$.\n\nGeneralizing Example \\ref{ground of formulas}, we have that $G(\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa))=\\mathcal{V}$: we proceed by induction to show that $B(\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa))={T}(\\Sigma^{\\kappa}, \\mathcal{V})\\setminus \\mathcal{V}$, what is equivalent. If $\\alpha$ is of order $0$, either we have $\\alpha=\\sigma^{\\beta}$, for a $\\sigma\\in\\Sigma_{0}$ and $\\beta\\in\\kappa$, and therefore $\\alpha\\in B(\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa))$; or we have that $\\alpha=p\\in \\mathcal{V}$, and if there exists $\\sigma\\in\\Sigma_{m}$ and $\\alpha_{1}, \\dotsc   , \\alpha_{m}\\in {T}(\\Sigma^{\\kappa}, \\mathcal{V})$ such that \n\\[p\\in \\sigma_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}(\\alpha_{1}, \\dotsc   , \\alpha_{m})\\]\nwe have $p=\\sigma^{\\beta}(\\alpha_{1}, \\dotsc  ,\\alpha_{m})$, for $\\beta\\in\\kappa$, which is absurd given the structure of $F(\\Sigma^{\\kappa}, \\mathcal{V})$, forcing us to conclude that $x\\notin B(\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa))$. If $\\alpha$ is of order $n+1$, we have that $\\alpha=\\sigma^{\\beta}(\\alpha_{1}, \\dotsc  ,\\alpha_{m})$ for a $\\sigma\\in\\Sigma_{m}$, $\\beta\\in\\kappa$ and $\\alpha_{1}, \\dotsc   , \\alpha_{m}$ of order at most $n$, and therefore we have $\\alpha$ in $\\sigma_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}(\\alpha_{1}, \\dotsc   , \\alpha_{m})$, meaning that $\\alpha\\in B(\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa))$.\n\n\\begin{definition}\nGiven a $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and a set $S\\subseteq A$, we define the sets $\\langle S\\rangle_{m}$\\label{mth generated} by induction: $\\langle S\\rangle _{0}=S\\cup\\bigcup_{\\sigma\\in \\Sigma_{0}}\\sigma_{\\mathcal{A}}$; and assuming we have defined $\\langle S\\rangle_{m}$, we make\n\\[\\langle S\\rangle_{m+1}=\\langle S\\rangle_{m}\\cup \\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in \\langle S\\rangle_{m} \\big\\}.\\]\nThe set generated\\index{Generated}\\label{generated} by $S$, denoted by $\\langle S\\rangle$, is then defined as $\\langle S\\rangle=\\bigcup_{m\\in\\mathbb{N}}\\langle S\\rangle_{m}$.\n\nWe say $\\mathcal{A}$ is generated by $S$ if $\\langle S\\rangle=A$.\n\\end{definition}\n\n\n\n\\begin{lemma}\\label{sub mF is gen ground}\nEvery submultialgebra $\\mathcal{A}$ of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ is generated by $G(\\mathcal{A})$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $a$ is an element of $\\mathcal{A}$ not contained in $\\langle G(\\mathcal{A})\\rangle$ of minimum order: since $a$ cannot belong to $G(\\mathcal{A})\\cup\\bigcup_{\\sigma\\in\\Sigma_{0}}\\sigma_{\\mathcal{A}}=\\langle G(\\mathcal{A})\\rangle_{0}$, there exist $n>0$, $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}\\in A$ such that $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$.\n\nSince $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\subseteq \\sigma_{\\textbf{mF}(\\Sigma, X, \\kappa)}(a_{1}, \\dotsc   , a_{n})$ we derive that $a_{1}$ to $a_{n}$ are of order less than that of $a$: by our hypothesis, there must exist $m_{1}, \\dotsc   , m_{n}$ such that $a_{j}\\in \\langle G(\\mathcal{A})\\rangle_{m_{j}}$ for all $j\\in\\{1, \\dotsc   , n\\}$; taking $m=\\max\\{m_{1}, \\dotsc   , m_{n}\\}$, $a_{1}, \\dotsc   , a_{n}\\in \\langle G(\\mathcal{A})\\rangle_{m}$, and therefore\n\\[a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\subseteq \\langle G(\\mathcal{A})\\rangle_{m+1},\\]\nwhich contradicts our assumption and proves the lemma.\n\\end{proof}\n\n\\begin{theorem}\\label{sub mF is cdf-gen ground}\nEvery submultialgebra $\\mathcal{A}$ of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ is $\\textbf{cdf}$-generated by $G(\\mathcal{A})$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ be a submultialgebra of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$, let $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$ be any $\\Sigma$-multialgebra, let $f:G(\\mathcal{A})\\rightarrow B$ be a function and $C$ a collection of choices from $\\mathcal{A}$ to $\\mathcal{B}$. We define $f_{C}:\\mathcal{A}\\rightarrow\\mathcal{B}$ by induction on $\\langle G(\\mathcal{A})\\rangle_{m}$:\n\n\\begin{enumerate} \\item if $a\\in \\langle G(\\mathcal{A})\\rangle_{0}$ and $a\\in G(\\mathcal{A})$, we define $f_{C}(a)=f(a)$;\n\n\\item if $a\\in \\langle G(\\mathcal{A})\\rangle_{0}$ and $a\\in\\sigma_{\\mathcal{A}}$, for a $\\sigma\\in\\Sigma_{0}$, we define $f_{C}(a)=C\\sigma(a)$;\n\n\\item if $f_{C}$ is defined for all elements of $\\langle G(\\mathcal{A})\\rangle_{m}$, $a_{1}, \\dotsc   , a_{n}\\in \\langle G(\\mathcal{A})\\rangle_{m}$ and $\\sigma\\in\\Sigma_{n}$, for every element $a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ we define \n\\[f_{C}(a)=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}(a).\\]\n\\end{enumerate}\n\nFirst, we must prove that $f_{C}$ is well defined. There are two possibly problematic cases to consider for an element $a\\in A$: the one in which $a\\in G(\\mathcal{A})$ and there are $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}\\in A$ for which $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$, corresponding to $a$ falling simultaneously in the cases $(1)$ and $(2)$, or $(1)$ and $(3)$ of the definition; and the one where there are $\\sigma\\in\\Sigma_{n}$, $\\theta\\in\\Sigma_{m}$, $a_{1}, \\dotsc   , a_{n}\\in A$ and $b_{1}, \\dotsc   , b_{m}\\in A$ such that $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ and $a\\in\\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{m})$, corresponding to the cases $(2)$ and $(3)$, $(2)$ and $(2)$, or $(3)$ and $(3)$ occurring simultaneously.\n\nThe first case is not possible, since $G(\\mathcal{A})\\subseteq A\\setminus\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ for every $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}\\in A$; in the second case, we find that \n\\[a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\cap\\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{m})\\subseteq\\sigma_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}(a_{1}, \\dotsc   , a_{n})\\cap \\theta_{\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)}( b_{1}, \\dotsc    , b_{m}),\\]\nso $n=m$, $\\sigma=\\theta$ and $a_{1}=b_{1}, \\dotsc   , a_{n}=b_{m}$, and therefore $f_{C}(a)$ is well-defined.\n\nSecond, we must prove that $f_{C}$ is defined over all of $A$: that is simple, for $f_{C}$ is defined over all of $\\langle G(\\mathcal{A})\\rangle$ and we established in Lemma~\\ref{sub mF is gen ground} that $A=\\langle G(\\mathcal{A})\\rangle$.\n\nSo $f_{C}:A\\rightarrow B$ is a well-defined function: it remains to be shown that it is a homomorphism; given $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}$, we see that\n\\[f_{C}(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}))=\\big\\{C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}(a)\\ : \\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\big\\}\\subseteq\\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})),\\]\nwhile we also have that $f_{C}$ clearly extends both $f$ and all $C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}, \\dotsc   , f_{C}(a_{n})}$. \n\nTo finish the proof, suppose $g:\\mathcal{A}\\rightarrow\\mathcal{B}$ is another homomorphism extending both $f$ and all $C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{g(a_{1}, \\dotsc   , g(a_{n})}$: we will prove that $g=f_{C}$ again by induction on the $m$ of $\\langle G(\\mathcal{A})\\rangle_{m}$. For $m=0$, an element $a\\in \\langle G(\\mathcal{A})\\rangle_{0}$ is either in $G(\\mathcal{A})$, when we have $g(a)=f(a)=f_{C}(a)$, or in $\\sigma_{\\mathcal{A}}$ for a $\\sigma\\in\\Sigma_{0}$, when $g(a)=C\\sigma(a)=f_{C}(a)$.\n\nSuppose $g$ is equal to $f_{C}$ in $\\langle G(\\mathcal{A})\\rangle_{m}$ and take an $a\\in \\langle G(\\mathcal{A})\\rangle_{m+1}\\setminus \\langle G(\\mathcal{A})\\rangle_{m}$: we have that there exist $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc   , a_{n}\\in \\langle G(\\mathcal{A})\\rangle_{m}$ such that $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$, and then \n\\[g(a)=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{g(a_{1}), \\dotsc   , g(a_{n})}(a)=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}(a)=f_{C}(a),\\]\nproving that $g=f_{C}$ and that, in fact, $f_{C}$ is unique. This means that $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $G(\\mathcal{A})$.\n\\end{proof}\n\nThe following lemma may be found in section 2 of \\cite{CFG}.\n\n\\begin{lemma}\\label{image of hom is sub}\nIf $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$ are $\\Sigma$-multialgebras and $f:\\mathcal{A}\\rightarrow\\mathcal{B}$ is a homomorphism, $\\mathcal{C}=(f(A), \\{\\sigma_{\\mathcal{C}}\\}_{\\sigma\\in\\Sigma})$ such that\n\\[\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n})=\\bigcup\\{f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})) : f(a_{1})=c_{1}, \\dotsc  , f(a_{n})=c_{n}\\}\\]\nis a $\\Sigma$-submultialgebra of $\\mathcal{B}$, while $f:\\mathcal{A}\\rightarrow\\mathcal{C}$ is an epimorphism.\\footnote{A epimorphism between $\\Sigma$-multialgebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$ is defined, as usual, as a homomorphism $\\varphi:A\\rightarrow B$ between $\\mathcal{A}$ and $\\mathcal{B}$ that is surjective.} The $\\Sigma$-multialgebra $\\mathcal{C}$ is known as the direct image\\index{Direct image} of $\\mathcal{A}$ trough $f$.\n\\end{lemma}\n\n\\begin{proof}\nFirst of all, $A$ is not empty, and therefore so is $f(A)$.\n\nTake $c_{1}, \\dotsc  , c_{n}\\in f(A)$: there must exist $a_{1}, \\dotsc  , a_{n}\\in A$ such that $f(a_{1})=c_{1}, \\dotsc ,$\\\\ $f(a_{n})=c_{n}$, and since $\\mathcal{A}$ is a multialgebra, $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\neq\\emptyset$, implying that $f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))$ is not empty and, therefore, that $\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n})$ is non-empty, given it contains $f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))$. It is obvious that, as defined, $\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n})$ is a subset of $f(A)$, and so we can deduce that $\\mathcal{C}$ is a $\\Sigma$-multialgebra.\n\nNow, given $f:\\mathcal{A}\\rightarrow\\mathcal{B}$ is a homomorphism, for all $a_{1}, \\dotsc  , a_{n}\\in A$ we have $f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\subseteq \\sigma_{\\mathcal{B}}(f(a_{1}), \\dotsc  , f(a_{n}))$, meaning\n\\[\\sigma_{\\mathcal{C}}(c_{1}, \\dotsc  , c_{n})=\\bigcup\\{f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})) : f(a_{1})=c_{1}, \\dotsc  , f(a_{n})=c_{n}\\}\\subseteq\\]\n\\[\\bigcup\\{\\sigma_{\\mathcal{B}}(f(a_{1}), \\dotsc  , f(a_{n})) : f(a_{1})=c_{1}, \\dotsc  , f(a_{n})=c_{n}\\}=\\sigma_{\\mathcal{B}}(c_{1}, \\dotsc  , c_{n}),\\]\nor what is equivalent, that $\\mathcal{C}$ is a submultialgebra of $\\mathcal{B}$.\n\nFinally, $f:A\\rightarrow f(A)$ is still a well-defined function, obviously surjective: for any $n$-ary $\\sigma$ and elements $a_{1}$ through $a_{n}$ of $A$, one has\n\\[f(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\subseteq \\bigcup\\{f(\\sigma_{\\mathcal{A}}(a'_{1}, \\dotsc  , a'_{n})) : f(a'_{1})=f(a_{1}), \\dotsc  , f(a'_{n})=f(a_{n})\\}=\\]\n\\[\\sigma_{\\mathcal{C}}(f(a_{1}), \\dotsc  , f(a_{n}))\\]\nand that, in conclusion, $f$ is a homomorphism.\n\\end{proof}\n\n\\begin{theorem}\\label{cdf-gen is iso to sub mF}\nIf the multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ over $\\Sigma$ is $\\textbf{cdf}$-generated by $X$, then $\\mathcal{A}$ is isomorphic to a submultialgebra of $\\textbf{mF}(\\Sigma, X, |A|)$ containing $X$.\n\\end{theorem}\n\n\\begin{proof}\nTake $f:X\\rightarrow F(\\Sigma^{|A|}, X)$ to be the inclusion (such that $f(x)=x$), and take a collection of choices $C$ such that, for $\\sigma\\in\\Sigma_{n}$, $a_{1}, \\dotsc   , a_{n}\\in A$ and $\\alpha_{1}, \\dotsc   , \\alpha_{n}\\in F(\\Sigma^{|A|}, X)$, \n\\[C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{\\alpha_{1}, \\dotsc   , \\alpha_{n}}:\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\rightarrow\\sigma_{\\textbf{mF}(\\Sigma, X, |A|)}(\\alpha_{1}, \\dotsc   , \\alpha_{n})\\]\nis an injective function. Such collection of choices exist since $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\subseteq A$ and\\\\ $\\sigma_{\\textbf{mF}(\\Sigma, X, |A|)}(\\alpha_{1}, \\dotsc   , \\alpha_{n})$ is of cardinality $|A|$. Now, since $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $X$, there exists a homomorphism $f_{C}:\\mathcal{A}\\rightarrow\\textbf{mF}(\\Sigma, X, |A|)$ extending $f$ and each $C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}$. \n\n\nLet $\\mathcal{B}=(f_{C}(A), \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$ be the direct image of $\\mathcal{A}$ trough $f_{C}$, so that $f_{C}:\\mathcal{A}\\rightarrow\\mathcal{B}$ is an epimorphism, what is possible given Lemma \\ref{image of hom is sub}: notice too that \n\\[X=X\\cap f_{C}(A)=G(\\textbf{mF}(\\Sigma, X, |A|))\\cap f_{C}(A)\\subseteq G(\\mathcal{B})\\]\nbecause $\\mathcal{B}$ is a submultialgebra of $\\textbf{mF}(\\Sigma, X, |A|)$. Take any $g:G(\\mathcal{B})\\rightarrow A$ such that $g(x)=x$, for every $x\\in X$. And take a collection of choices $D$ from $\\mathcal{B}$ to $\\mathcal{A}$ such that, for any $\\sigma\\in\\Sigma_{n}$, $b_{1}, \\dotsc   , b_{n}\\in f_{C}(A)$ and $a_{1}, \\dotsc   , a_{n}\\in A$, the function\n\\[D\\sigma_{b_{1}, \\dotsc   , b_{n}}^{a_{1}, \\dotsc   , a_{n}}:\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc   , b_{n})\\rightarrow\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\]\nsatisfies that, if $a \\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ is such that $C\\sigma_{a_{1}, \\dotsc  , a_{n}}^{b_{1}, \\dotsc  , b_{n}}(a) \\in \\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})$, then\\\\ $D\\sigma_{b_{1}, \\dotsc   , b_{n}}^{a_{1}, \\dotsc   , a_{n}}(C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{b_{1}, \\dotsc   , b_{n}}(a))=a$. Given that $C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{b_{1}, \\dotsc   , b_{n}}$ is injective, this condition is well-defined.\n\nSince $\\mathcal{B}$ is $\\textbf{cdf}$-generated by $G(\\mathcal{B})$, we know to exist a homomorphism $g_{D}:\\mathcal{B}\\rightarrow \\mathcal{A}$ extending $g$ and the functions $D\\sigma_{b_{1}, \\dotsc   , b_{n}}^{g_{D}(b_{1}), \\dotsc   , g_{D}(b_{n})}$.\n\nFinally, we take $g_{D}\\circ f_{C}:\\mathcal{A}\\rightarrow\\mathcal{A}$: it extends the injection $id=g\\circ f:X\\rightarrow A$, for which $id(x)=x$; it also extends the collection of choices $E$ defined by\n\\[E\\sigma_{a_{1}, \\dotsc  , a_{n}}^{a'_{1}, \\dotsc  , a'_{n}}=D\\sigma_{f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})}^{a'_{1}, \\dotsc  , a'_{n}}\\circ C_{a_{1}, \\dotsc  , a_{n}}^{f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})}:\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\rightarrow\\sigma_{\\mathcal{A}}(a'_{1}, \\dotsc  , a'_{n}),\\]\nfor $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  a_{n}, a'_{1}, \\dotsc  , a'_{n}\\in A$. \nThis way, $E\\sigma_{a_{1}, \\dotsc  , a_{n}}^{a_{1}, \\dotsc  , a_{n}}$ is the identity on $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$: indeed, for any $a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$, \n\\[C\\sigma_{a_{1}, \\dotsc  , a_{n}}^{f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})}(a)=f_{C}(a)\\]\nby definition of $f_{C}$, and, given $f_{C}$ is a homomorphism, $f_{C}(a)\\in \\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n}))$, meaning\\\\ $C\\sigma_{a_{1}, \\dotsc  , a_{n}}^{f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})}(a) \\in \\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n}))$; then\n\\[E\\sigma_{a_{1}, \\dotsc  , a_{n}}^{a_{1}, \\dotsc  , a_{n}}(a) = D\\sigma_{f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})}^{a_{1}, \\dotsc  , a_{n}}(C_{a_{1}, \\dotsc  , a_{n}}^{f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})}(a)) = a\\]\nby definition of $D$.\n\nBut notice that the identical homomorphism $Id_{\\mathcal{A}}:\\mathcal{A}\\rightarrow \\mathcal{A}$ also extends both $id$ and $E$ and, given the unicity of such extensions on the definition of being $\\textbf{cdf}$-generated, we obtain that $Id_{\\mathcal{A}}=g_{D}\\circ f_{C}$. Of course, the fact that $f_{C}:\\mathcal{A}\\rightarrow\\mathcal{B}$ has a left inverse implies that is injective, and by definition of $\\mathcal{B}$ it is also surjective, meaning it is a bijective function; moreover, $g_{D}$ is the inverse function of $f_{C}$. Finally, for $\\sigma\\in \\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\\[f_{C}(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\subseteq \\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})),\\]\nsince $f_{C}$ is a homomorphism; however, given $g_{D}$ is also a homomorphism.\n\\[g_{D}(\\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n})))\\subseteq \\sigma_{\\mathcal{A}}(g_{D}\\circ f_{C}(a_{1}), \\dotsc  , g_{D}\\circ f_{C}(a_{n}))=\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}),\\]\nand by applying $f_{C}$ to both sides, one obtains\n\\[\\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n}))=f_{C}(g_{D}(\\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc  , f_{C}(a_{n}))))\\subseteq f_{C}(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})),\\]\nwhat proves $f_{C}$ is a full homomorphism, that is, an isomorphism.\n\\end{proof}\n\nNotice that, \\textit{mutatis mutandis}, the previous proof shows that if $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is $\\textbf{cdf}$-generated by $X$, then $\\mathcal{A}$ is isomorphic to a submultialgebra of $\\textbf{mF}(\\Sigma, X, M(\\mathcal{A}))$, where\n\\[M(\\mathcal{A})=\\sup \\big\\{|\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})| \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in A \\big\\}.\\]\n\\label{M(A)}Quite obviously, $M(\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa))=\\kappa$ and $M(\\mathcal{A})\\leq \\kappa$ for any submultialgebra of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$. The value $M(\\mathcal{A})$ has been regarded for some time, in the literature of multialgebras, as one of their fundamental aspects, see for example ~\\cite{CuponaMadarasz}; unfortunately, their definition of homomorphism is grossly different from ours, meaning their results are not applicable to or studies.\n\nNotice, furthermore, that written in classical terms, the previous Theorems~\\ref{sub mF is cdf-gen ground} and~\\ref{cdf-gen is iso to sub mF} state something quite well known: an algebra is absolutely free if, and only if, it is isomorphic to some algebra of formulas over the same signature.\n\n\\begin{corollary}\\label{cdf-gen is cdf-gen by ground}\nEvery $\\textbf{cdf}$-generated multialgebra $\\mathcal{A}$ is generated by its ground $G(\\mathcal{A})$.\n\\end{corollary}\n\n\\begin{proof}\nSince every $\\textbf{cdf}$-generated multialgebra is isomorphic to a submultialgebra of some\\\\ $\\textbf{mF}(\\Sigma, X, \\kappa)$, from Theorem \\ref{cdf-gen is iso to sub mF}, and every submultialgebra of $\\textbf{mF}(\\Sigma, X, \\kappa)$ is generated by its ground, by Lemma \\ref{sub mF is gen ground}, the result follows.\n\\end{proof}\n\n\\begin{corollary}\nEvery $\\textbf{cdf}$-generated multialgebra $\\mathcal{A}$ is $\\textbf{cdf}$-generated by its ground $G(\\mathcal{A})$.\n\\end{corollary}\n\n\\begin{definition}\nA $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is said to be disconnected\\index{Disconnected} if, for every $\\sigma\\in\\Sigma_{n}, \\theta\\in \\Sigma_{m}$, $a_{1}, \\dotsc    , a_{n}, b_{1}, \\dotsc    , b_{m}\\in A$,\n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\cap \\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{m})\\neq\\emptyset\\]\nimplies that $n=m$, $\\sigma=\\theta$ and $a_{1}=b_{1}, \\dotsc   , a_{n}=b_{m}$.\n\\end{definition}\n\n\n\\begin{example}\n$\\textbf{F}(\\Sigma, \\mathcal{V})$ is disconnected.\n\\end{example}\n\n\\begin{example}\nAll directed forests of height $\\omega$, when considered as $\\Sigma_{s}$-multialgebras, are disconnected, given that no two arrows point to the same element.\n\\end{example}\n\nIt is clear that if $\\mathcal{B}$ is a submultialgebra of $\\mathcal{A}$ and $\\mathcal{A}$ is disconnected, then $\\mathcal{B}$ is also disconnected, since if $\\sigma_{\\mathcal{B}}(a_{1}, \\dotsc   , a_{n})\\cap \\theta_{\\mathcal{B}}(b_{1}, \\dotsc   , b_{m})\\neq\\emptyset$, for $a_{1}, \\dotsc   , a_{n}, b_{1}, \\dotsc   , b_{m}\\in B$, given that $\\sigma_{\\mathcal{B}}(a_{1}, \\dotsc   , a_{n})\\subseteq\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ and $\\theta_{\\mathcal{B}}(b_{1}, \\dotsc   , b_{m})\\subseteq \\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{m})$, we find that \n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\cap \\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{m})\\neq\\emptyset\\]\nand therefore $n=m$, $\\sigma=\\theta$ and $a_{1}=b_{1}, \\dotsc   , a_{n}=b_{m}$.\n\nWe noticed before that $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ is disconnected, and by Theorem \\ref{cdf-gen is iso to sub mF} we obtain that every $\\textbf{cdf}-$generated algebra is disconnected. This has a deeper meaning: being disconnected is an attempt to measure how free of identities a multialgebra is. After all, having no two multioperations to coincide, on any elements, is strongly indicative that the multialgebra does not satisfy any identities. The fact that our candidates for an absolutely free multialgebra (the submultialgebras of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$) are all disconnected suggests that they are well-deserving of the ``weakly free'' title.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Being disconnected and generated by the ground}\n\nWe have offered, so far, two characterizations of the multialgebras we chose to call weakly free: first of all, they are submultialgebras of some $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$, and perhaps this is to be taken as their definition; second, they are $\\textbf{cdf}$-generated. Now, we look at other possible characterizations of being weakly free that could lead to possible future definitions of relatively free multialgebras. \n\nAlgebras of formulas satisfy no identities, what would partially correspond here to the concept of being disconnected. However, there is one property that is maybe more representative of our intuition of formulas (which are, up to isomorphism, the elements of all absolutely free algebras): whenever one deals with formulas, one starts by defining them from elements that are as simple as possible (variables), and continues indefinitely by combining them trough operations (connectives). \n\nThe concept of simplest (or indecomposable) element, here, is replaced by that of being an element of the ground, so one would expect that being generated by it plays some sort of role in the objects we have defined so far: a multialgebra which is generated by a set has all of its elements either on the set, or as the result of increasingly more complex (multi)operations performed on that very set.\n\n\n\\begin{lemma}\\label{X in ground if cdf-gen by X}\nIf $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $X$, then $X\\subseteq G(\\mathcal{A})$.\n\\end{lemma}\n\n\\begin{proof}\nIf $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $X$, then $\\mathcal{A}$ is isomorphic to a submultialgebra of $\\textbf{mF}(\\Sigma, X, |A|)$ containing $X$, from Theorem \\ref{cdf-gen is iso to sub mF}: let us assume that $\\mathcal{A}$ is equal to this submultialgebra, without loss of generality.\n\nThen we have $X=G(\\textbf{mF}(\\Sigma, X, |A|))\\cap A\\subseteq G(\\mathcal{A})$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{no proper cdf-gen sets}\nIf $\\mathcal{A}$ is $\\textbf{cdf}$-generated by both $X$ and $Y$, with $X\\subseteq Y$, then $X=Y$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $X\\neq Y$ and let $y\\in Y\\setminus X$: take a $\\Sigma$-multialgebra $\\mathcal{B}$ over the same signature as $\\mathcal{A}$ such that $|B|\\geq 2$, and a collection of choices $C$ from $\\mathcal{A}$ to $\\mathcal{B}$.\n\nTake also two functions $g, h:Y\\rightarrow B$ such that $g|_{X}=h|_{X}$ and $g(y)\\neq h(y)$, which is possible since $|B|\\geq 2$: given that $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $Y$, there exist unique homomorphisms $g_{C}$ and $h_{C}$ extending both, respectively, $g$ and $C$ and $h$ and $C$.\n\nHowever, $g_{C}$ and $h_{C}$ extend both $g|_{X}:X\\rightarrow B$ and $C$, and since $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $X$, we find that $g_{C}=h_{C}$. This is not possible, since $g_{C}(y)\\neq h_{C}(y)$, what must imply that $Y\\setminus X=\\emptyset$ and therefore $X=Y$.\n\\end{proof}\n\n\\begin{theorem}\\label{ground is only cdf-gen set}\nEvery $\\textbf{cdf}$-generated multialgebra $\\mathcal{A}$ is uniquely $\\textbf{cdf}$-generated by its ground.\n\\end{theorem}\n\n\\begin{proof}\nFrom Theorem \\ref{cdf-gen is cdf-gen by ground}, $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $G(\\mathcal{A})$, and from Lemma \\ref{X in ground if cdf-gen by X}, if $\\mathcal{A}$ is also $\\textbf{cdf}$-generated by $X$, then $X\\subseteq G(\\mathcal{A})$. By Lemma \\ref{no proper cdf-gen sets}, this implies that $X=G(\\mathcal{A})$.\n\\end{proof}\n\n\nWe have proved so far that, if $\\mathcal{A}$ is $\\textbf{cdf}$-generated, then $\\mathcal{A}$ is generated by its ground and disconnected. We would like to prove that this is enough to characterize a $\\textbf{cdf}$-generated multialgebra: that is, if $\\mathcal{A}$ is generated by its ground and disconnected, then it is $\\textbf{cdf}$-generated, exactly by its ground.\n\nThe idea is similar to the one we used to prove, in Theorem \\ref{sub mF is cdf-gen ground}, that all submultialgebras of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ are $\\textbf{cdf}$-generated: take a multialgebra $\\mathcal{A}$ that is both generated by its ground $G(\\mathcal{A})$, which we will denote by $X$, and disconnected, and fix a multialgebra $\\mathcal{B}$ over the same signature, a function $f:X\\rightarrow B$ and a collection of choices $C$ from $\\mathcal{A}$ to $\\mathcal{B}$.\n\nWe define a function $f_{C}:A\\rightarrow B$ using induction on the $\\langle X\\rangle_{m}$: for $m=0$, either we have an element $x\\in X$, when we define $f_{C}(x)=f(x)$, or we have an $a\\in\\sigma_{\\mathcal{A}}$, for a $\\sigma\\in\\Sigma_{0}$, when we define $f_{C}(a)=C\\sigma(a)$. Notice how, up to this point, there are no contradictions on the definition, given an element cannot belong both to $X$ and to a $\\sigma_{\\mathcal{A}}$, since $X=G(\\mathcal{A})$.\n\nSuppose we have successfully defined $f_{C}$ on $\\langle X\\rangle_{m}$ and take an $a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ for $a_{1}, \\dotsc   , a_{n}\\in\\langle X\\rangle_{m}$. We then define \n\\[f_{C}(a)=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}(a).\\]\nAgain the function remains well-defined: $a$ cannot belong to $X$, since $X=G(\\mathcal{A})$, and cannot belong to a $\\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{p})$ unless $p=n$, $\\theta=\\sigma$ and $b_{1}=a_{1}, \\dotsc    ,b_{p}=a_{n}$, since $\\mathcal{A}$ is disconnected.\n\nClearly $f_{C}$ is a homomorphism, since the image of $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ under $f_{C}$ is contained in $\\sigma_{\\mathcal{B}}(f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n}))$, and $f_{C}$ extends both $f$ and $C$.\n\n\\begin{lemma}\\label{gen by ground and disc implies cdf-gen}\nIf a multialgebra $\\mathcal{A}$ is both generated by its ground $X$ and disconnected, $\\mathcal{A}$ is $\\textbf{cdf}$-generated by $X$.\n\\end{lemma}\n\n\\begin{proof}\nIt remains for us to show that $f_{C}$, as defined above, is the only homomorphism extending $f$ and $C$. Suppose $g$ is another such homomorphism and we shall proceed yet again by induction.\n\nOn $\\langle X\\rangle_{0}$, we have that $f_{C}(x)=f(x)=g(x)$ for all $x\\in X$; and for $a\\in\\sigma_{\\mathcal{A}}$ and $\\sigma\\in\\Sigma_{0}$ we have that \n\\[f_{C}(a)=C\\sigma(a)=g(a),\\]\nhence $f_{C}$ and $g$ coincide on $\\langle X\\rangle_{0}$. Suppose that $f_{C}$ and $g$ are equal on $\\langle X\\rangle_{m}$ and take $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ for $a_{1}, \\dotsc   , a_{n}\\in \\langle X\\rangle_{m}$: we have by induction hypothesis that\n\\[f_{C}(a)=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{f_{C}(a_{1}), \\dotsc   , f_{C}(a_{n})}(a)=C\\sigma_{a_{1}, \\dotsc   , a_{n}}^{g(a_{1}), \\dotsc   , g(a_{n})}(a)=g(a),\\]\nwhich concludes our proof.\n\\end{proof}\n\n\\begin{theorem}\\label{cdf-gen iff gen by ground and disc}\nA multialgebra $\\mathcal{A}$ is $\\textbf{cdf}$-generated if, and only if, $\\mathcal{A}$ is generated by its ground and disconnected.\n\\end{theorem}\n\nWe have introduced several concepts that, although dissimilar in their definitions, are intrinsically connected; so it becomes important to analyze whether they are indeed distinct: are there multialgebras that are disconnected but not generated by their grounds? Are there multialgebras that are generated by their grounds but not disconnected? Or does being generated by its ground implies being disconnected, or vice-versa? We show below that this is not the case, for we provide examples answering positively both previous questions.\n\n\\begin{example}\\label{C}\nTake the signature $\\Sigma_{s}$ with a single unary operator, first defined in Example \\ref{s}. Consider the $\\Sigma_{s}-$multialgebra $\\mathcal{C}=(\\{-1,1\\}, \\{s_{\\mathcal{C}}\\})$ such that $s_{\\mathcal{C}}(-1)=\\{1\\}$ and $s_{\\mathcal{C}}(1)=\\{-1\\}$ (that is, $s_{\\mathcal{C}}(x)=\\{-x\\}$).\n\nWe state that $\\mathcal{C}$ is disconnected, but not generated by its ground. $\\mathcal{C}$ is clearly disconnected since $s_{\\mathcal{C}}(-1)\\cap s_{\\mathcal{C}}(1)=\\emptyset$; now, $B(\\mathcal{C})=s_{\\mathcal{C}}(-1)\\cup s_{\\mathcal{C}}(1)=\\{-1,1\\}$, and so $G(\\mathcal{C})=\\emptyset$. Since $\\Sigma_{0}=\\emptyset$, $\\bigcup_{\\sigma\\in\\Sigma_{0}}\\sigma_{\\mathcal{C}}=\\emptyset$ and therefore $\\langle G(\\mathcal{C})\\rangle_{n}=\\emptyset$ for every $n\\in\\mathbb{N}$, so that $G(\\mathcal{C})$ does not generate $\\mathcal{C}$.\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n    -1 \\arrow[rr, bend left=50, \"s_{\\mathcal{C}}\"]  && 1 \\arrow[ll, bend left=50, \"s_{\\mathcal{C}}\"]\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{C}$}\n\\end{figure}\n\\end{example}\n\n\\begin{example}\\label{B}\nTake the signature $\\Sigma_{s}$ with a single unary operator. Consider the $\\Sigma_{s}-$multialge\\-bra $\\mathcal{B}=(\\{0, 1\\}, \\{s_{\\mathcal{B}}\\})$ such that $s_{\\mathcal{B}}(0)=\\{1\\}$ and $s_{\\mathcal{B}}(1)=\\{1\\}$ (that is, $s_{\\mathcal{B}}(x)=\\{1\\}$).\n\nThen $\\mathcal{B}$ is clearly not disconnected, since $s_{\\mathcal{B}}(0)\\cap s_{\\mathcal{B}}(1)=\\{1\\}$, yet $\\mathcal{B}$ is generated by its ground: $B(\\mathcal{B})=\\{1\\}$ and so $G(\\mathcal{B})=\\{0\\}$, and we see that $\\langle G(\\mathcal{B})\\rangle_{1}$ is already $\\{0,1\\}$.\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n    0 \\arrow[r, \"s_{\\mathcal{B}}\"]  & 1 \\arrow[loop right, out=30, in=-30, distance=3em]{}{s_{\\mathcal{B}}}\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{B}$}\n\\end{figure}\n\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Being disconnected and having a strong basis}\n\nNow, we define the notion of a strong basis (a minimum generating set), and prove that, on one hand, being generated by the ground implies having a strong basis, what means that being disconnected and generated by the ground implies being disconnected and having a strong basis; reciprocally, we also prove having a strong basis and being disconnected implies being disconnected and generated by the ground (although having a strong basis does not imply being generated by the ground). This provides a third characterization of our weakly free multialgebras.\n\nOur motivation, when coining the definition of a strong basis, was to be able to weaken that very condition: after all, absolutely free algebras (that is, algebras freely generated on the variety of all algebras on a given signature) are easier to define than the relatively free ones (which are the algebras freely generated on any variety one wants to consider), so it is natural that we start this study with ``absolutely free'' multialgebras. However, we still would like to be able, in the future, to define what should be a relatively free multialgebra (whatever a variety of multialgebras may be); to this end, weakening a strong basis to be a minimal (instead of minimum) generating set makes sense, given many relatively free algebras, on domains such as that of vector spaces, indeed have basis.\n\n\n\\begin{definition}\nWe say $B\\subseteq A$ is a strong basis\\index{Strong basis} of the $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ if it is the minimum of the set $\\mathcal{G}=\\{S\\subseteq A\\ : \\  \\langle S\\rangle=A\\}$ when ordered by inclusion.\n\\end{definition}\n\n\\begin{example}\nThe set of variables $\\mathcal{V}$ is a strong basis of $\\textbf{F}(\\Sigma, \\mathcal{V})$.\n\\end{example}\n\n\\begin{example}\nThe set of elements without predecessor of a directed forest of height $\\omega$ is a strong basis of the forest, when considered as a $\\Sigma_{s}$-multialgebra.\n\\end{example}\n\n\\begin{lemma}\\label{ground cap span is in set}\nFor every subset $S$ of the universe of a $\\Sigma$-multialgebra $\\mathcal{A}$, $G(\\mathcal{A})\\cap \\langle S\\rangle\\subseteq S$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $x\\in G(\\mathcal{A})\\cap \\langle S\\rangle$: if $x\\notin S$, we will show that $x$ cannot be in $\\langle S\\rangle$, which contradicts our assumption. Indeed, if $x\\notin S$ then \n\\[x\\notin\\langle S\\rangle_{0}=S\\cup \\bigcup_{\\sigma\\in\\Sigma_{0}}\\sigma_{\\mathcal{A}},\\]\nsince $x\\notin S$, and $x\\in G(\\mathcal{A})$ implies that \n\\[x\\in A\\setminus B(\\mathcal{A}) \\subseteq A\\setminus \\bigcup_{\\sigma\\in\\Sigma_{0}}\\sigma_{\\mathcal{A}}.\\]\n\nNow, for induction hypothesis, suppose that $x\\notin\\langle S\\rangle_{m}$; then\n\\[x\\notin \\langle S\\rangle_{m+1}=\\langle S\\rangle_{m}\\cup\\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in \\langle S\\rangle_{m} \\big\\}\\]\nsince $x\\notin \\langle S\\rangle_{m}$, and $x\\in G(\\mathcal{A})$ implies that \n\\[x\\in A\\setminus B(\\mathcal{A}) \\subseteq A\\setminus \\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in \\langle S\\rangle_{m} \\big\\}.\\]\n\\end{proof}\n\n\\begin{theorem}\\label{ground is in basis}\nIf the $\\Sigma$-multialgebra $\\mathcal{A}$ has a strong basis $B$, $G(\\mathcal{A})\\subseteq B$.\n\\end{theorem}\n\n\\begin{proof}\nBy lemma \\ref{ground cap span is in set}, $G(\\mathcal{A})=G(\\mathcal{A})\\cap A=G(\\mathcal{A})\\cap\\langle B\\rangle\\subseteq B$.\n\\end{proof}\n\nNotice lemma \\ref{ground cap span is in set} leads us too to the fact that, if $\\mathcal{A}$ is generated by its ground, then it has the ground as a strong basis: this is because, if $\\langle S\\rangle=A$, $G(\\mathcal{A})=G(\\mathcal{A})\\cap\\langle S\\rangle\\subseteq S$, and therefore $G(\\mathcal{A})$ becomes a minimum generating set.\n\n\\begin{definition}\nIf $B$ is a strong basis of a disconnected $\\Sigma$-multialgebra $\\mathcal{A}$, we define the $B$-order\\index{Order, $B$-}\\label{Border} of an element $a\\in A$ as the natural number \n\\[o_{B}(a)=\\min \\big\\{k\\in\\mathbb{N}\\ : \\  a\\in\\langle B\\rangle_{k}\\big\\}.\\]\n\\end{definition}\n\nThis is a clear generalization of the order, or complexity, of a formula: in fact, the order of a formula in ${T}(\\Sigma, \\mathcal{V})$ is exactly its $\\mathcal{V}$-order.\n\n\\begin{proposition}\nIf $a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ and $o_{B}(a)\\geq 1$, then $o_{B}(a_{1}), \\dotsc   , o_{B}(a_{n})<o_{B}(a)$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $o_{B}(a)$ equals $m+1$, with $m\\in\\mathbb{N}$, implying that\n\\[a\\in \\langle B\\rangle_{m+1}=\\langle B\\rangle_{m}\\cup \\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in \\langle B\\rangle_{m} \\big\\};\\]\nsince $m+1=\\min\\{k\\in\\mathbb{N}\\ : \\  a\\in\\langle B\\rangle_{k}\\}$, we have that $a\\notin \\langle B\\rangle_{m}$ and therefore\n\\[a\\in \\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in \\langle B\\rangle_{m} \\big\\}.\\]\nFinally, we obtain that there exist $p\\in\\mathbb{N}$, $\\theta\\in\\Sigma_{p}$ and $b_{1}, \\dotsc   , b_{p}\\in \\langle B\\rangle_{m}$ such that $a\\in \\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{p})$. Since $a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$, this implies that  $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})\\cap\\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{p})\\neq\\emptyset$, and therefore $p=n$, $\\theta=\\sigma$ and $b_{1}=a_{1}, \\dotsc , b_{p}=a_{n}$, so that $o_{B}(a_{1}), \\dotsc   , o_{B}(a_{n})\\leq m$.\n\\end{proof}\n\nBut what if $a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$ and $o_{B}(a)=0$, implying $a\\in B$? We state this case cannot occur, for if it does, \n\\[B^{*}=\\big(B\\cup\\{a_{1}, \\dotsc   , a_{n}\\}\\big)\\setminus\\{a\\}\\]\ngenerates $A$, while clearly not containing $B$, even in the case where $n=0$. We have that $a\\in \\langle B^{*}\\rangle_{1}$, since $a_{1}, \\dotsc   , a_{n}\\in B^{*}$ and $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$, and given that $B\\setminus\\{a\\}\\subseteq B^{*}$, it follows that $B\\subseteq \\langle B^{*}\\rangle_{1}$, and so $\\langle B\\rangle_{0}\\subseteq\\langle B^{*}\\rangle_{1}$. \n\nIt is then true that, for every $m\\in\\mathbb{N}$, $\\langle B\\rangle_{m}\\subseteq \\langle B^{*}\\rangle_{m+1}$: if this is true for $m$, let $b\\in \\langle B\\rangle_{m+1}$, and then either $b\\in \\langle B\\rangle_{m}$, so that $b\\in \\langle B^{*}\\rangle_{m+1}\\subseteq \\langle B^{*}\\rangle_{m+2}$, or there exist $\\theta\\in\\Sigma_{p}$ and $b_{1}, \\dotsc   , b_{p}\\in \\langle B\\rangle_{m}$ such that $b\\in\\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{p})$; in this case, since $\\langle B\\rangle_{m}\\subseteq\\langle B^{*}\\rangle_{m+1}$, we have that\n\\[b\\in \\theta_{\\mathcal{A}}(b_{1}, \\dotsc   , b_{p})\\subseteq \\bigcup \\big\\{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n}) \\ : \\  n\\in\\mathbb{N}, \\, \\sigma\\in\\Sigma_{n}, \\, a_{1}, \\dotsc   , a_{n}\\in \\langle B^{*}\\rangle_{m+1} \\big\\} \\subseteq\\langle B^{*}\\rangle_{m+2},\\]\nso once again $b\\in\\langle B^{*}\\rangle_{m+2}$. Since $\\langle B\\rangle=\\bigcup_{m\\in\\mathbb{N}}\\langle B\\rangle_{m}$ equals $A$, we have that $\\langle B^{*}\\rangle$ also equals $A$, as we previously stated. This is absurd, since $B$ is the minimum of $\\{S\\subseteq A\\ : \\  \\langle S\\rangle=A\\}$ ordered by inclusion and $B\\not\\subseteq B^{*}$. The conclusion must be that if $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$, then $o_{B}(a_{1}), \\dotsc   , o_{B}(a_{n})<o_{B}(a)$, regardless of the value of $o_{B}(a)$.\n\n\\begin{lemma}\\label{ground is basis if disc}\nIf $\\mathcal{A}$ is disconnected and has a strong basis $B$, $B=G(\\mathcal{A})$ and so $\\mathcal{A}$ is generated by its ground.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $a\\in B\\setminus G(\\mathcal{A})$: since $a$ is in the build of $\\mathcal{A}$, there exist $\\sigma\\in\\Sigma_{n}$ and elements $a_{1}, \\dotsc   , a_{n}\\in A$ such that $a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc   , a_{n})$. If $n>0$, $o_{B}(a)>o_{B}(a_{1})\\geq 0$, which contradicts the fact that $a\\in B$ and therefore $o_{B}(a)=0$.\n\nIf $n=0$, it is clear that $B^{*}=B\\setminus \\{a\\}$ is a generating set smaller than $B$: generating set because, if $a\\in \\sigma_{\\mathcal{A}}$, $a\\in \\bigcup_{\\sigma\\in\\Sigma_{0}}\\sigma_{\\mathcal{A}}$ and therefore $B\\subseteq \\langle B^{*}\\rangle_{0}$, so that $\\langle B\\rangle_{m}\\subseteq\\langle B^{*}\\rangle_{m+1}$ and $\\bigcup_{m\\in\\mathbb{N}}\\langle B\\rangle_{m}=\\bigcup_{m\\in\\mathbb{N}}\\langle B^{*}\\rangle_{m}$. This is also a contradiction, since $B$ is a strong basis.\n\\end{proof}\n\n\\begin{theorem}\\label{theorem 3}\n$\\mathcal{A}$ is generated by its ground and disconnected if, and only if, it has a strong basis and it is disconnected.\n\\end{theorem}\n\n\\begin{proof}\nWe already proved, in Lemma \\ref{ground is basis if disc}, that if $\\mathcal{A}$ is disconnected and has a strong basis $B$, then it is generated by its ground and disconnected. Reciprocally, if $\\mathcal{A}$ is disconnected and generated by its ground, first of all it is clearly disconnected.\n\nNow, if $\\langle G(\\mathcal{A})\\rangle=A$ one has that $G(\\mathcal{A})\\subseteq S$ for every $S\\in \\{S\\subseteq A\\ : \\  \\langle S\\rangle=A\\}$, by Lemma \\ref{ground cap span is in set}. Therefore, the ground is a strong basis.\n\\end{proof}\n\nOnce again, we ask ourselves whether the concepts we have defined in this section are truly independent: does being disconnected imply having a strong basis? Does having a strong basis imply being disconnected? We show that neither is the case by providing examples of a multialgebra that is disconnected but does not have a strong basis and one of a multialgebra that has a strong basis but is not disconnected.\n\n\\begin{example}\nTake the $\\Sigma_{s}-$multialgebra $\\mathcal{C}$ from Example \\ref{C}.\n\nWe know that $\\mathcal{C}$ is disconnected, but we also state that it does not have a strong basis: in fact, we see that the set $\\{S\\subseteq \\{-1,1\\}\\ :\\  \\langle S\\rangle=\\{-1,1\\}\\}$ is exactly $\\{\\{-1\\}, \\{1\\}, \\{-1,1\\}\\}$, and this set has no minimum.\n\\end{example}\n\n\\begin{example}\nTake the $\\Sigma_{s}-$multialgebra $\\mathcal{B}$ from Example \\ref{B}.\n\nAs we saw before, $\\mathcal{B}$ is not disconnected, but we state that it has a strong basis: $B=\\{0\\}$ generates $\\mathcal{B}$ and, since $\\{1\\}$ does not generate the multialgebra, we find that $B$ is a minimum generating set.\n\\end{example}\n\nIn these two examples we presented a multialgebra ($\\mathcal{B}$) which has a strong basis and is generated by its ground, and one multialgebra ($\\mathcal{C}$) which does not have a strong basis and is not generated by its ground. Clearly being generated by its ground implies having a strong basis, so it is natural to hypothesize that having a strong basis and being generated by its ground could be equivalent notions; however, as we show in the example below, having a basis does not imply being generated by its ground.\n\n\\begin{example}\nTake the signature $\\Sigma_{s}$ with a single unary operator from Example \\ref{s}. Consider the $\\Sigma_{s}-$multialgebra $\\mathcal{M}=(\\{-1, 0, 1\\}, \\{s_{\\mathcal{M}}\\})$ such that $s_{\\mathcal{M}}(0)=\\{0\\}$, $s_{\\mathcal{M}}(1)=\\{1\\}$ and $s_{\\mathcal{M}}(-1)=\\{1\\}$ (that is, $s_{\\mathcal{M}}(x)=\\{|x|\\}$, where $|X|$ denotes the absolute value of $x$).\n\nWe have that $G(\\mathcal{M})=\\{-1\\}$ and that $\\langle\\{-1\\}\\rangle=\\{-1, 1\\}$, so that $\\mathcal{M}$ is not generated by its ground. But we state that $\\{-1, 0\\}$ is a strong basis: first of all, it clearly generates $\\mathcal{M}$; furthermore, the generating sets of $\\mathcal{M}$ are only $\\{-1, 0\\}$ and $\\{-1, 0, 1\\}$, so that $\\{-1, 0\\}$ is in fact the smallest generating set.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n    -1 \\arrow[rr, bend right=50]{}{s_{\\mathcal{M}}}  & 0 \\arrow[u, loop]{}{s_{\\mathcal{M}}} & 1 \\arrow[loop right, out=30, in=-30, distance=3em]{}{s_{\\mathcal{M}}}\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{M}$}\n\\end{figure}\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Being disconnected and chainless}\n\nThe last equivalence to being a submultialgebra of $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ we give depends on the notion of being chainless, which is very graph-theoretical in nature. Think of a tree that ramifies ever downward: one can pick any vertex and proceed, against the arrows, upwards until an element without predecessor is reached. More than that, it is not possible to find an infinite path, starting in any one vertex, by always going against the arrows: such a path, if it existed, would be what we shall call a chain. A multialgebra without chains is, very naturally, chainless.\n\nAs it was in the case of strong bases, there isn't a parallel concept to being chainless among the theory of universal algebra: it seems that this concept is far more natural when dealing with multioperations, although it can be easily applied to algebras if one wishes to do so. A close, although not equivalent, entity are the branches in the formation trees of formulas: if allowed to grow infinitely, these would became chains.\n\nThe main result of this section is, perhaps, the fact that being chainless implies being generated by its ground, which, as we know, implies in turn having a strong basis. Of course, when adding disconnectedness to the equation, all three concepts become equivalent to one another and to the fact that the multialgebra at hand is weakly free. We have, then, the following schematic diagram, that offers an overview of the aforementioned results.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tikzcd}[arrows=Rightarrow, every arrow\/.append style={shift right=3ex}]\n\\text{$\\mathcal{A}$ is chainless} \\arrow[Rightarrow]{d}\\\\\n\\text{$\\mathcal{A}$ is generated by $G(\\mathcal{A})$} \\arrow[Rightarrow]{d}\\arrow[negated]{u}\\\\\n\\text{$\\mathcal{A}$ has a strong basis} \\arrow[negated]{u}\n  \\end{tikzcd}\n\\end{minipage}\n\\hspace{2cm}\n\\centering\n\\begin{minipage}[t]{6cm}\n\\centering\n\\begin{tikzcd}[arrows=Rightarrow, every arrow\/.append style={shift right=3ex}]\n\\begin{tabular}{c}$\\mathcal{A}$ is chainless\\\\ and disconnected\\end{tabular} \\arrow[Rightarrow]{d}\\\\\n\\begin{tabular}{c}$\\mathcal{A}$ is generated by $G(\\mathcal{A})$\\\\ and disconnected\\end{tabular} \\arrow[Rightarrow]{d}\\arrow{u}\\\\\n\\begin{tabular}{c}$\\mathcal{A}$ has a strong basis\\\\ and is disconnected\\end{tabular} \\arrow{u}\n  \\end{tikzcd}\n\\end{minipage}\n\\end{figure}\n\n\n\nGiven a function $\\tau:\\{1, \\dotsc   , n\\}\\rightarrow\\{1, \\dotsc   , n\\}$ in $S_{n}$, the group of permutations on $n$ elements (meaning $\\tau$ is bijective), the action of $\\tau$ in an $n$-tuple $(x_{1}, \\dotsc   , x_{n})\\in X^{n}$ is given by\n\\[\\tau(x_{1}, \\dotsc   , x_{n})=(x_{\\tau(1)}, \\dotsc   , x_{\\tau(n)}).\\]\nGiven $1\\leq i, j\\leq n$, we define $[i,j]$ to be the permutation such that $[i,j](i)=j$, $[i,j](j)=i$ and, for $k\\in\\{1, \\dotsc   , n\\}$ different from $i$ and $j$, $[i,j](k)=k$.\n\n\\begin{definition}\nGiven a $\\Sigma$-multialgebra $\\mathcal{A}$, a sequence $\\{a_{n}\\}_{n\\in\\mathbb{N}}\\subseteq A$ is said to be a chain\\index{Chain} if, for every $n\\in\\mathbb{N}$, there exist a natural number $m_{n}\\in\\mathbb{N}$, a functional symbol $\\sigma^{n}\\in\\Sigma_{m_{n}}$, a permutation $\\tau_{n}\\in S_{m_{n}}$ and elements $a_{1}^{n}, \\dotsc   , a_{m_{n}-1}^{n}\\in A$ such that \n\\[a_{n}\\in \\sigma^{n}_{\\mathcal{A}}(\\tau_{n}(a_{n+1}, a_{1}^{n}, \\dotsc   , a_{m_{n}-1}^{n})).\\]\n\nA $\\Sigma$-multialgebra is said to be chainless\\index{Chainless} when it has no chains.\n\\end{definition}\n\n\\begin{example}\nTake a directed forest of height $\\omega$ and add to it a loop, that is, choose a vertex $v$ and add an arrow from $v$ to $v$: then $\\{a_{n}\\}_{n\\in\\mathbb{N}}$, such that $a_{n}=v$ for every $n\\in\\mathbb{N}$, is a chain.\n\\end{example}\n\n\\begin{example}\n$\\textbf{F}(\\Sigma, \\mathcal{V})$ is chainless.\n\\end{example}\n\n\\begin{lemma}\\label{chainless implies gen by ground}\nIf $\\mathcal{A}$ is chainless, then it is generated by its ground.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that this not hold, so $A\\setminus\\langle G(\\mathcal{A})\\rangle$ is not empty, and must therefore contain some element $a_{0}$. We create a chain $\\{a_{n}\\}_{n\\in\\mathbb{N}}$ by induction, being the case $n=0$ already done.\n\nSo, suppose we have created a finite sequence of elements $a_{0}, \\dotsc   , a_{k}\\in A\\setminus \\langle G(\\mathcal{A})\\rangle$ such that, for each $0\\leq n< k$, there exist a positive integer $m_{n}\\in\\mathbb{N}\\setminus\\{0\\}$, a functional symbol $\\sigma^{n}\\in\\Sigma_{m_{n}}$, a permutation $\\tau_{n}\\in S_{m_{n}}$ and elements $a_{1}^{n}, \\dotsc   , a_{m_{n}-1}^{n}\\in A$ such that \n\\[a_{n}\\in \\sigma^{n}_{\\mathcal{A}}(\\tau_{n}(a_{n+1}, a_{1}^{n}, \\dotsc   , a_{m_{n}-1}^{n})).\\]\n\nSince $a_{k}\\in A\\setminus \\langle G(\\mathcal{A})\\rangle$, we have that $a_{k}$ is not an element of the ground; so, there must exist $m_{k}\\in\\mathbb{N}$, a functional symbol $\\sigma^{k}\\in\\Sigma_{m_{k}}$ and elements $b_{1}^{k}, \\dotsc   , b_{m_{k}}^{k}\\in A$ such that\n\\[a_{k}\\in\\sigma^{k}_{\\mathcal{A}}(b_{1}^{k}, \\dotsc   , b_{m_{k}}^{k}).\\]\nNow, if all $b_{1}^{k}, \\dotsc   , b_{m_{k}}^{k}$ belonged to $\\langle G(\\mathcal{A})\\rangle$, so would $a_{k}$: there must be an element $a_{k+1}\\in\\{b_{1}^{k}, \\dotsc   , b_{m_{k}}^{k}\\}$, say $b_{l}^{k}$, such that $a_{k+1}\\in A\\setminus \\langle G(\\mathcal{A})\\rangle$. We then define $a_{i}^{k}$ as $b_{j}^{k}$, for $i\\in\\{1, \\dotsc   , m_{k}-1\\}$ and $j=\\min\\{i\\leq p\\leq m_{k}\\ : \\  p\\neq l\\}$, and \n\\[\\tau_{k}=[l-1, l]\\circ\\cdots\\circ[1,2],\\]\nand then it is clear that $\\{a_{n}\\}_{n\\in\\mathbb{N}}$ becomes a chain, with the extra condition that $\\{a_{n}\\}_{n\\in\\mathbb{N}}\\subseteq A\\setminus \\langle G(\\mathcal{A})\\rangle$. Since $\\mathcal{A}$ is chainless, we reach a contradiction, so we must have instead that $A\\setminus \\langle G(\\mathcal{A})\\rangle=\\emptyset$, and therefore $\\mathcal{A}$ is generated by its ground.\n\\end{proof}\n\nSo, a disconnected, chainless multialgebra is, by Lemma \\ref{chainless implies gen by ground}, disconnected and generated by its ground. We state, that, in fact, the reciprocal also holds, when we arrive at yet another characterization of being an weakly free multialgebra.\n\nSo, suppose $\\mathcal{A}$ is disconnected and generated by its ground, and let $\\{a_{n}\\}_{n\\in\\mathbb{N}}$ be a chain in $\\mathcal{A}$: clearly no $a_{n}$ can belong to the ground, since \n\\[a_{n}\\in \\sigma^{n}_{\\mathcal{A}}(\\tau_{n}(a_{n+1}, a_{1}^{n}, \\dotsc   , a_{m_{n}-1}^{n})),\\]\nand therefore $o_{G(\\mathcal{A})}(a_{n+1})<o_{G(\\mathcal{A})}(a_{n})$, that is, the $G(\\mathcal{A})$-order of $a_{n+1}$ is less than the $G(\\mathcal{A})$-order of $a_{n}$; we reach a contradiction, since if $o_{G(\\mathcal{A})}(a_{0})=m$, then $o_{G(\\mathcal{A})}(a_{m+1})<0$, what is impossible. $\\mathcal{A}$ must then be chainless.\n\n\\begin{theorem}\\label{generated by ground and disc. iff chainless and disc.}\n$\\mathcal{A}$ is generated by its ground and disconnected if, and only if, it is chainless and disconnected.\n\\end{theorem}\n\nFinally, Theorems \\ref{sub mF is cdf-gen ground}, \\ref{cdf-gen is iso to sub mF}, \\ref{cdf-gen iff gen by ground and disc}, \\ref{theorem 3} and \\ref {generated by ground and disc. iff chainless and disc.} can be summarized as follows; the object we decided to call a weakly free multialgebra is then a multialgebra which satisfies any, and therefore all, of the conditions found in the following theorem.\n\n\\begin{theorem}\nAre equivalent:\n\n\\begin{enumerate} \\item $\\mathcal{A}$ is a submultialgebra of some $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$;\n\n\\item $\\mathcal{A}$ is $\\textbf{cdf}$-generated;\n\n\\item $\\mathcal{A}$ is generated by its ground and disconnected;\n\n\\item $\\mathcal{A}$ has a strong basis and is disconnected;\n\n\\item $\\mathcal{A}$ is chainless and disconnected.\n\\end{enumerate}\n\\end{theorem}\n\nAn important point to stress is that, although not all concepts present in the previous theorem have natural counterparts in universal algebra, by defining them for algebras, presented as multialgebras, we find that all of the conditions in the theorem are valid only for algebras of formulas. This follows easily from the fact that the only $\\textbf{cdf}$-generated algebras are the algebras of formulas themselves.\n\nNow, a few examples concerning being chainless, disconnected, having a strong basis and being generated by the ground will be given. Essentially, they show that the notion of being chainless is independent from that of being disconnected, and that despite being chainless implies being generated by the ground (and so having a strong basis), the reciprocal does not hold, and there exist multialgebras generated by the ground which are not chainless.\n\n\\begin{example}\nTake the signature $\\Sigma_{s}$ from Example \\ref{s}, and consider the $\\Sigma_{s}-$multialgebra $\\mathcal{Y}=(\\mathbb{N}\\cup\\{a,b\\}, \\{s_{\\mathcal{Y}}\\})$ such that $s_{\\mathcal{Y}}(n)=\\{n+1\\}$, for $n\\in\\mathbb{N}$, and $s_{\\mathcal{Y}}(a)=s_{\\mathcal{Y}}(b)=\\{0\\}$.\n\nWe see that $\\mathcal{Y}$ is chainless since, given a chain $\\{a_{n}\\}_{n\\in\\mathbb{N}}$, it must be contained in the build of $\\mathcal{Y}$, that is, $\\mathbb{N}$: but then $a_{n+1}=a_{n}-1$, what is a contradiction, since there are only a finite number of elements with index smaller than that of $a_{0}$. At the same time, $\\mathcal{Y}$ is not disconnected, since $s_{\\mathcal{Y}}(a)=s_{\\mathcal{Y}}(b)$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\na\\arrow[dr]{}{s_{\\mathcal{Y}}} & & & \\\\\n    & 0\\arrow[r]{}{s_{\\mathcal{Y}}} & 1\\arrow[r]{}{s_{\\mathcal{Y}}} & \\cdots\\\\\nb\\arrow[ur]{}{s_{\\mathcal{Y}}} & & & \\\\\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{Y}$}\n\\end{figure}\n\\end{example}\n\n\n\\begin{example}\nTake the $\\Sigma_{s}-$multialgebra $\\mathcal{C}$ from Example \\ref{C}.\n\nWe know that $\\mathcal{C}$ is disconnected, but we state that it is not chainless: in fact,\\\\ $\\{(-1)^{n}\\}_{n\\in\\mathbb{N}}$ and $\\{(-1)^{n+1}\\}_{n\\in\\mathbb{N}}$ are chains in $\\mathcal{C}$.\n\\end{example}\n\n\\begin{example}\nTake the $\\Sigma_{s}-$multialgebra $\\mathcal{B}$ from Example \\ref{B}.\n\nWe have already established that $\\mathcal{B}$ has a basis and is generated by its ground, $\\{0\\}$, yet it is not chainless: $\\{ 1\\}_{n\\in\\mathbb{N}}$ is a chain in $\\mathcal{B}$.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Freely generated multialgebras}\\label{Freely generated multialgebra}\n\nSo, as we mentioned time and time again, the algebras of formulas, in the study of universal algebras, are specially significant given they are the only absolutely free algebras, meaning they are the only ones with the unique extension property in the variety of all algebras over their signature.\n\nSo we turn now to a somewhat folkloric result: the unique extension property, relative to the class of all multialgebras on a given signature, is satisfied by no one; alternatively, this statement can be reworded as to say that the forgetful functor from the category of multialgebras to $\\textbf{Set}$ does not have a left adjoint, what implies in particular that this category has no initial objects. \n\nOf course, such a result is stated in various ways, depending on your definition of homomorphism and, perhaps most importantly, even on your own definition of multialgebra. So, we offer what we believe to be a simplified proof of the result for the category $\\textbf{MAlg}(\\Sigma)$ as we have defined it. We proceed by contradiction, and start defining the multialgebras which do satisfy the unique extension property.\n\n\\begin{definition}\nA $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is freely generated by $X$, for a $X\\subseteq A$, if for every $\\Sigma$-multialgebra $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$ and map $f:X\\rightarrow B$ there exists only one homomorphism $\\overline{f}:\\mathcal{A}\\rightarrow\\mathcal{B}$ extending $f$.\n\\end{definition}\n\nIn other words, if $j:X\\rightarrow A$ is the inclusion, there exists only one homomorphism $\\overline{f}:\\mathcal{A} \\to \\mathcal{B}$ commuting the following diagram in {\\bf Set}.\n\\[\n\\begin{tikzcd}\n & A \\arrow{dr}{\\overline{f}} \\\\\nX \\arrow{ur}{j} \\arrow{rr}{f} && B\n\\end{tikzcd}\n\\]\n\n\n\\begin{lemma}\\label{bijective homomorphism is full}\nIf $\\varphi$ is a bijective $\\Sigma$-homomorphism from $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\})$ to $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\})$ whose inverse is also a homomorphism, then $\\varphi$ is a isomorphism.\n\\end{lemma}\n\n\\begin{proof}\nWe only have to prove that $\\varphi$ is a full homomorphism, so take $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$: from the fact $\\varphi$ is a homomorphism we know that \n\\[\\{\\varphi(a) \\ : \\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}));\\]\nand since $\\varphi^{-1}$ is also a homomorphism, we have that\n\\[\\{\\varphi^{-1}(b) \\ : \\  b\\in\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\}\\subseteq \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}).\\]\nSince $\\varphi$ is a bijection, we may apply it to the last equation while preserving the inclusion, thus getting \n\\[\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\subseteq \\{\\varphi(a) \\ : \\  a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}.\\]\n\\end{proof}\n\n\\begin{proposition}\nIf there exist $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ which are freely generated by, respectively, $X$ and $Y$ such that $|X|=|Y|$, then $\\mathcal{A}$ and $\\mathcal{B}$ are isomorphic.\n\\end{proposition}\n\n\\begin{proof}\nSince $X$ and $Y$ are of the same cardinality, there exists bijective functions $f:X\\rightarrow Y$ and $g:Y\\rightarrow X$ inverses of each other. Take the extensions $\\overline{f}:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\overline{g}:\\mathcal{B}\\rightarrow\\mathcal{A}$ and we have that $\\overline{g}\\circ\\overline{f}$ is a homomorphism extending $g\\circ f=Id_{X}$, the identity on $X$.\n\nSince the identical homomorphism $Id_{\\mathcal{A}}:\\mathcal{A}\\rightarrow \\mathcal{A}$ also extends $Id_{X}$, we have that $Id_{\\mathcal{A}}=\\overline{g}\\circ\\overline{f}$. In a similar way we have that $Id_{\\mathcal{B}}=\\overline{f}\\circ\\overline{g}$, so that $\\mathcal{A}$ and $\\mathcal{B}$ are isomorphic by use of Lemma \\ref{bijective homomorphism is full}.\n\\end{proof}\n\nThis way we can refer ourselves to the unique $\\Sigma$-multialgebra freely generated by $X$, up to isomorphisms.\n\nWe will denote by $\\mathcal{U}:\\textbf{MAlg}(\\Sigma)\\rightarrow \\textbf{Set}$ the forgetful functor, which takes a multialgebra to its universe and a homomorphism to itself, seem now merely as a function between sets.\n\n\n\\begin{lemma}\\label{adjoint of forget}\nIf there exist, for every set $X$, a $\\Sigma$-multialgebra freely generated by $X$, then the functor $F:\\textbf{Set}\\rightarrow \\textbf{MAlg}(\\Sigma)$, associating a set $X$ to a $\\Sigma$-multialgebra freely generated by $X$, which we will denote $FX$, and a function $f:X\\rightarrow Y$ to the only homomorphism $\\overline{f}:FX\\rightarrow FY$ extending $f$, is a left adjoint of $\\mathcal{U}$.\n\\end{lemma}\n\n\\begin{proof}\nFor $X$ a set and $\\mathcal{A}$ a $\\Sigma$-multialgebra with universe $A$ we consider\n\\[\\Phi_{\\mathcal{A}, X}:Hom_{\\textbf{Set}}(X,\\mathcal{U}\\mathcal{A})\\rightarrow Hom_{\\textbf{MAlg}(\\Sigma)}(FX, \\mathcal{A})\\]\nassociating a map $f:X\\rightarrow A$ to the only homomorphism $\\overline{f}:FX\\rightarrow \\mathcal{A}$ extending $f$. Each $\\Phi_{\\mathcal{A}, X}$ is clearly a bijection given that $FX$ is freely generated by $X$.\n\nNow, given sets $X$ and $Y$, $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, a function $f:Y\\rightarrow X$ and a homomorphism $h:\\mathcal{A}\\rightarrow \\mathcal{B}$, we have only left to prove that the following diagram commutes in {\\bf Set}.\n\\[ \\begin{tikzcd}[sep=huge]\nHom_{\\textbf{Set}}(X, \\mathcal{U}\\mathcal{A}) \\arrow{r}{\\Phi_{\\mathcal{A}, X}} \\arrow{d}{Hom(f, \\mathcal{U}\\varphi)}& Hom_{\\textbf{MAlg}(\\Sigma)}(FX, \\mathcal{A}) \\arrow{d}{Hom(Ff, \\varphi)} \\\\%\nHom_{\\textbf{Set}}(Y, \\mathcal{U}\\mathcal{B}) \\arrow{r}{\\Phi_{\\mathcal{B}, Y}}& Hom_{\\textbf{MAlg}(\\Sigma)}(FY, \\mathcal{B})\n\\end{tikzcd}\n\\]\n\nSo we take a function $g:X\\rightarrow \\mathcal{U}\\mathcal{A}$: taking the top-right edges of the diagram we have $\\Phi_{\\mathcal{A}, X}g=\\overline{g}$ and $Hom(Ff, h)\\overline{g}=h\\circ \\overline{g}\\circ Ff$; on the bottom-left ones, $Hom(f, \\mathcal{U}h)g=\\mathcal{U}h\\circ g\\circ f$ and $\\Phi_{\\mathcal{B}, Y}(\\mathcal{U}h\\circ g\\circ f)=\\overline{\\mathcal{U}h\\circ g\\circ f}$.\n\nNow both $h\\circ \\overline{g}\\circ Ff$ and $\\overline{\\mathcal{U}h\\circ g\\circ f}$ are homomorphisms from $FY$ to $\\mathcal{B}$ extending $\\mathcal{U}h\\circ g\\circ f:Y\\rightarrow \\mathcal{U}\\mathcal{B}$: for the second one this is obvious, for the first we take an element $y\\in Y$ and see that\n\\[h\\circ \\overline{g}\\circ Ff(y)=h\\circ\\overline{g}\\circ f(y)=h\\circ g\\circ f(y)=\\mathcal{U}h\\circ g\\circ f(y)\\]\nsince, respectively, $Ff=\\overline{f}$, which extends $f$; $\\overline{g}$ extends $g$, which is defined on $X$ that contains $f(y)$; and $\\mathcal{U}h=h$, seen only as a function between sets.\n\nGiven that $FY$ is freely generated over $Y$, we have that $h\\circ \\overline{g}\\circ Ff=\\overline{\\mathcal{U}\\varphi\\circ g\\circ f}$ and the diagram in fact commutes.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{no f-gen multi}\nGiven a non-empty signature $\\Sigma$ and a set $X$, there does not exist a $\\Sigma$-multialgebra freely generated by $X$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is freely generated by $X$ and let $\\mathcal{V}$ be a set that properly contains $X$, meaning $\\mathcal{V}\\neq \\emptyset$ and therefore that $\\textbf{F}(\\Sigma, \\mathcal{V})$ is well defined; then take the identity function $j: X\\rightarrow F(\\Sigma, \\mathcal{V})$, such that $j(x)=x$ for every $x\\in X$, and the homomorphism $\\overline{j}:\\mathcal{A}\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$ extending $j$.\n\nNow, take the identity function $id:\\mathcal{V}\\rightarrow F(\\Sigma^{2}, \\mathcal{V})$ and the collections of choices $C$ and $D$ from $\\textbf{F}(\\Sigma, \\mathcal{V})$ to $\\textbf{mF}(\\Sigma, \\mathcal{V}, 2)$ such that, for $\\sigma\\in\\Sigma_{n}$,\n\\[C\\sigma_{\\alpha_{1}, \\dotsc   , \\alpha_{n}}^{\\beta_{1}, \\dotsc   , \\beta_{n}}(\\sigma(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))=\\sigma^{0}(\\beta_{1}, \\dotsc  ,\\beta_{n})\\]\nand\n\\[D\\sigma_{\\alpha_{1}, \\dotsc   , \\alpha_{n}}^{\\beta_{1}, \\dotsc   , \\beta_{n}}(\\sigma(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))=\\sigma^{1}(\\beta_{1}, \\dotsc  ,\\beta_{n}),\\]\nand consider the only homomorphisms $id_{C}, id_{D}:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{mF}(\\Sigma, \\mathcal{V}, 2)$ extending, respectively, $id$ and $C$, and $id$ and $D$, which we know to exist given $\\textbf{F}(\\Sigma, \\mathcal{V})$ is \\textbf{cdf}-generated by $\\mathcal{V}$. Since $id_{C}\\circ \\overline{j}, id_{D}\\circ\\overline{j}:\\mathcal{A}\\rightarrow \\textbf{mF}(\\Sigma, \\mathcal{V}, 2)$ both extend the function $j':X\\rightarrow {T}(\\Sigma^{2}, \\mathcal{V})$ such that $j'(x)=x$ for every $x\\in X$ (remember $\\mathcal{V}$ properly contains $X$), we have $id_{C}\\circ \\overline{j}=id_{D}\\circ\\overline{j}$.\n\nNow, if $\\alpha\\in {T}(\\Sigma, \\mathcal{V})\\setminus \\mathcal{V}$, we have that there exist $\\sigma\\in \\Sigma_{n}$, for $n\\in\\mathbb{N}$, and elements $\\alpha_{1}, \\dotsc   , \\alpha_{n}\\in F(\\Sigma, \\mathcal{V})$ such that $\\alpha=\\sigma(\\alpha_{1}, \\dotsc  ,\\alpha_{n})$; in this case, \n\\[id_{C}(\\alpha)=\\sigma^{0}(id_{C}(\\alpha_{1}), \\dotsc  ,id_{C}(\\alpha_{n}))\\neq \\sigma^{1}(id_{D}(\\alpha_{1}), \\dotsc  ,id_{D}(\\alpha_{n}))=id_{D}(\\alpha),\\]\ngiven that the main connectives are distinct, and therefore implying that $id_{C}$ and $id_{D}$ are always different outside of $\\mathcal{V}$. \n\nSince $id_{C}\\circ \\overline{j}=id_{D}\\circ\\overline{j}$, we must have that $\\overline{j}(A)\\subseteq \\mathcal{V}$, and this is absurd since we are assuming $\\Sigma$ non-empty: if $\\Sigma_{0}\\neq\\emptyset$, for a $\\sigma\\in\\Sigma_{0}$ and $a\\in\\sigma_{\\mathcal{A}}$ we have that $\\overline{j}(a)=\\sigma$ in $F(\\Sigma, \\mathcal{V})$, which is not in $X$; if it is another $\\Sigma_{n}$ which is not empty, given $a\\in A$ (which exists given the universe of multialgebras are assumed to be non-empty) we have that, for $b\\in \\sigma_{\\mathcal{A}}(a, \\dotsc   , a)$, is valid that $\\overline{j}(b)=\\sigma(\\overline{j}(a), \\dotsc   , \\overline{j}(a))$, which is again not in $\\mathcal{V}$.\n\nWe must conclude that there are no freely generated multialgebras.\n\\end{proof}\n\n\\begin{corollary}\\label{no free obj}\nThe category $\\textbf{MAlg}(\\Sigma)$ does not have an initial object.\n\\end{corollary}\n\n\\begin{proof}\nWe state that if $\\mathcal{A}$ is an initial object, $\\mathcal{A}$ is freely generated by $\\emptyset$: in fact, for every $\\Sigma$-multialgebra $\\mathcal{B}$ and map $f:\\emptyset\\rightarrow B$, there exists a single homomorphism $!_{\\mathcal{B}}:\\mathcal{A}\\rightarrow \\mathcal{B}$ extending $f=\\emptyset$, that is, the only homomorphism between $\\mathcal{A}$ and $\\mathcal{B}$.\n\nBy Theorem~\\ref{no f-gen multi}, freely generated multialgebras do not exist, what ends the proof.\n\\end{proof}\n\n\\begin{theorem}\\label{no left adjoints}\nThe forgetful functor $\\mathcal{U}:\\textbf{MAlg}(\\Sigma)\\rightarrow\\textbf{Set}$ does not have a left adjoint.\n\\end{theorem}\n\n\\begin{proof}\nFor suppose we have a left adjoint $F:\\textbf{Set}\\rightarrow\\textbf{MAlg}(\\Sigma)$ of $\\mathcal{U}$, so that $F$ has a right adjoint and is therefore cocontinuous. Since $\\emptyset$ is the initial object in $\\textbf{Set}$, we have that $F\\emptyset$ must be an initial object in $\\textbf{MAlg}(\\Sigma)$, which by Corollary~\\ref{no free obj} does not exist.\n\\end{proof}\n\nAnswering a question posed at the end of Section \\ref{Formulas and how to interpret them}, we can prove with arguments similar to those found in Theorem \\ref{no f-gen multi} and Corollary \\ref{no free obj} that the category of $\\Sigma$-multialgebras equipped with full homomorphisms, $\\textbf{MAlg}_{=}(\\Sigma)$, does not have intial objects, and so its forgetful functor into $\\textbf{Set}$ does not have a left adjoint.\n\nIn fact, suppose $\\mathcal{A}$ is an initial object of $\\textbf{MAlg}_{=}(\\Sigma)$, and let $!$ be the only full homomorphism from $\\mathcal{A}$ to $\\textbf{mF}(\\Sigma, \\mathcal{V}, 2)$; we take the identity function $id:\\mathcal{V}\\rightarrow F(\\Sigma^{2}, \\mathcal{V})$ and the collections of choices $C$ and $D$ from $\\textbf{mF}(\\Sigma, \\mathcal{V}, 2)$ into itself such that\n\\[C\\sigma_{\\alpha_{1}, \\dotsc   , \\alpha_{n}}^{\\beta_{1}, \\dotsc   , \\beta_{n}}(\\sigma^{0}(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))=\\sigma^{0}(\\beta_{1}, \\dotsc  ,\\beta_{n}),\\quad C\\sigma_{\\alpha_{1}, \\dotsc   , \\alpha_{n}}^{\\beta_{1}, \\dotsc   , \\beta_{n}}(\\sigma^{1}(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))=\\sigma^{1}(\\beta_{1}, \\dotsc  ,\\beta_{n}),\\]\n\\[D\\sigma_{\\alpha_{1}, \\dotsc   , \\alpha_{n}}^{\\beta_{1}, \\dotsc   , \\beta_{n}}(\\sigma^{0}(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))=\\sigma^{1}(\\beta_{1}, \\dotsc  ,\\beta_{n})\\quad\\text{and}\\quad D\\sigma_{\\alpha_{1}, \\dotsc   , \\alpha_{n}}^{\\beta_{1}, \\dotsc   , \\beta_{n}}(\\sigma^{1}(\\alpha_{1}, \\dotsc  ,\\alpha_{n}))=\\sigma^{0}(\\beta_{1}, \\dotsc  ,\\beta_{n}),\\]\nand it is clear that $id_{C}$ and $id_{D}$ are both full homomorphisms satisfying $id_{C}\\circ !=id_{D}\\circ !$. Since $id_{C}$ and $id_{D}$ are always distinct over the build of $\\textbf{mF}(\\Sigma, \\mathcal{V}, 2)$, we get that the image of $!$ is contained in $\\mathcal{V}$, a contradiction as we saw in the proof of Theorem \\ref{no f-gen multi}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A categorical characterization}\n\nGiven the category $\\textbf{MAlg}(\\Sigma)$ whose objects are all $\\Sigma-$multialgebras and morphisms from $\\mathcal{A}$ to $\\mathcal{B}$ are all homomorphisms from $\\mathcal{A}$ to $\\mathcal{B}$, we would like to construct a related category $\\textbf{MG}(\\Sigma)$ whose objects are also exactly all $\\Sigma-$multialgebras, but where the $\\textbf{cdf}$-generated multialgebras can be construed as the image of an adjoint-having functor. The motivation behind the search for such a $\\textbf{MG}(\\Sigma)$ is that having such a category, and the associated pair of adjoint functors, would make weakly free multialgebras far more similar to absolutely free algebras than they are now: after all, the latter are images under the so called free functor, right adjoint of the forgetful functor from the category of $\\Sigma$-algebras $\\textbf{Alg}(\\Sigma)$ to the category of sets $\\textbf{Set}$. Unfortunately, since, as we showed in Theorem \\ref{no left adjoints}, the forgetful functor of $\\textbf{MAlg}(\\Sigma)$ does not have a left-adjoint, the task is not as simple as in the classical case and will require far more craftsmanship.\n\n\\begin{definition}\nGiven a homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ between $\\Sigma-$multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, we say that $\\varphi$ is \\textit{ground-preserving}, or that it preserves grounds, if $\\varphi(G(\\mathcal{A}))\\subseteq G(\\mathcal{B})$.\n\\end{definition}\n\n\\begin{proposition}\nTaking as objects all $\\Sigma-$multialgebras and as morphisms between the\\\\ $\\Sigma-$multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ all ground-preserving homomorphisms between $\\mathcal{A}$ and $\\mathcal{B}$, the resulting structure $\\textbf{MAlg}_{G}(\\Sigma)$ is a subcategory of $\\textbf{MAlg}(\\Sigma)$.\n\\end{proposition}\n\n\\begin{proof}\nClearly the identical homomorphism $Id_{\\mathcal{A}}$ is ground-preserving, so $Id_{\\mathcal{A}}$ is in\\\\ $Hom_{\\textbf{MAlg}_{G}(\\Sigma)}(\\mathcal{A}, \\mathcal{A})$.\n\nAnd if $\\varphi:\\mathcal{A}\\rightarrow \\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ are ground-preserving, then \n\\[\\psi\\circ\\varphi(G(\\mathcal{A}))=\\psi(\\varphi(G(\\mathcal{A}))\\subseteq\\psi(G(\\mathcal{B}))\\subseteq G(\\mathcal{C}),\\]\nso that $\\psi\\circ\\varphi$ is also ground-preserving and $\\textbf{MAlg}_{G}(\\Sigma)$ is in fact a subcategory of $\\textbf{MAlg}(\\Sigma)$.\n\\end{proof}\n\nNow, for ground-preserving homomorphisms $\\varphi, \\psi:\\mathcal{A}\\rightarrow\\mathcal{B}$, we define an equivalence relation \"${\\sim}$\" in $Hom_{\\textbf{MAlg}(\\Sigma)}(\\mathcal{A}, \\mathcal{B})$ by means of\n\\[\\varphi{\\sim}\\psi \\Leftrightarrow \\varphi|_{G(\\mathcal{A})}=\\psi|_{G(\\mathcal{A})}.\\]\nThat is in fact an equivalence relation since, given homomorphisms $\\varphi, \\psi, \\nu:\\mathcal{A}\\rightarrow\\mathcal{B}$ we have that:\n\\begin{enumerate}\n\\item $\\varphi{\\sim}\\varphi$, since $\\varphi|_{G(\\mathcal{A})}=\\varphi|_{G(\\mathcal{A})}$;\n\\item if $\\varphi{\\sim}\\psi$, then $\\varphi|_{G(\\mathcal{A})}=\\psi|_{G(\\mathcal{A})}$ and therefore $\\psi|_{G(\\mathcal{A})}=\\varphi|_{G(\\mathcal{A})}$, and so $\\psi{\\sim}\\varphi$;\n\\item if $\\varphi{\\sim}\\psi$ and $\\psi{\\sim}\\nu$, we have that $\\varphi|_{G(\\mathcal{A})}=\\psi|_{G(\\mathcal{A})}$ and $\\psi|_{G(\\mathcal{A})}=\\nu|_{G(\\mathcal{A})}$, and therefore $\\varphi|_{G(\\mathcal{A})}=\\nu|_{G(\\mathcal{A})}$ and $\\varphi{\\sim}\\nu$.\n\\end{enumerate}\nWe will denote the equivalence class by ${\\sim}$ with representative a homomorphism $\\varphi$ by $[\\varphi]$. Finally, we can define the morphisms from $\\mathcal{A}$ to $\\mathcal{B}$ in $\\textbf{MG}(\\Sigma)$\\label{MGSigma} to be\n\\[Hom_{\\textbf{MG}(\\Sigma)}(\\mathcal{A}, \\mathcal{B})=Hom_{\\textbf{MAlg}_{G}(\\Sigma)}(\\mathcal{A}, \\mathcal{B})\/{\\sim},\\]\nand their composition by, given $[\\varphi]\\in Hom_{\\textbf{MG}(\\Sigma)}(\\mathcal{A}, \\mathcal{B})$ and $[\\psi]\\in Hom_{\\textbf{MG}(\\Sigma)}(\\mathcal{B}, \\mathcal{C})$, \n\\[[\\psi]\\circ[\\varphi]=[\\psi\\circ\\varphi].\\]\nWe state that this definition of composition is well behaved: if $[\\varphi_{1}]=[\\varphi_{2}]$ and $[\\psi_{1}]=[\\psi_{2}]$ for $\\varphi_{1}, \\varphi_{2}:\\mathcal{A}\\rightarrow \\mathcal{B}$ and $\\psi_{1}, \\psi_{2}:\\mathcal{B}\\rightarrow\\mathcal{C}$ ground-preserving homomorphisms, we have that for an $a\\in G(\\mathcal{A})$ it is valid that\n\\[\\psi_{1}\\circ\\varphi_{1}(a)=\\psi_{1}(\\varphi_{1}(a))=\\psi_{1}(\\varphi_{2}(a));\\]\nsince $\\varphi_{2}$ is ground-preserving, we have that $\\varphi_{2}(a)\\in G(\\mathcal{B})$, implying\n\\[\\psi_{1}(\\varphi_{2}(a))=\\psi_{2}(\\varphi_{2}(a))=\\psi_{2}\\circ\\varphi_{2}(a),\\]\nand therefore $\\psi_{1}\\circ\\varphi_{1}|_{G(\\mathcal{A})}=\\psi_{2}\\circ\\varphi_{2}|_{G(\\mathcal{A})}$, meaning $[\\psi_{1}\\circ\\varphi_{1}]=[\\psi_{2}\\circ\\varphi_{2}]$.\n\n\\begin{theorem}\n$\\textbf{MG}(\\Sigma)$ is a category.\n\\end{theorem}\n\n\\begin{proof}\nBy the above remark $Hom_{\\textbf{MG}(\\Sigma)}$ has a well-defined composition: to see that this composition is also associative, take ground-preserving homomorphisms $\\varphi:\\mathcal{A}\\rightarrow \\mathcal{B}$, $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ and $\\theta:\\mathcal{C}\\rightarrow \\mathcal{D}$: then\n\\[[\\theta]\\circ([\\psi]\\circ[\\varphi])=[\\theta]\\circ[\\psi\\circ\\varphi]=[\\theta\\circ(\\psi\\circ\\varphi)]=[(\\theta\\circ\\psi)\\circ\\varphi]=[\\theta\\circ\\psi]\\circ[\\varphi]=([\\theta]\\circ[\\psi])\\circ[\\varphi].\\]\n\nFurthermore, $[Id_{\\mathcal{A}}]$, for $Id_{\\mathcal{A}}\\in Hom_{\\textbf{MAlg}(\\Sigma)}(\\mathcal{A}, \\mathcal{A})$ the usual identical homomorphism, is the identity on $Hom_{\\textbf{MG}(\\Sigma)}(\\mathcal{A}, \\mathcal{A})$, since for any other $\\Sigma-$multialgebra $\\mathcal{B}$ and ground-preserving homomorphisms $\\varphi:\\mathcal{A}\\rightarrow \\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{A}$ we have\n\\[[\\varphi]\\circ[Id_{\\mathcal{A}}]=[\\varphi\\circ Id_{\\mathcal{A}}]=[\\varphi]\\]\nand\n\\[[Id_{\\mathcal{A}}]\\circ[\\psi]=[Id_{\\mathcal{A}}\\circ\\psi]=[\\psi].\\]\n\\end{proof}\n\nOur goal is to prove that the functor associating a set $X$ to a $\\textbf{cdf}$-generated multialgebra over $X$ is the adjoint of a \"forgetful functor\" on $\\textbf{MG}(\\Sigma)$: the clear problem is that the morphisms on this category are not functions, but rather equivalence classes of functions.  \n\nSo we define the functor $U:\\textbf{MG}(\\Sigma)\\rightarrow \\textbf{Set}$, which we shall call forgetful, by $U\\mathcal{A}=G(\\mathcal{A})$ and, for a morphism $[\\varphi]:\\mathcal{A}\\rightarrow\\mathcal{B}$, by $U[\\varphi]=\\varphi|_{G(\\mathcal{A})}$, which is well-defined since $\\varphi$ is ground-preserving.\n\n$U$ is indeed a functor since: if $[Id_{\\mathcal{A}}]$ is the identity on $\\mathcal{A}$, $U[Id_{\\mathcal{A}}]=Id_{\\mathcal{A}}|_{G(\\mathcal{A})}$, which is the identical function on $G(\\mathcal{A})$; and given homomorphisms $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$, \n\\[U[\\psi\\circ\\varphi]=\\psi\\circ\\varphi|_{G(\\mathcal{A})}=\\psi|_{\\varphi(G(\\mathcal{A}))}\\circ\\varphi|_{G(\\mathcal{A})}=\\psi|_{G(\\mathcal{B})}\\circ\\varphi|_{G(\\mathcal{A})}=U[\\psi]\\circ U[\\varphi].\\]\n\n\\begin{theorem}\nThe functor \n\\[F_{\\kappa}:\\textbf{Set}\\rightarrow\\textbf{MG}(\\Sigma)\\]\nassociating to a set $X$ the $\\Sigma-$multialgebra $\\textbf{mF}(\\Sigma, X, \\kappa)$ and to a function $f:X\\rightarrow Y$ the morphism $[f_{C}]$, for any collection of choices $C$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\textbf{mF}(\\Sigma, Y, \\kappa)$, is a left-adjoint of the forgetful functor $U:\\textbf{MG}(\\Sigma)\\rightarrow\\textbf{Set}$.\n\\end{theorem}\n\n\\begin{proof}\nFirst of all $F$ is well defined over morphisms given that for any two collection of choices $C$ and $D$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\textbf{mF}(\\Sigma, Y, \\kappa)$ we have that $f_{C}|_{X}=f=f_{D}|_{X}$, so $[f_{C}]=[f_{D}]$.\n\nIt is a functor since:\n\\begin{enumerate}\n\\item given the identity between sets $Id_{X}:X\\rightarrow X$, the morphism $F_{\\kappa}Id_{X}$, when restricted to $X$, equals $Id_{\\textbf{mF}(\\Sigma, X, \\kappa)}$, and therefore $F_{\\kappa}Id_{X}=[Id_{\\textbf{mF}(\\Sigma, X, \\kappa)}]$;\n\\item given functions $f:X\\rightarrow Y$ and $g:Y\\rightarrow Z$, \n\\[F_{\\kappa}g\\circ F_{\\kappa}f=[g_{D}]\\circ[f_{C}]=[g_{D}\\circ f_{C}],\\]\nfor collections of choices $C$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\textbf{mF}(\\Sigma, Y, \\kappa)$ and $D$ from $\\textbf{mF}(\\Sigma, Y, \\kappa)$ to $\\textbf{mF}(\\Sigma, Z, \\kappa)$; for any collection of choices $E$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\textbf{mF}(\\Sigma, Z, \\kappa)$ we have that $(g\\circ f)_{E}$, when restricted to the ground, is the same as $g_{D}\\circ f_{C}$, that is, $g\\circ f$, and therefore \n\\[F_{\\kappa}g\\circ F_{\\kappa}f=[(g\\circ f)_{E}]=F_{\\kappa}(g\\circ f).\\]\n\\end{enumerate}\n\nWe define the bijection $\\Phi_{X,\\mathcal{A}}:Hom_{\\textbf{Set}}(X, U\\mathcal{A})\\rightarrow Hom_{\\textbf{MG}(\\Sigma)}(F_{\\kappa}X, \\mathcal{A})$ by, given a function $f:X\\rightarrow U\\mathcal{A}$ (where one should remember that $U\\mathcal{A}=G(\\mathcal{A})$), $\\Phi_{X, \\mathcal{A}}f=[f_{C}]$ for any collection of choices $C$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\mathcal{A}$: again, this is clearly well-defined since if $D$ is another collection of choices we have $f_{C}|_{X}=f=f_{D}|_{X}$.\n\nTo see that such $\\phi_{X \\mathcal{A}}$ is really a bijection is easy:\n\\begin{enumerate}\n\\item given two functions $f, g:X\\rightarrow G(\\mathcal{A})$, if $\\Phi_{X, \\mathcal{A}}f=\\Phi_{X, \\mathcal{A}}g$, then $[f_{C}]=[g_{D}]$, for any collections of choices $C$ and $D$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\mathcal{A}$, and therefore $f=f_{C}|_{X}=g_{D}|_{X}=g$;\n\\item for any homomorphism $\\varphi:\\textbf{mF}(\\Sigma, X, \\kappa)\\rightarrow\\mathcal{A}$ and any collection of choices $C$ from $\\textbf{mF}(\\Sigma, X,\\kappa)$ to $\\mathcal{A}$, once we define $f=\\varphi|_{X}$ we have that $[\\varphi]=[f_{C}]=F_{\\kappa}f$.\n\\end{enumerate}\n\nNow we must prove that for any sets $X$ and $Y$, any $\\Sigma-$multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, any function $f:Y\\rightarrow X$ and morphism $[\\varphi]:\\mathcal{A}\\rightarrow\\mathcal{B}$ we have that the following diagram commutes.\n\\[ \\begin{tikzcd}[row sep=5em, column sep=3em]\nHom_{\\textbf{Set}}(X, U\\mathcal{A}) \\arrow{r}{\\Phi_{X, \\mathcal{A}}} \\arrow{d}{Hom(f, U[\\varphi])}& Hom_{\\textbf{MG}(\\Sigma)}(F_{\\kappa}X, \\mathcal{A}) \\arrow{d}{Hom(F_{\\kappa}f, [\\varphi])} \\\\%\nHom_{\\textbf{Set}}(Y, U\\mathcal{B}) \\arrow{r}{\\Phi_{Y, \\mathcal{B}}}& Hom_{\\textbf{MG}(\\Sigma)}(F_{\\kappa}Y, \\mathcal{B})\n\\end{tikzcd}\n\\]\nGiven a map $g:X\\rightarrow U\\mathcal{A}$, $\\Phi_{X, \\mathcal{A}}g=[g_{C}]$ for some collection of choices $C$ from $\\textbf{mF}(\\Sigma, X, \\kappa)$ to $\\mathcal{A}$, and \n\\[Hom(F_{\\kappa}f, [\\varphi])\\Phi_{X,\\mathcal{A}}g=Hom(F_{\\kappa}f, [\\varphi])[g_{C}]=[\\varphi]\\circ[g_{C}]\\circ [f_{D}]=[\\varphi\\circ g_{C}\\circ f_{D}],\\]\nfor some collection of choices $D$ from $\\textbf{mF}(\\Sigma, Y, \\kappa)$ to $\\textbf{mF}(\\Sigma, X, \\kappa)$; on the other branch of the diagram, we have that $Hom(f, U[\\varphi])g=U[\\varphi]\\circ g\\circ f$ and therefore\n\\[\\Phi_{Y, \\mathcal{B}} (Hom(f, U[\\varphi])g)=\\Phi_{Y, \\mathcal{B}}(U[\\varphi]\\circ g\\circ f)=[(U[\\varphi]\\circ g\\circ f)_{E}]\\]\nfor some collection of choices $E$ from $\\textbf{mF}(\\Sigma, Y, \\kappa)$ to $\\mathcal{B}$. Now, is enough to prove that $\\varphi\\circ g_{C}\\circ f_{D}$ and $(U[\\varphi]\\circ g\\circ f)_{E}$ are equal over the ground of $\\textbf{mF}(\\Sigma, Y, \\kappa)$, that is, $Y$. This is easy, since for an element $y\\in Y$ we have that\n\\[\\varphi\\circ g_{C}\\circ f_{D}(y)=\\varphi\\circ g_{C}(f(y))=\\varphi(g(f(y))\\]\nand that\n\\[(U[\\varphi]\\circ g\\circ f)_{E}(y)=U[\\varphi]\\circ g\\circ f(y)=U[\\varphi]\\circ g(f(y))=U[\\varphi](g(f(y)))=\\varphi(g(f(y))).\\]\n\\end{proof}\n\nNotice that two multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ $\\textbf{cdf}$-generated respectively by sets $X$ and $Y$ of the same cardinality are isomorphic in $\\textbf{MG}(\\Sigma)$: for, given a bijective function $f:X\\rightarrow Y$ and collections of choices $C$ from $\\mathcal{A}$ to $\\mathcal{B}$ and $D$ from $\\mathcal{B}$ to $\\mathcal{A}$, we have that $[f^{-1}_{D}]\\circ[f_{C}]=[Id_{\\mathcal{A}}]$, since $f^{-1}\\circ f$ is the identity on $X$, and $[f_{C}]\\circ[f^{-1}_{D}]=[Id_{\\mathcal{B}}]$. \n\nIn general, this makes the classes of isomorphisms of multialgebras in $\\textbf{MG}(\\Sigma)$ too coarse, meaning that this category can not tell apart multialgebras that, intuitively, we consider very much distinct, such as $\\textbf{mF}(\\Sigma, X, \\kappa)$ and $\\textbf{mF}(\\Sigma, X, \\beta)$ for distinct cardinals $\\kappa$ and $\\beta$; of course, this is not to say that $\\textbf{MG}(\\Sigma)$ doesn't play a role in attempting to categorify weakly free multialgebras, just that this category is not completely successful at that task. Of course, before anything definitive is said about $\\textbf{MG}(\\Sigma)$'s success, or lack of it, as an ambient category for further studies on non-deterministic algebraization, a better understanding on the relationship between legal valuations, and grounds and collections of choices is necessary: after all, while the former are already quite well established as the central object of study of non-deterministic semantics, the latter have only recently been formally defined and, despite the results outlined here, are arguably not very well understood.\n\nNow, while still in the topic of categories, we may represent those found in this chapter through the following diagram.\n\n\\[ \\begin{tikzcd}\n\\textbf{MG}(\\Sigma)\\arrow[transform canvas={yshift=.9ex}]{ddrr}{U} & & \\textbf{MAlg}_{G}(\\Sigma)\\arrow[swap]{ll}{Q}\\arrow[hook]{rr}\\arrow[dashed]{dd} && \\textbf{MAlg}(\\Sigma) \\arrow{ddll}{\\mathcal{U}}\\\\\n&&&&\\\\\n&& \\textbf{Set}\\arrow[transform canvas={yshift=-.9ex}]{uull}{F_{\\kappa}}&&\n\\end{tikzcd}\n\\]\n\n\nClearly $\\textbf{MAlg}_{G}(\\Sigma)$ is a subcategory of $\\textbf{MAlg}(\\Sigma)$, sharing all of its objects, while the dashed arrow represents the forgetful functor from $\\textbf{MAlg}\n_{G}(\\Sigma)$ into $\\textbf{Set}$, of which we have nothing interesting to say. We can also define the functor $Q:\\textbf{MAlg}_{G}(\\Sigma)\\rightarrow\\textbf{MG}(\\Sigma)$, that is the identity on objects and takes ground-preserving homomorphism to their equivalence classes, to better understand the relation between $\\textbf{MAlg}_{G}(\\Sigma)$ and $\\textbf{MG}(\\Sigma)$: $Q$ is surjective on both objects and morphisms, but is only one-to-one on objects. Finally, from Theorem \\ref{no left adjoints}, $\\mathcal{U}$ does not have a left adjoint.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{2}\n\\chapter{Multialgebras as partially ordered algebras}\\label{Chapter3}\\label{Chapter 3}\n\nIt is a fundamental result (see \\cite{Oosten} for a proof) that there exists a (bijective) correspondence between sets and complete, atomic Boolean algebras (\\textit{CABA}'s\\index{CABA}): on one hand, a set is taken to its powerset (which is certainly a \\textit{CABA}), while on the other direction a \\textit{CABA} is taken to its set of atomic elements; accordingly, a map $f:X\\rightarrow Y$ is taken to the map $\\mathcal{P}(f):\\mathcal{P}(Y)\\rightarrow\\mathcal{P}(X)$ such that $\\mathcal{P}(f)(B)=\\{a\\in A\\ :\\  f(a)\\in B\\}$, for any $B\\subseteq Y$, and a continuous homomorphism of Boolean algebras $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ is taken to the function $A\\varphi$, from atoms of $\\mathcal{B}$ to atoms of $\\mathcal{A}$, such that $A\\varphi(b)=a$ is the unique atom of $\\mathcal{A}$ satisfying that $b\\leq \\varphi(a)$, for any atom $b$ of $\\mathcal{B}$.\n\nThese two assignments can be made into functors, giving rise to an equivalence of $\\textbf{Set}^{op}$ and $\\textbf{CABA}$\\label{CABA}, the category with \\textit{CABA}'s as objects and continuous homomorphisms of Boolean algebras as morphisms. This is part of a broader area of study, known by Stone dualities, which studies relationships between partially ordered sets and topological spaces (or rather, their respective categories), and was established by Stone (\\cite{Stone}) and his representation theorem, which states that every Boolean algebra is isomorphic to a field of sets, specifically the algebra of open and close subsets of its Stone space, a topological space where points are ultrafilters of the original Boolean algebra: an ultrafilter of a Boolean algebra $\\mathcal{A}$ is any non-empty subset $U$ of its universe such that \n\\begin{enumerate}\n\\item $a, b\\in U$ imply the existence of $c\\in U$ satisfying $c\\leq a\\wedge b$; \n\\item $a\\in U$ and $a\\leq b$ imply $b\\in U$; \n\\item and, for every element $a$ of $\\mathcal{A}$, either $a\\in U$ or $\\neg a\\in U$. \n\\end{enumerate}\nThe topology of the Stone space $S(\\mathcal{A})$ of $\\mathcal{A}$ is generated by the sets of ultrafilters sharing an element $a$ of $\\mathcal{A}$, that is, $\\{U\\in S(\\mathcal{A}): a\\in U\\}$. Of course, Stone' theorem corresponds to an equivalence between the category $\\textbf{BA}$ of Boolean algebras and that of Stone spaces, with continuous functions between them as morphisms.\n\nIn this search of dualities with categories of partially ordered sets, we focus on a more palpable equivalence, closely related to the one between $\\textbf{Set}^{op}$ and $\\textbf{CABA}$: we shift from $\\textbf{Set}$ to the category $\\textbf{MAlg}(\\Sigma)$ of multialgebras over a given signature $\\Sigma$ by adding further structure to $\\textbf{Set}$, that is, multioperations; correspondingly, we replace $\\textbf{CABA}$ by a category of \\textit{CABA}'s equipped with $\\Sigma$-operations compatible with their orders.\n\nWe make only small adjustments to these categories, in order to better suit our needs (although results for the aforementioned categories do exist): since we are most interested in non-partial multialgebras, we consequently exchange \\textit{CABA}'s by posets corresponding to powersets with the empty-set removed (that is, \\textit{CABA}'s without minimum elements, an apparently paradoxical concept we make precise further ahead). This way, a multialgebra with universe $A$ is taken to an algebra over the set of non-empty subsets of $A$, with order given by inclusion and operations given by ``accumulating'' the operations of the multialgebra: that is, an $n$-ary operator $\\sigma$, when evaluated on non-empty subsets $A_{i}$ of $A$, returns the non-empty set\n\\[\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times \\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n});\\]\nreciprocally, an ``almost'' Boolean' algebra, as we have loosely described, is taken to its set of atomic elements, transformed into a multialgebra by addition of multioperations returning, for $\\sigma$ of arity $n$ and atoms $a_{i}$, \n\\[\\{\\text{$a$ is an atom} : a\\leq \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}.\\]\n\nIn non-deterministic semantics (\\cite{AvronLev}), all these considerations involving a category equivalent to that of multialgebras offer an alternative: many logicians, mainly due to philosophical objections, are reluctant to use multialgebras in order to characterize a given logic; the equivalence we here present shows one can, if one chooses to, replace those semantics based on multialgebras by, arguably more complex, semantics involving both an algebra and an underlying order, which are however very classically behaved; of course, adding order-theoretic elements to algebraic semantics is nothing new, see \\cite{Raftery} for an example, and the already existing theory of order-algebraizable logics may offer new insights into non-deterministic semantics through the paradigm here presented.\n\n\n\n\n\n\n\n\\section{Complete, atomic and bottomless Boolean algebras}\\label{bottomless Boolean algebras}\n\nWe have discussed Boolean algebras in some depth in Section \\ref{Lattices, and Boolean... }, but we return to the topic now with a more order-theoretic mindset. A partially ordered set\\index{Partially ordered set}, or \\textit{poset}\\index{Poset}, is a pair $(A, \\leq)$, with $A$ a set and $\\leq$ a relation on $A$ (meaning it is a subset of $A\\times A$) such that:\n\\begin{enumerate}\n\\item $\\leq$ is reflexive, meaning that for any $a\\in A$, $(a, a)$ is in the relation, what we write as $a\\leq a$;\n\\item $\\leq$ is anti-symmetric, meaning that if $a\\leq b$ and $b\\leq a$, then $a=b$, for any $a, b\\in A$;\n\\item $\\leq$ is transitive, what means that if $a\\leq b$ and $b\\leq c$, then $a\\leq c$, for any $a, b, c\\in A$.\n\\end{enumerate}\nWhenever $a\\leq b$, we may also write $b\\geq a$; if $a\\leq b$ and we know $a\\neq b$, one can also write $a<b$, what may also be denoted by $b>a$.\n\n\\begin{definition}\nGiven a poset $(A, \\leq)$, an element $a\\in A$ is:\n\\begin{enumerate}\n\\item a maximum if, for all $b\\in A$, $b\\leq a$;\n\\item a minimum if, for all $b\\in A$, $a\\leq b$;\n\\item maximal if, for all $b\\in A$, $a\\leq b$ implies $a=b$;\n\\item minimal if, for all $b\\in A$, $b\\leq a$ implies $a=b$.\n\\end{enumerate}\n\nFurthermore, given a subset $S\\subseteq A$, we say $a\\in A$ is:\n\\begin{enumerate}\n\\item an upper bound for $S$ if, for any $s\\in S$, $s\\leq a$;\n\\item a lower bound for $S$ if, for any $s\\in S$, $a\\leq s$;\n\\item the supremum of $S$, when we write $a=\\sup S$, if it is the minimum of all upper bounds for $S$, meaning that:\n\\begin{enumerate}\n\\item for any $s\\in S$, $s\\leq a$;\n\\item if $b\\in A$ is such that, for any $s\\in S$, $s\\leq b$, then $a\\leq b$;\\footnote{Notice that there is indeed only one supremum of a set, as well as only one infimum: if $a$ and $b$ are both minimum upper bounds for $S$, the fact that $a$ is a minimum upper bound and that $b$ is an upper bound gives us $a\\leq b$; reciprocally, the fact $b$ is a minimum upper bound and $a$ is an upper bound implies $b\\leq a$, and so $a=b$. The same reasoning applies to infima.}\n\\end{enumerate}\n\\item the infimum of $S$, when we write $a=\\inf S$, if it is the maximum of all lower bounds for $S$, meaning that:\n\\begin{enumerate}\n\\item for any $s\\in S$, $a\\leq s$;\n\\item if $b\\in A$ is such that, for any $s\\in S$, $a\\leq s$, then $b\\leq a$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{definition}\n\nOf course, we may define maxima, minima, maximal and minimal elements for a subset $S\\subseteq A$ of $A$, by merely restricting the order of $(A, \\leq)$ to $S$.\n\nA Boolean algebra, to which we have already given on Section \\ref{Lattices, and Boolean... } a purely algebraic formulation, is a partially-ordered sets $(A, \\leq)$ such that: there are a maximum (denoted by $1$) and a minimum ($0$) elements, which we shall assume distinct; for every pair of elements $a, b\\in A$, the set $\\{a, b\\}$ has a supremum, denoted by $a\\vee b$, and an infimum, denoted by $a\\wedge b$; and every element $a$ has a complement $b$, which satisfies\n\\[b=\\inf\\{c\\in A:  \\sup\\{a, c\\}=1\\}\\]\nand\n\\[b=\\sup\\{c\\in A:  \\inf\\{a, c\\}=0\\}.\\]\n\nA poset $(A, \\leq)$ is said to be complete\\index{Poset, complete} if every $S\\subseteq A$ has a supremum and an infimum. \n\n\\begin{lemma}\n\\begin{enumerate} \n\\item Every Boolean algebra $(A, \\leq)$ is distributive, meaning \n\\[a\\vee (b\\wedge c)=(a\\vee b)\\wedge(a\\vee c)\\quad\\text{and}\\quad a\\wedge(b\\vee c)=(a\\wedge b)\\vee(a\\wedge c),\\]\nfor any $a, b, c\\in A$;\n\n\\item every complete Boolean algebra $(A, \\leq)$ is infinite distributive, meaning that for any $S\\cup\\{a\\}\\subseteq A$, \n\\[\\sup\\{\\inf\\{a, s\\}: s\\in S\\}=\\inf\\{a, \\sup S\\}\\quad\\text{and}\\quad\\inf\\{\\sup\\{a, s\\}: s\\in S\\}=\\sup\\{a, \\inf S\\}.\\]\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe only prove the first equation related to distributivity, given that the proof for the other is very similar. Denote $p=a\\vee (b\\wedge c)$, meaning \n\\[p=\\sup\\{a, \\inf\\{b, c\\}\\},\\]\nand $q=(a\\vee b)\\wedge (a\\vee c)$, that is \n\\[q=\\inf\\{\\sup\\{a, b\\}, \\sup\\{a, c\\}\\};\\]\nsince $\\inf\\{b, c\\}\\leq b$ and $\\inf\\{b, c\\}\\leq c$, $p=\\sup\\{a, \\inf\\{b, c\\}\\}\\leq \\sup\\{a, b\\}$ and, analogously, $p\\leq \\sup\\{a, c\\}$, implying $p$ is a lower bound for $\\{\\sup\\{a, b\\}, \\sup\\{a, c\\}\\}$. Since $q$ is the largest of these lower bounds, we get $p\\leq q$.\n\nReciprocally, $p=\\sup\\{a, \\inf\\{b, c\\}\\}$ implies $a\\leq p$ and $\\inf\\{b, c\\}\\leq p$: if $\\inf\\{b, c\\}=p$, this means $a\\leq \\inf\\{b, c\\}$ and therefore $a\\leq b$ and $a\\leq c$, meaning that $\\sup\\{a, b\\}=b$ and $\\sup\\{a, c\\}=c$ and therefore $q=\\inf\\{b, c\\}=p$; if, otherwise, $\\inf\\{b, c\\}<p$, $p$ is not a lower bound for $\\{b,c\\}$ and so either $b\\leq p$ or $c\\leq p$, implying either $\\sup\\{a, b\\}\\leq p$ or $\\sup\\{a, c\\}\\leq p$ and consequently $q=\\inf\\{\\sup\\{a, b\\}, \\sup\\{a, c\\}\\}\\leq p$, from what we get $p=q$.\n\nMore generally, we now prove one of the equalities of infinite-distributivity, the other being analogous. We know both \n\\[p=\\sup\\{\\inf\\{a, s\\}: s\\in S\\}\\quad\\text{and}\\quad q=\\inf\\{a, \\sup S\\}\\]\nexist, given by hypothesis our Boolean algebra is complete. Now, for any $s\\in S$, $s\\leq \\sup S$, and therefore $\\inf\\{a, s\\}\\leq \\inf\\{a, \\sup S\\}=q$, implying $q$ is an upper bound for $\\{\\inf\\{a, s\\}: s\\in S\\}$ and therefore $p\\leq q$, given $p$ is the least upper bound for that set.\n\nReciprocally, $q=\\inf\\{a, \\sup S\\}$ implies $q\\leq a$ and $q\\leq \\sup S$: if $q=\\sup S$, this means $\\sup S\\leq a$, and therefore $s\\leq a$ for every $s\\in S$, meaning $\\inf\\{a, s\\}=s$, for every $s\\in S$ and so $p=\\sup\\{s: s\\in S\\}=\\sup S$, giving us the desired equality; otherwise, $q<\\sup S$, and there must exist $s\\in S$ such that $q\\leq s$. Since we also have $q\\leq a$, $q\\leq\\inf\\{a, s\\}$, and since $p$ is an upper bound for $\\{\\inf\\{a, s\\}: s\\in S\\}$, $p\\geq q$, what finishes the proof.\n\\end{proof}\n\nAn element $a$ of a Boolean algebra is an atom\\index{Atom} if it is minimal in $A\\setminus\\{0\\}$, what means that if $b\\leq a$, then either $b=0$ or $b=a$; the set of atoms smaller than $a$ will be denoted by $A_{a}$; and a Boolean algebra is said to be atomic if, for every one of its elements $a$, $a=\\sup A_{a}$.\n\nComplete, atomic Boolean algebras are essentially powersets (\\cite{Oosten}): if one takes, for a Boolean algebra $\\mathcal{A}=(A, \\leq)$, the set $A_{1}$ of all of its atoms (what is valid given all atoms are smaller than $1$), one sees $\\mathcal{A}$ is equivalent to $\\mathcal{P}(A_{1})$, the powerset of $A_{1}$, where an element $a\\in A\\setminus\\{0\\}$ is taken, by this isomorphism, to $A_{a}$ (and $0$ to $\\emptyset$). Reciprocally, the complete, atomic Boolean algebra associated to a set $X$ is precisely $\\mathcal{P}(X)$. For more information, look at Theorem $2.4$ of \\cite{Oosten}.\n\nUseful to our intents and purposes are Boolean algebras that are, simultaneously, complete, atomic and bottomless, meaning they lack a minimum element: this may seem a contradiction, given we assumed Boolean algebras to have such elements, but this can be adequately formalized.\n\n\\begin{definition}\nGiven a partially ordered set $\\mathcal{A}=(A, \\leq_{\\mathcal{A}})$, we define\\label{A0} \n\\[\\mathcal{A}_{0}=(A\\cup\\{0\\}, \\leq_{\\mathcal{A}_{0}}),\\]\nwhere we assume $0\\notin A$, as the poset such that $a\\leq_{\\mathcal{A}_{0}} b$ if and only if:\n\n\\begin{enumerate} \n\\item either $a\\leq_{\\mathcal{A}} b$;\n\n\\item or $a=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\nThe non-empty partially ordered set $\\mathcal{A}$ is a complete, atomic and bottomless Boolean algebra\\index{Boolean algebra, Complete, atomic and bottomless} if, and only if, $\\mathcal{A}_{0}$ is a complete, atomic Boolean algebra.\n\\end{definition}\n\nNotice that, since $\\mathcal{P}(\\emptyset)$ only has $\\emptyset$ as element, for any complete, atomic and bottomless Boolean algebra $\\mathcal{A}$ we can not have $\\mathcal{A}_{0}$ equivalent to $\\mathcal{P}(\\emptyset)$, given $\\mathcal{A}$ has at least one element and therefore $\\mathcal{A}_{0}$ must have at least two. This means complete, atomic and bottomless Boolean algebras correspond to the powerset of non-empty sets with $\\emptyset$ removed.\n\n\\begin{proposition}\\label{A0 is poset}\nIf $\\mathcal{A}$ is a partially ordered set, so it is $\\mathcal{A}_{0}$.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item For any $a\\in A\\cup\\{0\\}$, we have that: if $a\\in A$, since $\\mathcal{A}$ is a partially ordered set, $a\\leq_{\\mathcal{A}}a$, and therefore $a\\leq_{\\mathcal{A}_{0}}a$; if $a=0$, we immediately have that $a\\leq_{\\mathcal{A}_{0}}a$, and therefore $\\leq_{\\mathcal{A}_{0}}$ is reflexive.\n\\item Suppose $a\\leq_{\\mathcal{A}_{0}}b$ and $b\\leq_{\\mathcal{A}_{0}}a$: if both $a$ and $b$ are in $A$, this means $a\\leq_{\\mathcal{A}}b$ and $b\\leq_{\\mathcal{A}}a$, and therefore $a=b$; if $a=0$, $b\\leq_{\\mathcal{A}_{0}}a$ implies that $b=0$, and therefore $a=b$; if $b=0$, $a\\leq_{\\mathcal{A}_{0}}b$ implies that $a=0$, and from that $a=b$; if $a=b=0$, there is nothing to be done, following from that that $\\leq_{\\mathcal{A}_{0}}$ is anti-symmetric.\n\\item Suppose $a\\leq_{\\mathcal{A}_{0}}b$ and $b\\leq_{\\mathcal{A}_{0}}c$: if $c=0$, $b\\leq_{\\mathcal{A}_{0}}c$ implies $b=0$, and then $a\\leq_{\\mathcal{A}_{0}}b$ implies $a=0$, and so $a\\leq_{\\mathcal{A}_{0}}c$; if $b=0$, $a\\leq_{\\mathcal{A}_{0}}b$ implies $a=0$, and so $a\\leq_{\\mathcal{A}_{0}}c$; obviously, if $a=0$ we promptly have that $a\\leq_{\\mathcal{A}_{0}}c$; so suppose $a, b, c\\in A$ as our last case, and then $a\\leq_{\\mathcal{A}_{0}}b$ implies $a\\leq_{\\mathcal{A}}b$ while $b\\leq_{\\mathcal{A}_{0}}c$ implies $b\\leq_{\\mathcal{A}}c$, and since $\\mathcal{A}$ is a partially ordered set we get that $a\\leq_{\\mathcal{A}}c$ and then $a\\leq_{\\mathcal{A}_{0}}c$, which proves $\\leq_{\\mathcal{A}_{0}}$ is transitive.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lemma}\\label{sup of low bounds is low bound}\nGiven a partially ordered set $(A, \\leq)$, for elements $a, b\\in A$ we have that the supremum of the lower bounds of $\\{a,b\\}$, if it exists, is itself a lower bound for $\\{a,b\\}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $S$ be he supremum of the lower bounds of $\\{a, b\\}$: by definition, this means that for any upper bound $d$ for the set $\\{c\\in A:  c\\leq a, c\\leq b\\}$ of lower bounds, $S\\leq d$; but, since $a$ and $b$ are such upper bounds, we find that $S\\leq a$ and $S\\leq b$.\n\\end{proof}\n\n\\begin{theorem}\\label{list}\nA partially ordered set $(A, \\leq)$ which satisfies all the following conditions is a complete, atomic and bottomless Boolean algebra.\n\n\\begin{enumerate} \n\\item It has a maximum element $1$.\n\\item All non-empty subsets $S$ of $A$ have a supremum.\n\\item For every $a\\in A$ different from $1$ there exists $b\\in A$, named the complement of $a$, such that \n\\[b=\\inf\\{c\\in A:  \\sup\\{a, c\\}=1\\}\\]\nand\n\\[b=\\sup\\{c\\in A:  \\text{$\\inf\\{a, c\\}$ does not exist}\\},\\]\nproperty we call being semi-complemented.\n\\item Denoting by $A_{a}$ the set of minimal elements smaller than $a$, $a=\\sup A_{a}$ (whenever a poset satisfies this condition, we will call it atomic).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nSuppose that $\\mathcal{A}=(A, \\leq_{\\mathcal{A}})$ is a partially ordered set satisfying the aforementioned conditions. Since $\\mathcal{A}$ is a partially ordered set, so is $\\mathcal{A}_{0}$ from Proposition \\ref{A0 is poset}. The maximum $1$ of $\\mathcal{A}$ remains a maximum in $\\mathcal{A}_{0}$, while $0$ becomes a minimum. For non-zero elements $a$ and $b$, the supremum of the two in $\\mathcal{A}_{0}$ remains the same as in $\\mathcal{A}$, while if $a=0$ or $b=0$ the supremum is simply the largest of the two. \n\nIf $a$ or $b$ equals $0$, the infimum is $0$, while if $a, b\\in A$ there are two cases to consider: if $\\inf\\{a, b\\}$ was defined in $\\mathcal{A}$, it remains the same in $\\mathcal{A}_{0}$; if the infimum was not defined in $\\mathcal{A}$, this means that there were no lower bounds for both $a$ and $b$, since otherwise we would have\n\\[\\inf\\{a,b\\}=\\sup\\{c\\in A:  c\\leq_{\\mathcal{A}}a\\quad\\text{and}\\quad c\\leq_{\\mathcal{A}}b\\}\\]\nby Lemma \\ref{sup of low bounds is low bound}, and therefore the infimum of both in $\\mathcal{A}_{0}$ is $0$. Every element $a\\in A\\setminus\\{1\\}$ already has a complement $b$ in $\\mathcal{A}$ such that $b=\\inf\\{c\\in A:  \\sup\\{a, c\\}=1\\}$ and\n\\[b=\\sup\\{c\\in A:  \\text{$\\inf\\{a, c\\}$ does not exist}\\};\\]\nof course the first equality keeps on holding in $\\mathcal{A}_{0}$, while the second becomes, remembering that the non-defined infima in $\\mathcal{A}$ become $0$ in $\\mathcal{A}_{0}$,\n\\[b=\\sup\\{c\\in A:  \\inf\\{a, c\\}=0\\};\\]\nthe complement of $1$ is clearly $0$ and vice-versa. This, of course, proves $\\mathcal{A}_{0}$ is a Boolean algebra.\n\nSince $\\mathcal{A}$ is closed under suprema of non-empty sets and $\\sup\\emptyset=0$ in $\\mathcal{A}_{0}$, it is clear that $\\mathcal{A}_{0}$ is closed under any suprema; it is clear that, for any set $S$, $\\inf S=(\\sup\\{s^{c}: s\\in S\\})^{c}$, where $a^{c}$ denotes the complement of $s$. Clearly $\\mathcal{A}_{0}$ remains atomic, since $\\mathcal{A}$ is atomic, what finishes the proof that the previous list of conditions imply $\\mathcal{A}$ is a complete, atomic and bottomless Boolean algebra.\n\\end{proof}\n\n\\begin{theorem}\nThe reciprocal of Theorem \\ref{list} holds, meaning that complete, atomic and bottomless Boolean algebras satisfy the list of conditions found in that theorem.\n\\end{theorem}\n\n\\begin{proof}\nGiven a partially ordered set $\\mathcal{A}$, suppose $\\mathcal{A}_{0}$ is a complete, atomic Boolean algebra.\n\n\\begin{enumerate} \n\\item The maximum $1$ of $\\mathcal{A}_{0}$ is still a maximum in $\\mathcal{A}$.\n\\item The supremum of any non-empty set in $\\mathcal{A}$ is just its supremum in $\\mathcal{A}_{0}$.\n\\item Given any element $a\\neq 1$, its complement $b$ in $\\mathcal{A}_{0}$ ends up being also its complement in $\\mathcal{A}$: clearly \n\\[b=\\inf\\{c\\in A:  \\sup\\{a,c\\}=1\\}.\\]\nNow, $\\inf\\{a,c\\}$ does not exist in $\\mathcal{A}$ if, and only if, $\\inf\\{a,c\\}=0$ in $\\mathcal{A}_{0}$: we already proved that if $\\inf\\{a,c\\}$ does not exist in $\\mathcal{A}$ then $\\inf\\{a,c\\}=0$ in $\\mathcal{A}_{0}$, remaining to show the reciprocal; if $\\inf\\{a, c\\}$ existed in $\\mathcal{A}$, since $\\inf\\{a, c\\}=0$ in $\\mathcal{A}_{0}$ and given the unicity of the infimum one would find that $\\inf\\{a, c\\}=0$ in $\\mathcal{A}$, contradicting the fact that $0$ is not in $\\mathcal{A}$. This way, we find that in $\\mathcal{A}$\n\\[b=\\sup\\{c\\in A:  \\text{$\\inf\\{a,c\\}$ does not exist}\\},\\]\nas necessary.\n\\item Clearly $\\mathcal{A}_{0}$ being atomic implies $\\mathcal{A}$ is atomic.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{proposition}\\label{sem-inf-dist}\nIf $(A, \\leq_{\\mathcal{A}})$ is a complete, atomic and bottomless Boolean algebra, for any $S\\subseteq A$, if \n\\[S^{a}=\\{s\\in S: \\text{$\\inf\\{a, s\\}$ exists}\\}\\neq\\emptyset,\\]\nthen\n\\[\\sup\\{\\inf\\{a, s\\}: s\\in S^{a}\\}=\\inf\\{a, \\sup S\\};\\]\nif $S^{a}=\\emptyset$, $\\inf\\{a, \\sup S\\}$ also does not exist.\n\\end{proposition}\n\n\\begin{proof}\nIf $S^{a}=\\emptyset$ this means that $\\inf\\{a,s\\}=0$ for every $s\\in S$ in $\\mathcal{A}_{0}$, and therefore\\\\ $\\inf\\{a, \\sup S\\}=0$, so that the same infimum no longer exists in $\\mathcal{A}$.\n\nIf $S^{a}\\neq\\emptyset$, all infima and suprema in $\\sup\\{\\inf\\{a,s\\}: s\\in S^{a}\\}$ and $\\inf\\{a, \\sup S\\}$ exist in $\\mathcal{A}$ and are therefore equal to their counterparts in $\\mathcal{A}_{0}$; given $\\sup\\{\\inf\\{a, s\\}: s\\in S^{a}\\}=\\sup\\{\\inf\\{a, s\\}: s\\in S\\}$ in $\\mathcal{A}_{0}$, since $s\\in S\\setminus S^{a}$ implies $\\inf\\{a, s\\}=0$, by the infinite-distributivity of $\\mathcal{A}_{0}$ one proves the desired result.\n\\end{proof}\n\nTo summarize the developments on this chapter so far, atomic and bottomless Bool\\-ean algebra are powersets (of non-empty sets) with their empty-set removed. The relevance of removing the empty-set from consideration is that the multialgebras we will work with are not partial, meaning the empty-set is never the result of an operation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A first attempt}\\label{A first attempt}\n\nFor simplicity, let us denote the set of non-empty subsets of $A$ by $\\mathcal{P}^{*}(A)$, that is, $\\mathcal{P}(A)\\setminus\\{\\emptyset\\}=\\mathcal{P}^{*}(A)$.\n\nConsider the categories $\\textbf{Alg}(\\Sigma)$\\label{AlgSigma} of $\\Sigma$-algebras, with homomorphisms between $\\Sigma$-algebras as morphisms, and $\\textbf{MAlg}(\\Sigma)$ of $\\Sigma$-multialgebras, with homomorphisms between $\\Sigma$-multialgebras as morphisms.\n\nConsider the transformation $\\mathcal{P}:\\textbf{MAlg}(\\Sigma)\\rightarrow\\textbf{Alg}(\\Sigma)$ taking:\n\\begin{enumerate}\n\\item a $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ to the $\\Sigma$-algebra \n\\[\\mathcal{P}(\\mathcal{A})=(\\mathcal{P}^{*}(A), \\{\\sigma_{\\mathcal{P}(\\mathcal{A})}\\}_{\\sigma\\in\\Sigma}),\\]\nwhere $\\mathcal{P}(A)$ is the powerset of $A$ and, for a $\\sigma\\in\\Sigma_{n}$ and nonempty $A_{1}, \\dotsc  , A_{n}\\subseteq A$,\n\\[\\sigma_{\\mathcal{P}(\\mathcal{A})}(A_{1}, \\dotsc  , A_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n});\\]\n\\item for $\\mathcal{A}$ and $\\mathcal{B}$ $\\Sigma$-multialgebras, a homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ to the function $\\mathcal{P}(\\varphi):\\mathcal{P}(A)\\rightarrow\\mathcal{P}(B)$ such that, for a $\\emptyset\\neq A'\\subseteq A$, \n\\[\\mathcal{P}(\\varphi)(A')=\\{\\varphi(a)\\in B\\ :\\  a\\in A'\\}.\\]\n\\end{enumerate}\n\nOne could hope that $\\mathcal{P}(\\varphi)$ is a homomorphism between $\\Sigma$-algebras, and perhaps that $\\mathcal{P}$ is a functor from $\\textbf{MAlg}(\\Sigma)$ to $\\textbf{Alg}(\\Sigma)$, but this does not happen: we have instead an inclusion.\n\n\\begin{lemma}\\label{homomorphisms lead to almost homomorphisms}\nFor $\\mathcal{A}$ and $\\mathcal{B}$ two $\\Sigma$-multialgebras and $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ a homomorphism, $\\mathcal{P}(\\varphi)$ satisfies\n\\[\\mathcal{P}(\\varphi)(\\sigma_{\\mathcal{P}(\\mathcal{A})}(A_{1}, \\dotsc  , A_{n}))\\subseteq \\sigma_{\\mathcal{P}(\\mathcal{B})}(\\mathcal{P}(\\varphi)(A_{1}), \\dotsc  , \\mathcal{P}(\\varphi)(A_{n}))\\]\nfor all $\\sigma\\in\\Sigma$ and nonempty $A_{1}, \\dotsc  , A_{n}\\subseteq A$; if $\\varphi$ is a full homomorphism, $\\mathcal{P}(\\varphi)$ is a homomorphism.\n\n\\end{lemma}\n\n\\begin{proof}\nGiven $\\sigma\\in \\Sigma_{n}$ and nonempty $A_{1}, \\dotsc  , A_{n}\\subseteq A$, we have that\n\\[\\sigma_{\\mathcal{P}(\\mathcal{B})}(\\mathcal{P}(\\varphi)(A_{1}), \\dotsc  , \\mathcal{P}(\\varphi)(A_{n}))=\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\mathcal{P}(\\varphi)(A_{1})\\times\\cdots\\times \\mathcal{P}(\\varphi)(A_{n})}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})=\\]\n\\[\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\{\\varphi(a)\\ :\\  a\\in A_{1}\\}\\times\\cdots\\times \\{\\varphi(a)\\ :\\  a\\in A_{n}\\}}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})),\\]\nwhich clearly contains\n\\[\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\{\\varphi(a)\\ :\\  a\\in \\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\{\\varphi(a)\\ :\\  a\\in \\sigma_{\\mathcal{P}(\\mathcal{A})}(A_{1}, \\dotsc  , A_{n})\\}=\\mathcal{P}(\\varphi)(\\sigma_{\\mathcal{P}(\\mathcal{A})}(A_{1}, \\dotsc  , A_{n})),\\]\nso that $\\mathcal{P}(\\varphi)$ satisfies the required property; if $\\varphi$ is a full homomorphism, $\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))=\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}$, and the inclusion between the equations above becomes an equality.\n\\end{proof}\n\nSo, let us restrict $\\mathcal{P}$ for a moment to the category $\\textbf{MAlg}_{=}(\\Sigma)$, of $\\Sigma$-multialgebras with only full homomorphisms between them as morphisms, and let us call this new transformation $\\mathcal{P}_{=}:\\textbf{MAlg}_{=}(\\Sigma)\\rightarrow \\textbf{Alg}(\\Sigma)$.\n\n\\begin{proposition}\n$\\mathcal{P}_{=}$ is, in fact, a functor.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Given a $\\Sigma$-multialgebra $\\mathcal{A}$, let $Id_{\\mathcal{A}}:\\mathcal{A}\\rightarrow\\mathcal{A}$ be the identity homomorphism. For any $\\emptyset\\neq A'\\subseteq A$, we have that \n\\[\\mathcal{P}_{=}(Id_{\\mathcal{A}})(A')=\\{Id_{\\mathcal{A}}(a)\\ :\\  a\\in A'\\}=\\{a\\ :\\  a\\in A'\\}=A',\\]\nand therefore $\\mathcal{P}_{=}(Id_{\\mathcal{A}})$ is the identity homomorphism on $\\mathcal{P}^{*}(\\mathcal{A})$.\n\n\\item Let $\\mathcal{A}$, $\\mathcal{B}$ and $\\mathcal{C}$ be $\\Sigma$-multialgebras and $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ be full homomorphisms between them. We have that, for every $\\emptyset\\neq A'\\subseteq A$, \n\\[\\mathcal{P}_{=}(\\psi)\\circ\\mathcal{P}_{=}(\\varphi)(A')=\\mathcal{P}_{=}(\\psi)(\\{\\varphi(a)\\in B\\ :\\  a\\in A'\\})=\\{\\psi(\\varphi(a))\\in C\\ :\\  a\\in A'\\}=\\]\n\\[\\mathcal{P}_{=}(\\psi\\circ\\varphi)(A'),\\]\nand so $\\mathcal{P}_{=}(\\psi)\\circ\\mathcal{P}_{=}(\\varphi)=\\mathcal{P}_{=}(\\psi\\circ\\varphi)$.\n\\end{enumerate}\n\\end{proof}\n\nWhat we actually want is an equivalence of categories, but we can be certain that $\\mathcal{P}_{=}$ will not provide this equivalence: unfortunately, $\\mathcal{P}_{=}$ is not injective on objects. Take the signature $\\Sigma_{s}$ with a single unary operator $s$, and consider the $\\Sigma$-multialgebras $\\mathcal{A}=(\\{0,1\\}, \\{s_{\\mathcal{A}}\\})$ and $\\mathcal{B}=(\\{0,1\\}, \\{s_{\\mathcal{B}}\\})$ such that: $s_{\\mathcal{A}}(0)=\\{1\\}$ and $s_{\\mathcal{A}}(1)=\\{1\\}$; and $s_{\\mathcal{B}}(0)=s_{\\mathcal{B}}(1)=\\{0,1\\}$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tikzcd}\n    0 \\arrow[r, \"s_{\\mathcal{A}}\"]  & 1 \\arrow[loop right, out=30, in=-30, distance=3em]{}{s_{\\mathcal{A}}}\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{A}$}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tikzcd}\n    0 \\arrow[rr, bend left=50, \"s_{\\mathcal{B}}\"]\\arrow[loop right, out=210, in=150, distance=3em]{}{s_{\\mathcal{B}}}  && 1 \\arrow[ll, bend left=50, \"s_{\\mathcal{B}}\"]\\arrow[loop right, out=30, in=-30, distance=3em]{}{s_{\\mathcal{B}}}\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{B}$}\n\\end{minipage}\n\\end{figure}\n\nClearly the two of then are not isomorphic, given that the result of an operation in $\\mathcal{A}$ always has cardinality $1$ and in $\\mathcal{B}$ always has cardinality $2$.\n\nHowever, we have that $s_{\\mathcal{P}_{=}(\\mathcal{A})}(\\{0\\})$, $s_{\\mathcal{P}_{=}(\\mathcal{A})}(\\{1\\})$, and $s_{\\mathcal{P}_{=}(\\mathcal{A})}(\\{0, 1\\})$ all equal $\\{1\\}$, while $s_{\\mathcal{P}_{=}(\\mathcal{B})}(\\{0\\})$, $s_{\\mathcal{P}_{=}(\\mathcal{B})}(\\{1\\})$, and $s_{\\mathcal{P}_{=}(\\mathcal{B})}(\\{0, 1\\})$ all equal $\\{0,1\\}$.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n  & \\{0,1\\} \\arrow[dr, \"s_{\\mathcal{P}_{=}(\\mathcal{A})}\"] &\\\\\n\\{0\\}\\arrow[rr, \"s_{\\mathcal{P}_{=}(\\mathcal{A})}\"] & & \\{1\\}\\arrow[loop right, out=30, in=-30, distance=3em]{}{s_{\\mathcal{P}_{=}(\\mathcal{A})}}\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{P}_{=}(\\mathcal{A})$}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n  & \\{0,1\\} \\arrow[loop, swap, \"s_{\\mathcal{P}_{=}(\\mathcal{B})}\"] &\\\\\n\\{0\\}\\arrow[ur, \"s_{\\mathcal{P}_{=}(\\mathcal{B})}\"] & & \\{1\\}\\arrow[ul, swap, \"s_{\\mathcal{P}_{=}(\\mathcal{B})}\"]\n  \\end{tikzcd}\n\\caption*{The $\\Sigma_{s}$-multialgebra $\\mathcal{P}_{=}(\\mathcal{B})$}\n\\end{figure}\n\nTaking the function $\\varphi:\\mathcal{P}^{*}(A)\\rightarrow\\mathcal{P}^{*}(B)$ such that $\\varphi(\\{0\\})=\\{0\\}$, $\\varphi(\\{1\\})=\\{0,1\\}$, and $\\varphi(\\{0,1\\})=\\{1\\}$, we see that it is a bijection and a homomorphism, and therefore $\\varphi:\\mathcal{P}_{=}(\\mathcal{A})\\rightarrow\\mathcal{P}_{=}(\\mathcal{B})$ is an isomorphism.\n\n\nCuriously, we can also quite easily show that $\\mathcal{P}_{=}$ does not have a right adjoint: if it did, that functor would itself posses a left adjoint, and therefore be would be continuous, preserving, among other constructions, terminal objects; and, while $\\textbf{Alg}(\\Sigma)$ has as terminal objects the $\\Sigma$-algebras with one element (where all operations return that very element), $\\textbf{MAlg}_{=}(\\Sigma)$ does not have them.\\footnote{Assuming, what is quite reasonable, that $\\Sigma$ is not empty.} In fact, for any $\\Sigma$-multialgebra $\\mathcal{A}$, by taking a cardinal $\\kappa$ greater than the cardinality of the universe of $\\mathcal{A}$ we see that there can be no full homomorphism from $\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ to $\\mathcal{A}$.\n\n\n\n\n\n\n\n\n\n\\section{An improvement}\n\nThe problem with our definition of $\\mathcal{P}_{=}$ is that it disregards the structure of the universe of $\\mathcal{P}(\\mathcal{A})$, which admits an order. So, we change our target category to reflect this structure, allowing the objects to carry with them such orderings.\n\n\\begin{definition}\nGiven a signature $\\Sigma$, a $(\\Sigma, \\leq)$-algebra\\index{Algebra, $(\\Sigma, \\leq)$-} $\\mathcal{A}$ is a triple \n\\[(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{A}})\\]\nsuch that:\n\\begin{enumerate}\n\\item $(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is a $\\Sigma$-algebra;\n\\item $(A,\\leq_{\\mathcal{A}})$ is a complete, atomic and bottomless Boolean algebra;\n\\item if $A_{a}$ is the set of minimal elements of $(A, \\leq_{\\mathcal{A}})$ (atoms) below $a$, for all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}$ we have that\n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sup\\{\\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n})\\ :\\  (b_{1}, \\dotsc  , b_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}\\}.\\]\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\nFor $\\mathcal{A}$ a $(\\Sigma, \\leq)$-algebra, $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}, b_{1}, \\dotsc  , b_{n}\\in A$ such that $a_{1}\\leq_{\\mathcal{A}} b_{1}, \\dotsc  , a_{n}\\leq_{\\mathcal{A}} b_{n}$,\n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\leq_{\\mathcal{A}} \\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n}).\\]\n\\end{proposition}\n\n\\begin{proof}\nSince, for every $i\\in\\{1, \\dotsc  , n\\}$, $a_{i}\\leq_{\\mathcal{A}}b_{i}$, we have that $A_{a_{i}}\\subseteq A_{b_{i}}$, and therefore $A_{a_{1}}\\times\\cdots\\times A_{a_{n}}\\subseteq A_{b_{1}}\\times\\cdots\\times A_{b_{n}}$; this way,\n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sup\\{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})\\ :\\  (c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}\\}\\leq_{\\mathcal{A}}\\]\n\\[\\sup\\{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})\\ :\\  (c_{1}, \\dotsc  , c_{n})\\in A_{b_{1}}\\times\\cdots\\times A_{b_{n}}\\}=\\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n}).\\]\n\\end{proof}\n\nFor a $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\sigma})$, we define $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$\\label{Functor PB} as the $(\\Sigma, \\leq)$-algebra \n\\[(\\mathcal{P}^{*}(A), \\{\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})})\\]\nsuch that $(\\mathcal{P}^{*}(A), \\{\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}\\}_{\\sigma\\in\\Sigma})$ is exactly $\\mathcal{P}(\\mathcal{A})$ (defined in the beginning of Section \\ref{A first attempt}) and, for nonempty subsets $A_{1}$ and $A_{2}$ of $A$, $A_{1}\\leq_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})} A_{2}$ if and only if $A_{1}\\subseteq A_{2}$. Since:\n\\begin{enumerate}\n\\item $\\mathcal{P}(\\mathcal{A})$ is a $\\Sigma$-algebra;\n\\item $(\\mathcal{P}^{*}(A), \\leq_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})})$ is a complete, atomic and bottomless Boolean algebra (since $\\mathcal{P}(A)$ is a complete, atomic Boolean algebra);\n\\item and, for $\\sigma\\in\\Sigma_{n}$ and $\\emptyset\\neq A_{1}, \\dotsc  , A_{n}\\subseteq A$, since the atoms of $A_{i}$ are exactly $A_{A_{i}}=\\{\\{a\\}\\ :\\  a\\in A_{i}\\}$,\n\\[\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(A_{1}, \\dotsc  , A_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\]\n\\[\\bigcup_{(\\{a_{1}\\}, \\dotsc  , \\{a_{n}\\})\\in A_{A_{1}}\\times\\cdots\\times A_{A_{n}}}\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(\\{a_{1}\\}, \\dotsc  , \\{a_{n}\\});\\]\n\\end{enumerate}\nwe truly have that $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$ is a $(\\Sigma, \\leq)$-algebra.\n\n\\begin{definition}\nGiven $(\\Sigma, \\leq)$-algebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{A}})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{B}})$, a function $\\varphi:A\\rightarrow B$ is said to be a $(\\Sigma, \\leq)$-homomorphism\\index{Homomorphism, $(\\sigma, \\leq)$-}, in which case we write $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$, when:\n\\begin{enumerate}\n\\item for all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$ we have that \n\\[\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\leq_{\\mathcal{B}}\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}));\\]\n\\item $\\varphi$ is continuous, that is, for every non-empty subset $A'\\subseteq A$, \n\\[\\varphi(\\sup A')=\\sup\\{\\varphi(a)\\ :\\  a\\in A'\\};\\]\n\\item $\\varphi$ maps minimal elements of $(A, \\leq_{\\mathcal{A}})$ to minimal elements of $(B, \\leq_{\\mathcal{B}})$.\n\\end{enumerate}\n\\end{definition}\n\nNotice that a $(\\Sigma, \\leq)-$homomorphism is essentially an \"almost $\\Sigma-$homomorphism\" which is also continuous and minimal-elements-preserving; notice too that a $(\\Sigma, \\leq)$-homomor\\-phism is order preserving: if $a\\leq_{\\mathcal{A}}b$, then $b=\\sup\\{a,b\\}$, and therefore $\\varphi(b)=\\sup\\{\\varphi(a), \\varphi(b)\\}$, meaning that $\\varphi(a)\\leq_{\\mathcal{B}}\\varphi(b)$.\n\n\n\n\n\\subsection{$\\mathcal{P}_{\\mathcal{B}}$ is a functor}\n\n\\begin{proposition}\nWhen we take as objects all $(\\Sigma, \\leq)$-algebras and as morphisms all the $(\\Sigma, \\leq)$-homomorphisms between them, the resulting object is a category, denoted by $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$\\label{AlgBSigma}.\n\\end{proposition}\n\n\\begin{proof}\nIt is enough to show the composition of $(\\Sigma, \\leq)$-homomorphisms returns a $(\\Sigma, \\leq)$-homo\\-morphism, that this composition is associative and has an identity element for every object in the category. Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$ and $\\mathcal{D}$ be $(\\Sigma, \\leq)-$algebras and $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$, $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ and $\\chi:\\mathcal{C}\\rightarrow\\mathcal{D}$ be $(\\Sigma, \\leq)$-homomorphisms.\n\n\\begin{enumerate}\n\\item\\begin{enumerate}\n\\item $\\psi\\circ\\varphi$ obviously is a function from $A$ to $C$, so let $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$: we have that, since $\\varphi$ is a $(\\Sigma, \\leq)$-homomorphism, $\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\leq_{\\mathcal{B}}\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))$, and since $\\psi$ is order-preserving\n\\[\\psi\\circ\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))=\\psi(\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})))\\leq_{\\mathcal{C}}\\psi(\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})));\\]\nand since $\\psi$ is a $(\\Sigma, \\leq)$-homomorphism,\n\\[\\psi(\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})))\\leq_{\\mathcal{C}}\\sigma_{\\mathcal{C}}(\\psi(\\varphi(a_{1})), \\dotsc  , \\psi(\\varphi(a_{n})))=\\sigma_{\\mathcal{C}}(\\psi\\circ\\varphi(a_{1}), \\dotsc  , \\psi\\circ\\varphi(a_{n})).\\]\n\n\\item Given a non-empty $A'\\subseteq A$, we have that $\\varphi(\\sup A')=\\sup\\{\\varphi(a)\\ :\\  a\\in A'\\}$ and, denoting $\\{\\varphi(a)\\ :\\  a\\in A'\\}$ by $B'$, we have that $\\psi(\\sup B')=\\sup\\{\\psi(b)\\ :\\  b\\in B'\\}$; since $\\sup B'=\\varphi(\\sup A')$, we obtain\n\\[\\psi\\circ\\varphi(\\sup A')=\\sup\\{\\psi(b)\\ :\\  b\\in B'\\}=\\sup\\{\\psi\\circ\\varphi(a)\\ :\\  a\\in A'\\},\\]\nwhich means that $\\psi\\circ\\varphi$ is continuous.\n\n\\item Finally, if $a\\in A$ is a minimal element, $\\varphi(a)\\in B$ is a minimal element, since $\\varphi$ preserves minimal elements, and for the same reason $\\psi\\circ\\varphi(a)=\\psi(\\varphi(a))\\in C$ remains a minimal element still, and from all of the above $\\psi\\circ\\varphi$ is a $(\\Sigma, \\leq)$-homomorphism.\n\\end{enumerate}\n\n\\item It is clear that $\\chi\\circ(\\psi\\circ\\varphi)=(\\chi\\circ\\psi)\\circ\\varphi$ since, as functions, both are the same.\n\n\\item Consider the identity function $Id_{\\mathcal{A}}:A\\rightarrow A$: it is a $(\\Sigma, \\leq)$-homomorphism, since for $n\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\\[Id_{\\mathcal{A}}(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))=\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sigma_{\\mathcal{A}}(Id_{\\mathcal{A}}(a_{1}), \\dotsc  , Id_{\\mathcal{A}}(a_{n}));\\]\nfor a non-empty $A'\\subseteq A$, \n\\[Id_{\\mathcal{A}}(\\sup A')=\\sup A'=\\sup\\{Id_{\\mathcal{A}}(a)\\ :\\  a\\in A'\\};\\]\nand for every minimal element $a$ of $\\mathcal{A}$, $Id_{\\mathcal{A}}(a)=a$ clearly remains a minimal element. Furthermore, $Id_{\\mathcal{A}}$ is clearly an identity for the composition, and is therefore an identity morphism.\n\\end{enumerate}\n\n\\end{proof}\n\nSo the transformation taking a $\\Sigma$-multialgebra $\\mathcal{A}$ to $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$ and a homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ to the $(\\Sigma, \\leq)$-homomorphism $\\mathcal{P}_{\\mathcal{B}}(\\varphi):\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})\\rightarrow\\mathcal{P}_{\\mathcal{B}}(\\mathcal{B})$ such that, for an $\\emptyset\\neq A'\\subseteq A$,\n\\[\\mathcal{P}_{\\mathcal{B}}(\\varphi)(A')=\\{\\varphi(a)\\in B\\ :\\  a\\in A'\\},\\]\nis a functor, of the form $\\mathcal{P}_{\\mathcal{B}}:\\textbf{MAlg}(\\Sigma)\\rightarrow \\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$. First we must show that $\\mathcal{P}_{\\mathcal{B}}(\\varphi)$ is, in fact, a $(\\Sigma, \\leq)$-homomorphism: for $\\sigma\\in\\Sigma_{n}$ and $\\emptyset\\neq A_{1}, \\dotsc  , A_{n}\\subseteq A$, as we saw in Lemma \\ref{homomorphisms lead to almost homomorphisms}, \n\\[\\mathcal{P}(\\varphi)(\\sigma_{\\mathcal{P}(\\mathcal{A})}(A_{1}, \\dotsc  , A_{n}))\\subseteq \\sigma_{\\mathcal{P}(\\mathcal{B})}(\\mathcal{P}(\\varphi)(A_{1}), \\dotsc  , \\mathcal{P}(\\varphi)(A_{n})),\\]\nand since $\\mathcal{P}(\\varphi)=\\mathcal{P}_{\\mathcal{B}}(\\varphi)$, we have that $\\mathcal{P}_{\\mathcal{B}}(\\varphi)$ satisfies the first condition for being a $(\\Sigma, \\leq)-$\\\\homomorphism; and, if $\\emptyset \\neq A''$ is a subset of $\\mathcal{P}(A)$, we have that\n\\[\\mathcal{P}_{\\mathcal{B}}(\\varphi)(\\sup A'')=\\{\\varphi(a)\\ :\\  a\\in \\sup A''\\}=\\{\\varphi(a)\\ :\\  a\\in \\bigcup_{A'\\in A''} A'\\}=\\]\n\\[\\bigcup_{A'\\in A''}\\{\\varphi(a)\\ :\\  a\\in A'\\}=\\bigcup_{A'\\in A''}\\mathcal{P}_{\\mathcal{B}}(\\varphi)(A')=\\sup \\{\\mathcal{P}_{\\mathcal{B}}(\\varphi)(A')\\ :\\  A'\\in A''\\},\\]\nwhat proves the satisfaction of the second condition; for the third condition, we remember that the minimal elements of $(\\mathcal{P}(A)\\setminus\\emptyset, \\subseteq)$ are the singletons, that is, sets of the form $\\{a\\}$ with $a\\in A$, and since $\\mathcal{P}_{\\mathcal{B}}(\\varphi)(\\{a\\})=\\{\\varphi(a)\\}$, $\\mathcal{P}_{\\mathcal{B}}(\\varphi)$ preserves minimal elements.\n\nNow we show that $\\mathcal{P}_{\\mathcal{B}}$ is, in fact, a functor.\n\\begin{enumerate}\n\\item Let $\\mathcal{A}$ be a $\\Sigma$-multialgebra and $Id_{\\mathcal{A}}$ be its identity homomorphism: we then have that, for every $\\emptyset\\neq A'\\subseteq A$, $\\mathcal{P}_{\\mathcal{B}}(Id_{\\mathcal{A}})(A')$ equals $\\{Id_{\\mathcal{A}}(a)\\ :\\  a\\in A'\\}=A'$, and therefore $\\mathcal{P}_{\\mathcal{B}}(Id_{\\mathcal{A}})$ is the identity $(\\Sigma, \\leq)$-homomorphism on $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$.\n\n\\item Let $\\mathcal{A}$, $\\mathcal{B}$ and $\\mathcal{C}$ be $\\Sigma$-multialgebras and $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ be homomorphisms: for every $\\emptyset\\neq A'\\subseteq A$, we have that\n\\[\\mathcal{P}_{\\mathcal{B}}(\\psi\\circ\\varphi)(A')=\\{\\psi\\circ\\varphi(a)\\ :\\  a\\in A'\\}=\\mathcal{P}_{\\mathcal{B}}(\\psi)(\\{\\varphi(a)\\ :\\  a\\in A'\\})=\\]\n\\[\\mathcal{P}_{\\mathcal{B}}(\\psi)(\\mathcal{P}_{\\mathcal{B}}(\\varphi)(A'))=\\mathcal{P}_{\\mathcal{B}}(\\psi)\\circ\\mathcal{P}_{\\mathcal{B}}(\\varphi)(A').\\]\n\\end{enumerate}\n\n\\begin{theorem}\n$\\mathcal{P}_{\\mathcal{B}}:\\textbf{MAlg}(\\Sigma)\\rightarrow \\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ is a functor.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\mathcal{P}_{\\mathcal{B}}$ is part of a monad}\n\nRemember that, for functors $F$ and $G$, both from the category $B$ to the category $C$, a natural transformation $\\eta$ from $F$ to $G$ (what we may denote by $\\eta:F\\rightarrow G$) is a  collection of morphisms $\\eta_{U}$ of $C$ indexed by the objects $U$ of $B$ such that: for objects $U$ and $V$ of $B$, $\\eta_{U}$ and $\\eta_{V}$ are morphisms from, respectively, $F(U)$ and $F(V)$ to, also respectively, $G(U)$ and $G(V)$ such that, for any morphism $f:U\\rightarrow V$,\n\\[\\eta_{V}\\circ F(f)=G(f)\\circ\\eta_{U}.\\]\n\nFor categories $B$, $C$ and $D$, and $\\eta:F\\rightarrow G$ a natural transformation between functors $F, G:B\\rightarrow C$, given a third functor $H:C\\rightarrow D$ we denote by $H\\eta$ the natural transformation between $H\\circ F$ and $H\\circ G$ given by, in an object $X$ of $C$, \n\\[(H\\eta)_{X}=H\\eta_{X};\\]\ngiven yet a fourth category $E$ and a functor $I:E\\rightarrow B$, we also define the natural transformation $\\eta I:F\\circ I\\rightarrow G\\circ I$ given by, for an object $Y$ of $E$,\n\\[(\\eta I)_{Y}=\\eta_{I(Y)}.\\]\n\n\\begin{definition}\nA monad\\index{Monad} (\\cite{CWM}), in a category $C$, is an endofunctor $P:C\\rightarrow C$ together with natural transformations $\\eta:1_{C}\\rightarrow P$ and $\\epsilon:P\\circ P\\rightarrow P$, where $1_{C}$ is the identity functor on $C$, such that the following conditions hold:\n\\begin{enumerate}\n\\item $\\epsilon\\circ P\\epsilon=\\epsilon\\circ \\epsilon P$;\n\\item $\\epsilon\\circ P\\eta=\\epsilon\\circ\\eta P=1_{P}$, where $1_{P}$ denotes the identity natural transformation on $P$.\n\\end{enumerate}\n\\end{definition}\n\nThe conditions required of a monad are equivalent to the commutativity of the following diagrams.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tikzcd}[row sep=large,column sep=large]\n   P^{3} \\arrow[rightarrow]{r}{P\\epsilon}\\arrow[rightarrow]{d}{\\epsilon P}  & P^{2} \\arrow[rightarrow]{d}{\\epsilon}\\\\\n   P^{2}\\arrow[rightarrow]{r}{\\epsilon} & P\n  \\end{tikzcd}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tikzcd}[row sep=large,column sep=large]\nP \\arrow[rightarrow]{r}{\\eta P}\\arrow[rightarrow]{d}{T\\eta}\\arrow[equal]{dr} & P^{2} \\arrow[rightarrow]{d}{\\epsilon}\\\\\nP^{2}\\arrow[rightarrow]{r}{\\epsilon} & P\n  \\end{tikzcd}\n\\end{minipage}\n\\end{figure}\n\nNow, we state that the functors $\\mathcal{P}_{\\mathcal{B}}$ and $\\mathcal{P}$ may be adequately adapted as to be part of a monad. Let $P:\\textbf{MAlg}(\\Sigma)\\rightarrow\\textbf{MAlg}(\\Sigma)$\\label{P} be the functor taking: a $\\Sigma$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ into the $\\Sigma$-multialgebra $P\\mathcal{A}=(\\mathcal{P}(A)\\setminus\\{\\emptyset\\}, \\{\\sigma_{P\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ such that, for non-empty subsets $A_{1}, \\dotsc  , A_{n}$ of $A$,\n\\[\\sigma_{P\\mathcal{A}}(A_{1}, \\dotsc  , A_{n})=\\{\\{a\\}\\ :\\  a\\in \\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\};\\]\nand, for $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, a homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ to the function $P\\varphi:\\mathcal{P}(A)\\setminus\\{\\emptyset\\}\\rightarrow\\mathcal{P}(B)\\setminus\\{\\emptyset\\}$ such that, for a non-empty $A'\\subseteq A$, \n\\[P\\varphi(A')=\\{\\varphi(a)\\in B\\ :\\  a\\in A'\\}.\\]\nNotice, first of all, that $P\\varphi$ is, indeed, a homomorphism from $P\\mathcal{A}$ to $P\\mathcal{B}$, whenever $\\varphi$ is a homomorphism from $\\mathcal{A}$ to $\\mathcal{B}$: for any non-empty subsets $A_{1}$ through $A_{n}$ of $A$ and a $\\sigma$ of arity $n$, one has that, for every $(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}$, $\\{\\varphi(a)\\ :\\  a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))$ since $\\varphi$ is a homomorphism, and therefore \n\\[\\{P\\varphi(A')\\ :\\  A'\\in \\sigma_{P\\mathcal{A}}(A_{1}, \\dotsc  , A_{n})\\}=\\{P\\varphi(\\{a\\})\\ :\\  a\\in\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\{\\{\\varphi(a)\\}\\ :\\  a\\in\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\{\\{\\varphi(a)\\} \\ :\\ a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq\\]\n\\[\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\{\\{b\\}\\ :\\  b\\in \\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))\\}=\\]\n\\[\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\{\\varphi(a_{1})\\ :\\  a_{1}\\in A_{1}\\}\\times\\cdots\\times\\{\\varphi(a_{n})\\ :\\  a_{n}\\in A_{n}\\}}\\{\\{b\\}\\ :\\  b\\in \\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})\\}=\\]\n\\[\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in P\\varphi(A_{1})\\times\\cdots\\times P\\varphi(A_{n})}\\{\\{b\\}\\ :\\  b\\in \\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})\\}=\\]\n\\[\\{\\{b\\}\\ :\\  b\\in\\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in P\\varphi(A_{1})\\times\\cdots\\times P\\varphi(A_{n})}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})\\}=\\sigma_{P\\mathcal{B}}(P\\varphi(A_{1}), \\dotsc  , P\\varphi(A_{n})).\\]\nIt is easy to see that the identity homomorphism of $\\mathcal{A}$ is mapped, by $P$, to the identity homomorphism of $P\\mathcal{A}$, and given homomorphisms $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$, we see that, for a non-empty subset $A'$ of $A$, one has\n\\[P(\\psi\\circ\\varphi)(A')=\\{\\psi\\circ\\varphi(a)\\ :\\  a\\in A'\\}=\\{\\psi(b)\\ :\\  b\\in P\\varphi(A')\\}=(P\\psi\\circ P\\varphi)(A'),\\]\nwhat finishes demonstrating that $P$ is indeed an endofunctor of $\\textbf{MAlg}(\\Sigma)$. \n\n$P$ is essentially equal to both $\\mathcal{P}$ and $\\mathcal{P}_{\\mathcal{B}}$, given that $\\Sigma$-algebras are multialgebras whose operations are always singletons and, disregarding its order, a $(\\Sigma, \\leq)$-algebra is nothing but a $\\Sigma$-algebra; the difference here is that, while the operations in $\\mathcal{P}(\\mathcal{A})$ return sets of elements of $\\mathcal{A}$, the operations in $P\\mathcal{A}$ returns sets of singletons of elements of $\\mathcal{A}$, precisely those elements obtained in the operation as performed in $\\mathcal{P}(\\mathcal{A})$.\n\nNow, to form a monad we need adequate natural transformations $\\eta:1_{\\textbf{MAlg}(\\Sigma)}\\rightarrow P$ and $\\epsilon:P\\circ P\\rightarrow P$, or what is equivalent, homomorphisms $\\eta_{\\mathcal{A}}:\\mathcal{A}\\rightarrow P\\mathcal{A}$ and $\\epsilon_{\\mathcal{A}}:PP\\mathcal{A}\\rightarrow P\\mathcal{A}$, for every $\\Sigma$-multialgebra $\\mathcal{A}$. And we simply take the obvious candidates $\\eta_{\\mathcal{A}}(a)=\\{a\\}$ and, for a non-empty set of non-empty subsets $\\{A_{i}\\}_{i\\in I}$ of $A$, $\\epsilon_{\\mathcal{A}}(\\{A_{i}\\}_{i\\in I})=\\bigcup_{i\\in I}A_{i}$.\n\n\\begin{proposition}\nFor any $\\Sigma$-multialgebra $\\mathcal{A}$, $\\eta_{\\mathcal{A}}$ and $\\epsilon_{\\mathcal{A}}$, as previously defined, are full homomorphisms.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Let $a_{1}, \\dotsc  , a_{n}$ be elements of $\\mathcal{A}$ and $\\sigma$ an $n$-ary operator: then \n\\[\\{\\eta_{\\mathcal{A}}(a)\\ :\\  a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\{\\{a\\}\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\sigma_{P\\mathcal{A}}(\\{a_{1}\\}, \\dotsc  , \\{a_{n}\\})=\\sigma_{P\\mathcal{A}}(\\eta_{\\mathcal{A}}(a_{1}), \\dotsc  , \\eta_{\\mathcal{A}}(a_{n})).\\]\n\n\\item Let $\\{A_{i}^{1}\\}_{i\\in I_{1}}$ through $\\{A_{i}^{n}\\}_{i\\in I_{n}}$ be elements of $PP\\mathcal{A}$ and $\\sigma$ be $n$-ary: we have that \n\\[\\bigcup_{(i_{1}, \\dotsc  , i_{n})\\in I_{1}\\times\\cdots\\times I_{n}}\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{i_{1}}^{1}\\times\\cdots\\times A_{i_{n}}^{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\bigcup_{i_{1}\\in I_{1}}A_{i_{1}}\\times\\cdots\\times\\bigcup_{i_{n}\\in I_{n}}A_{i_{n}}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\]\n\\[\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\epsilon_{\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}})\\times\\cdots\\times\\epsilon_{\\mathcal{A}}(\\{A_{i}^{n}\\}_{i\\in I_{n}})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}),\\]\nimplying that \n\\[\\sigma_{PP\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}}, \\dotsc  , \\{A_{i}^{n}\\}_{i\\in I_{n}})=\\{\\{A'\\}\\ :\\  A'\\in \\bigcup_{(i_{1}, \\dotsc  , i_{n})\\in I_{1}\\times\\cdots\\times I_{n}}\\sigma_{P\\mathcal{A}}(A_{i_{1}}^{1}, \\dotsc  , A_{i_{n}}^{n})\\}=\\]\n\\[\\{\\{\\{a\\}\\}\\ :\\  a\\in \\bigcup_{(i_{1}, \\dotsc  , i_{n})\\in I_{1}\\times\\cdots\\times I_{n}}\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{i_{1}}^{1}\\times\\cdots\\times A_{i_{n}}^{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\{\\{\\{a\\}\\}\\ :\\  a\\in \\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\epsilon_{\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}})\\times\\cdots\\times\\epsilon_{\\mathcal{A}}(\\{A_{i}^{n}\\}_{i\\in I_{n}})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\};\\]\nwith this, we obtain that \n\\[\\{\\epsilon_{\\mathcal{A}}(\\{A_{i}\\}_{i\\in I})\\ :\\  \\{A_{i}\\}_{i\\in I}\\in \\sigma_{PP\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}}, \\dotsc  , \\{A_{i}^{n}\\}_{i\\in I_{n}})\\}=\\]\n\\[\\{\\epsilon_{\\mathcal{A}}(\\{\\{a\\}\\})\\ :\\  a\\in \\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\epsilon_{\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}})\\times\\cdots\\times\\epsilon_{\\mathcal{A}}(\\{A_{i}^{n}\\}_{i\\in I_{n}})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\{\\{a\\}\\ :\\  a\\in \\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\epsilon_{\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}})\\times\\cdots\\times\\epsilon_{\\mathcal{A}}(\\{A_{i}^{n}\\}_{i\\in I_{n}})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\sigma_{P\\mathcal{A}}(\\epsilon_{\\mathcal{A}}(\\{A_{i}^{1}\\}_{i\\in I_{1}}), \\dotsc  ,\\epsilon_{\\mathcal{A}}(\\{A_{i}^{n}\\}_{i\\in I_{n}})).\\]\n\\end{enumerate}\n\\end{proof}\n\n\\begin{proposition}\nFor any $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, and homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$, \n\\begin{enumerate}\n\\item $P\\varphi\\circ \\eta_{\\mathcal{A}}=\\eta_{\\mathcal{B}}\\circ \\varphi$ and\n\\item $P\\varphi\\circ\\epsilon_{\\mathcal{A}}=\\epsilon_{\\mathcal{B}}\\circ PP\\varphi$,\n\\end{enumerate}\nmeaning $\\eta$ and $\\epsilon$ are natural transformations from, respectively, $Id_{\\textbf{MAlg}(\\Sigma)}$ to $P$, and $P\\circ P$ to $P$.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Let $a$ be an element of $\\mathcal{A}$: we have that $P\\varphi\\circ\\eta_{\\mathcal{A}}(a)=P\\varphi(\\eta_{\\mathcal{A}}(a))$, and since $\\eta_{\\mathcal{A}}(a)=\\{a\\}$, we have this equals $\\{\\varphi(a)\\}$; meanwhile, $\\eta_{\\mathcal{B}}\\circ\\varphi(a)=\\eta_{\\mathcal{A}}(\\varphi(a))=\\{\\varphi(a)\\}$, and as stated both expressions end up being equal.\n\n\\item Let $\\{A_{i}\\}_{i\\in I}$ be an element of $PP\\mathcal{A}$, meaning it is a non-empty set of non-empty subsets of $\\mathcal{A}$: $P\\varphi\\circ\\epsilon_{\\mathcal{A}}(\\{A_{i}\\}_{i\\in I})=P\\varphi(\\epsilon_{\\mathcal{A}}(\\{A_{i}\\}_{i\\in I}))$, and since $\\epsilon_{\\mathcal{A}}(\\{A_{i}\\}_{i\\in I})=\\bigcup_{i\\in I}A_{i}$, the whole expression simplifies to $\\{\\varphi(a)\\ :\\  a\\in \\bigcup_{i\\in I}A_{i}\\}$; at the other hand, \n\\[\\epsilon_{\\mathcal{B}}\\circ PP\\varphi(\\{A_{i}\\}_{i\\in I})=\\epsilon_{\\mathcal{B}}(\\{\\{\\varphi(a)\\ :\\  a\\in A_{i}\\}\\ :\\  i\\in I\\})=\\bigcup_{i\\in I}\\{\\varphi(a)\\ :\\  a\\in A_{i}\\}=\\]\n\\[\\{\\varphi(a)\\ :\\  a\\in \\bigcup_{i\\in I}A_{i}\\},\\]\ngiving us the desired equality.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theorem}\nThe triple formed by $P$, $\\eta$ and $\\epsilon$ constitutes a monad.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathcal{A}$ be a $\\Sigma$-multialgebra.\n\\begin{enumerate}\n\\item We must prove that $\\epsilon\\circ P\\epsilon=\\epsilon\\circ\\epsilon P$, what amounts to $\\epsilon_{\\mathcal{A}}\\circ P\\epsilon_{\\mathcal{A}}=\\epsilon_{\\mathcal{A}}\\circ \\epsilon_{P\\mathcal{A}}$, as homomorphisms from $P^{3}\\mathcal{A}$ to $P\\mathcal{A}$. So, let $\\{\\{A_{i}^{j}\\}_{i\\in I}\\}_{j\\in J}$ be an element of $P^{3}\\mathcal{A}$, where $I$ and $J$ are non-empty sets of indexes and all $A_{i}^{j}$ are non-empty subsets of $A$: \\[\\epsilon_{\\mathcal{A}}\\circ P\\epsilon_{\\mathcal{A}}(\\{\\{A_{i}^{j}\\}_{i\\in I}\\}_{j\\in J})=\\epsilon_{\\mathcal{A}}(\\{\\epsilon_{\\mathcal{A}}(\\{A_{i}^{j}\\ :\\  i\\in I\\})\\ :\\  j\\in J\\})=\\epsilon_{\\mathcal{A}}(\\{\\bigcup_{i\\in I}A_{i}^{j}\\ :\\  j\\in J\\})=\\]\n\\[\\bigcup_{j\\in J}\\bigcup_{i\\in I}A_{i}^{j},\\]\nwhile\n\\[\\epsilon_{\\mathcal{A}}\\circ\\epsilon_{P\\mathcal{A}}(\\{\\{A_{i}^{j}\\}_{i\\in I}\\}_{j\\in J})=\\epsilon_{\\mathcal{A}}(\\bigcup_{j\\in J}\\{A_{i}^{j}\\}_{i\\in I})=\\bigcup_{i\\in I}\\bigcup_{j\\in J}A_{i}^{j},\\]\nand it is clear both sets are the same.\n\n\\item It remains to be proven that $\\epsilon\\circ P\\eta=\\epsilon\\circ\\eta P=1_{P}$, meaning $\\epsilon_{\\mathcal{A}}\\circ \\eta_{P\\mathcal{A}}=\\epsilon_{\\mathcal{A}}\\circ P\\eta_{\\mathcal{A}}$, as homomorphisms from $P\\mathcal{A}$ to $P\\mathcal{A}$, and this equals the identity homomorphism on this multialgebra as well. So, we take a non-empty subset $A'$ of $A$, and we have that\n\\[\\epsilon_{\\mathcal{A}}\\circ\\eta_{P\\mathcal{A}}(A')=\\epsilon_{\\mathcal{A}}(\\{A'\\})=A',\\]\nwhile for the other other expression one derives\n\\[\\epsilon_{\\mathcal{A}}\\circ P\\eta_{\\mathcal{A}}(A')=\\epsilon_{\\mathcal{A}}(\\{\\eta_{\\mathcal{A}}(a)\\ :\\  a\\in A'\\})=\\epsilon_{\\mathcal{A}}(\\{\\{a\\}\\ :\\  a\\in A'\\})=\\bigcup_{a\\in A'}\\{a\\}=A',\\]\nwhat finishes the proof.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Multialgebras of atoms}\n\nGiven a $(\\Sigma, \\leq)$-algebra $\\mathcal{A}$, take the set $\\mathbb{A}((A, \\leq_{\\mathcal{A}}))$ of atoms of $(A,\\leq_{\\mathcal{A}})$, that is, the set of minimal elements of this poset.\n\nFor a $\\sigma\\in\\Sigma_{n}$ and atoms $a_{1}, \\dotsc  , a_{n}\\in \\mathbb{A}((A, \\leq_{\\mathcal{A}}))$, we define \n\\[\\sigma_{\\mathbb{A}(\\mathcal{A})}(a_{1}, \\dotsc  , a_{n})=\\{a\\in \\mathbb{A}((A, \\leq_{\\mathcal{A}}))\\ :\\  a\\leq_{\\mathcal{A}} \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=A_{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\neq\\emptyset.\\]\n\nThis way, $(\\mathbb{A}((A, \\leq_{\\mathcal{A}})), \\{\\sigma_{\\mathbb{A}(\\mathcal{A})}\\}_{\\sigma\\in\\Sigma})$ becomes a $\\Sigma$-multialgebra, that we will denote by $\\mathbb{A}(\\mathcal{A})$ and call the multialgebra of atoms\\index{Multialgebra of atoms}\\label{A} of $\\mathcal{A}$.\n\nGiven $(\\Sigma, \\leq)$-algebras $\\mathcal{A}$ and $\\mathcal{B}$ and a $(\\Sigma, \\leq)$-homomorphism $\\varphi:\\mathcal{A}\\rightarrow \\mathcal{B}$, we define $\\mathbb{A}(\\varphi):\\mathbb{A}((A, \\leq_{\\mathcal{A}}))\\rightarrow\\mathbb{A}((B, \\leq_{\\mathcal{B}}))$ as the restriction of $\\varphi$ to $\\mathbb{A}((A, \\leq_{\\mathcal{A}}))\\subseteq A$: it is well-defined since every $(\\Sigma, \\leq)$-homomorphism preserves minimal elements, that is, atoms.\n\nFor $\\sigma\\in\\Sigma_{n}$ and atoms $a_{1}, \\dotsc  , a_{n}\\in \\mathbb{A}((A, \\leq_{\\mathcal{A}}))$ we have that \n\\[\\{\\mathbb{A}(\\varphi)(a)\\ :\\  a\\in \\sigma_{\\mathbb{A}(\\mathcal{A})}(a_{1}, \\dotsc  , a_{n})\\}=\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathbb{A}(\\mathcal{A})}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\{\\varphi(a)\\in \\mathbb{A}((B, \\leq_{\\mathcal{B}}))\\ :\\  a\\leq_{\\mathcal{A}} \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\]\nand, since $a\\leq_{\\mathcal{A}} \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$ implies $\\varphi(a)\\leq_{\\mathcal{B}}\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))$ given $\\varphi$ is order preserving, which in turn implies $\\varphi(a)\\leq_{\\mathcal{B}}\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))$ since $\\varphi$ is an \"almost homomorphism\", we get that\n\\[\\{\\varphi(a)\\in \\mathbb{A}((B, \\leq_{\\mathcal{B}}))\\ :\\  a\\leq_{\\mathcal{A}} \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq\\{b\\in \\mathbb{A}((B, \\leq_{\\mathcal{B}}))\\ :\\  b\\leq_{\\mathcal{B}}\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})\\}=\\]\n\\[\\sigma_{\\mathbb{A}(\\mathcal{B})}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))=\\sigma_{\\mathbb{A}(\\mathcal{B})}(\\mathbb{A}(\\varphi)(a_{1}), \\dotsc  , \\mathbb{A}(\\varphi)(a_{n})),\\]\nwhat proves $\\mathbb{A}(\\varphi)$ is a homomorphism between $\\Sigma$-multialgebras, and we can write $\\mathbb{A}(\\varphi):\\mathbb{A}(\\mathcal{A})\\rightarrow\\mathbb{A}(\\mathcal{B})$.\n\nThe natural question is whether $\\mathbb{A}:\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)\\rightarrow\\textbf{MAlg}(\\Sigma)$ is a functor, to which the answer is yes.\n\n\\begin{enumerate}\n\\item Let $\\mathcal{A}$ be a $(\\Sigma, \\leq)$-algebra and $Id_{\\mathcal{A}}$ be its identity $(\\Sigma, \\leq)$-homomorphism: we then have that, for every $a\\in \\mathbb{A}((A, \\leq_{\\mathcal{A}}))$, $\\mathbb{A}(Id_{\\mathcal{A}})(a)=Id_{\\mathcal{A}}(a)=a$, and therefore $\\mathbb{A}(I_{\\mathcal{A}})$ is the identity $\\Sigma$-homomorphism of $\\mathbb{A}(\\mathcal{A})$.\n\\item Let $\\mathcal{A}$, $\\mathcal{B}$ and $\\mathcal{C}$ be $(\\Sigma, \\leq)$-algebras, and $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ and $\\psi:\\mathcal{B}\\rightarrow\\mathcal{C}$ be $(\\Sigma, \\leq)$-homomorphisms: for every $a\\in\\mathbb{A}((A, \\leq_{\\mathcal{A}}))$ we have that\n\\[\\mathbb{A}(\\psi\\circ\\varphi)(a)=\\psi\\circ\\varphi(a)=\\psi(\\varphi(a))=\\mathbb{A}(\\psi)(\\varphi(a))=\\mathbb{A}(\\psi)(\\mathbb{A}(\\varphi)(a))=(\\mathbb{A}(\\psi)\\circ\\mathbb{A}(\\varphi))(a).\\]\n\\end{enumerate}\n\n\n\\section{$\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ and $\\textbf{MAlg}(\\Sigma)$ are equivalent}\n\nNow, we aim to prove that $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ and $\\textbf{MAlg}(\\Sigma)$ are actually equivalent categories, the equivalence being given by the functors $\\mathcal{P}_{\\mathcal{B}}$ and $\\mathbb{A}$: so, let $Id_{\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)}$ and $Id_{\\textbf{MAlg}(\\Sigma)}$ be the identity functors on $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ and $\\textbf{MAlg}(\\Sigma)$ respectively, and we should prove that $\\mathcal{P}_{\\mathcal{B}}\\circ\\mathbb{A}$ and $Id_{\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)}$, and $\\mathbb{A}\\circ\\mathcal{P}_{\\mathcal{B}}$ and $Id_{\\textbf{MAlg}(\\Sigma)}$ are naturally isomorphic.\n\nEquivalently, $\\mathcal{P}_{\\mathcal{B}}$ and $\\mathbb{A}$ form an equivalence of categories if both are full and faithful and $\\mathbb{A}$ is a right adjoint of $\\mathcal{P}_{\\mathcal{B}}$, what we will prove instead.\n\n\n\\subsection{$\\mathcal{P}_{\\mathcal{B}}$ and $\\mathbb{A}$ are full and faithful}\n\nIt is easy to see $\\mathcal{P}_{\\mathcal{B}}$ is faithful: given $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ and homomorphisms $\\varphi, \\psi:\\mathcal{A}\\rightarrow\\mathcal{B}$, if $\\mathcal{P}_{\\mathcal{B}}(\\varphi)=\\mathcal{P}_{\\mathcal{B}}(\\psi)$, we have that for every $a\\in A$\n\\[\\{\\varphi(a)\\}=\\mathcal{P}_{\\mathcal{B}}(\\varphi)(\\{a\\})=\\mathcal{P}_{\\mathcal{B}}(\\psi)(\\{a\\})=\\{\\psi(a)\\},\\]\nand therefore $\\varphi=\\psi$.\n\n\\begin{proposition}\n$\\mathbb{A}$ is faithful.\n\\end{proposition}\n\n\\begin{proof}\nGiven $(\\Sigma, \\leq)$-algebras $\\mathcal{A}$ and $\\mathcal{B}$, and $(\\Sigma, \\leq)$-homomorphisms $\\varphi, \\psi:\\mathcal{A}\\rightarrow\\mathcal{B}$, suppose we have $\\mathbb{A}(\\varphi)=\\mathbb{A}(\\psi)$: then, for every $a\\in A$, we can write $a=\\sup A_{a}$, since $(A, \\leq_{\\mathcal{A}})$ is atomic.\n\nSince $\\varphi$ and $\\psi$ are continuous, $\\varphi(a)=\\sup\\{\\varphi(a')\\ :\\  a'\\in A_{a}\\}$ and $\\psi(a)=\\sup\\{\\psi(a')\\ :\\  a'\\in A_{a}\\}$; but, since $\\mathbb{A}(\\varphi)=\\mathbb{A}(\\psi)$, $\\varphi$ and $\\psi$ are the same when restricted to atoms, and therefore $\\{\\varphi(a')\\ :\\  a'\\in A_{a}\\}=\\sup\\{\\psi(a')\\ :\\  a'\\in A_{a}\\}$, what means $\\varphi(a)=\\psi(a)$ and, since $a$ is arbitrary, $\\varphi=\\psi$.\n\\end{proof}\n\nNow, given $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$ and a $(\\Sigma, \\leq)$-homomorphism $\\varphi:\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})\\rightarrow\\mathcal{P}_{\\mathcal{B}}(\\mathcal{B})$, to prove $\\mathcal{P}_{\\mathcal{B}}$ is also full we must find a homomorphism $\\psi:\\mathcal{A}\\rightarrow\\mathcal{B}$ such that $\\mathcal{P}_{\\mathcal{B}}(\\psi)=\\varphi$.\n\nFor every $a\\in A$, $\\{a\\}$ is an atom and, since $\\varphi$ preserves atoms, $\\varphi(\\{a\\})$ is an atom of $\\mathcal{P}_{\\mathcal{B}}(B)$, and therefore of the form $\\{b_{a}\\}$ for some $b_{a}\\in B$: we define $\\psi:\\mathcal{A}\\rightarrow\\mathcal{B}$ by $\\psi(a)=b_{a}$. First of all, we must show $\\psi$ is in fact a homomorphism, which is quite analogous to the proof of the same fact for $\\mathbb{A}(\\varphi)$: given $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\\[\\{\\psi(a)\\ :\\  a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\{b_{a}\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\sup\\{\\{b_{a}\\}\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\]\n\\[\\sup\\{\\varphi(\\{a\\})\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}=\\varphi(\\sup \\{\\{a\\}\\ :\\  a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\})=\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))=\\]\n\\[\\varphi(\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(\\{a_{1}\\}, \\dotsc  , \\{a_{n}\\}))\\subseteq \\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{B})}(\\varphi(\\{a_{1}\\}), \\dotsc  , \\varphi(\\{a_{n}\\}))= \\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{B})}(\\{b_{a_{1}}\\}, \\dotsc  , \\{b_{a_{n}}\\})=\\]\n\\[\\sigma_{\\mathcal{B}}(b_{a_{1}}, \\dotsc  , b_{a_{n}})=\\sigma_{\\mathcal{B}}(\\psi(a_{1}), \\dotsc  , \\psi(a_{n})).\\]\nNow, when we consider $\\mathcal{P}_{\\mathcal{B}}(\\psi)$, we see that, for every atom $\\{a\\}$ of $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$, $\\mathcal{P}_{\\mathcal{B}}(\\psi)(\\{a\\})=\\{b_{a}\\}=\\varphi(\\{a\\})$, and so the restrictions of $\\varphi$ and $\\mathcal{P}_{\\mathcal{B}}(\\psi)$ to atoms are the same, therefore $\\mathbb{A}(\\varphi)=\\mathbb{A}(\\mathcal{P}_{\\mathcal{B}}(\\psi))$. Since $\\mathbb{A}$ is faithful, we discover that $\\varphi=\\mathcal{P}_{\\mathcal{B}}(\\psi)$ and, as we stated before, $\\mathcal{P}_{\\mathcal{B}}$ is full.\n\nNow it remains to be shown that $\\mathbb{A}$ is also full: given $(\\Sigma, \\leq)$-algebras $\\mathcal{A}$ and $\\mathcal{B}$ and a homomorphism $\\varphi:\\mathbb{A}(\\mathcal{A})\\rightarrow\\mathbb{A}(\\mathcal{B})$; we then define $\\psi:\\mathcal{A}\\rightarrow\\mathcal{B}$ by\n\\[\\psi(a)=\\sup\\{\\varphi(c)\\ :\\  c\\in A_{a}\\}.\\]\nFirst of all, we must prove that $\\psi$ is a $(\\Sigma, \\leq)$-homomorphism, for which we shall need a few lemmas.\n\n\\begin{lemma}\\label{sup. of the union is sup. of sup.}\nGiven a family of subsets $\\{X_{i}\\}_{i\\in I}$ of a complete poset $(S, \\leq)$, suppose we define $x_{i}=\\sup X_{i}$ for $i\\in I$ and $X=\\bigcup_{i\\in I}X_{i}$: then $\\sup\\{x_{i}\\ :\\  i\\in I\\}=\\sup X$.\n\\end{lemma}\n\n\\begin{proof}\nWe define $a=\\sup\\{x_{i}\\ :\\  i\\in I\\}$ and $b=\\sup X$: first, we show that $a$ is an upper bound for $X$, so that $a\\geq b$. For every $x\\in X$, we have that, since $X=\\bigcup_{i\\in I}X_{i}$, there exists $j\\in I$ such that $x\\in X_{j}$, and therefore $x_{j}\\geq x$; since $a=\\sup\\{x_{i}\\ :\\  i\\in I\\}$, we have that $a\\geq x_{j}$, and by transitivity $a\\geq x$, and therefore $a$ is indeed an upper bound for $X$.\n\nNow we show that $b$ is an upper bound for $\\{x_{i}\\ :\\  i\\in I\\}$, and so $b\\geq a$ (and $a=b$). For every $i\\in I$, we have that $b$ is an upper bound for $X_{i}$  (since $X_{i}\\subseteq X$ and $b$ is an upper bound for $X$), and therefore $b\\geq x_{i}$, since $x_{i}$ is the smallest upper bound for $X_{i}$; it follows that $b$ is indeed an upper bound for $\\{x_{i}\\ :\\  i\\in I\\}$, what finishes the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{Union of atoms is atoms of sup.}\nIn a complete, atomic and bottomless Boolean algebra, given a non-empty set $C$, $\\bigcup_{c\\in C}A_{c}=A_{\\sup C}$.\n\\end{lemma}\n\n\\begin{proof}\nIf $d\\in A_{c}$ for a $c\\in C$, $d$ is an atom smaller than $c$; since $c\\leq_{\\mathcal{A}}\\sup C$, $d\\leq_{\\mathcal{A}}\\sup C$, and therefore belongs to $A_{\\sup C}$. So $\\bigcup_{c\\in C}A_{c}\\subseteq A_{\\sup C}$.\n\nReciprocally, suppose $d\\in A_{\\sup C}$: then $d$ is an atom which is smaller than $\\sup C$, and therefore $\\inf\\{d, \\sup C\\}=d$; it follows the subset $C^{d}\\subseteq C$ of $c\\in C$ such that $\\inf\\{d,c\\}$ exists is not empty, by the infinite-distributivity of $\\mathcal{A}$. But if $c\\in C^{d}$, $\\inf\\{d,c\\}$ exists, and since $d$ is an atom, we have that $d\\in A_{c}\\subseteq \\bigcup_{c\\in C}A_{c}$, and from that $\\bigcup_{c\\in C}A_{c}=A_{\\sup C}$.\n\\end{proof}\n\nSince $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$ equals the supremum of $\\{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})\\ :\\  (c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}\\}$, by Lemma \\ref{Union of atoms is atoms of sup.} we have that $A_{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}$ equals \n\\[\\bigcup_{(c_{1}, \\dotsc   ,c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})},\\]\nthat is, we have the following lemma.\n\n\\begin{lemma}\\label{atoms of operation are union of operations over atoms}\nFor a $\\sigma\\in\\Sigma_{n}$, and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\\[A_{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}=\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})}.\\]\n\\end{lemma}\n\n\\begin{theorem}\n$\\mathbb{A}$ is full.\n\\end{theorem}\n\n\\begin{proof}\nSo, first of all, we prove using Lemmas \\ref{sup. of the union is sup. of sup.}, \\ref{Union of atoms is atoms of sup.} and \\ref{atoms of operation are union of operations over atoms} that $\\psi$ is a $(\\Sigma, \\leq)$-homomor\\-phism.\n\\begin{enumerate}\n\\item First, it is clear that $\\psi$ maps atoms into atoms: if $a$ is an atom,\n\\[\\psi(a)=\\sup\\{\\varphi(c)\\ :\\  c\\in A_{a}\\}=\\sup\\{\\varphi(a)\\}=\\varphi(a),\\]\nwhich is an atom since $\\varphi$ is a map between $\\mathbb{A}(\\mathcal{A})$ and $\\mathbb{A}(\\mathcal{B})$.\n\n\n\\item $\\psi$ is continuous: for any non-empty set $C\\subseteq A$, we remember that $\\psi(c)=\\sup\\{\\varphi(d)\\ :\\  d\\in A_{c}\\}$, and from Lemma \\ref{sup. of the union is sup. of sup.} we get that\n\\[\\sup\\{\\psi(c)\\ :\\  c\\in C\\}=\\sup\\{\\sup\\{\\varphi(d)\\ :\\  d\\in A_{c}\\}\\ :\\  c\\in C\\}=\\sup \\bigcup_{c\\in C}\\{\\varphi(d)\\ :\\  d\\in A_{c}\\};\\]\nfrom Lemma \\ref{Union of atoms is atoms of sup.},\n\\[\\sup \\bigcup_{c\\in C}\\{\\varphi(d)\\ :\\  d\\in A_{c}\\}=\\sup\\{\\varphi(d)\\ :\\  d\\in A_{\\sup C}\\}=\\psi(\\sup C).\\]\n\n\\item \nSince $\\{\\varphi(a)\\ :\\  a\\in\\sigma_{\\mathbb{A}(\\mathcal{A})}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq \\sigma_{\\mathbb{A}(\\mathcal{B})}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))$, it follows from Lemma \\ref{atoms of operation are union of operations over atoms} that\n\\[\\psi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))=\\sup\\{\\varphi(c)\\ :\\  c\\in A_{\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\}=\\]\n\\[\\sup\\{\\varphi(c)\\ :\\  c\\in \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})}\\};\\]\nsince, for atoms $c_{1}, \\dotsc  , c_{n}$ of $\\mathcal{A}$, $\\sigma_{\\mathbb{A}(\\mathcal{A})}(c_{1}, \\dotsc  , c_{n})=A_{\\sigma_{\\mathcal{A}}(c_{1}, \\dotsc  , c_{n})}$, we have that the previous expression equals\n\\[\\sup\\{\\varphi(c)\\ :\\  c\\in \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}\\sigma_{\\mathbb{A}(\\mathcal{A})}(c_{1}, \\dotsc  , c_{n})\\}=\\]\n\\[\\sup\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}\\{\\varphi(c)\\ :\\  c\\in \\sigma_{\\mathbb{A}(\\mathcal{A})}(c_{1}, \\dotsc  , c_{n})\\},\\]\nwhich is less than or equal to \n\\[\\sup \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}\\sigma_{\\mathbb{A}(\\mathcal{B})}(\\varphi(c_{1}), \\dotsc  , \\varphi(c_{n}))\\]\nin $\\mathcal{B}$; since, for atoms $d_{1}, \\dotsc  , d_{n}$ of $\\mathcal{B}$, we have that $\\sigma_{\\mathbb{A}(\\mathcal{B})}(d_{1}, \\dotsc  , d_{n})=A_{\\sigma_{\\mathcal{B}}(d_{1}, \\dotsc  , d_{n})}$, this equals\n\\[\\sup \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{B}}(\\varphi(c_{1}), \\dotsc  , \\varphi(c_{n}))}=\\sup \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{B}}(\\psi(c_{1}), \\dotsc  , \\psi(c_{n}))};\\]\nnow, since $\\psi$ is continuous, $c_{i}\\leq_{\\mathcal{A}}a_{i}$, for every $i\\in \\{1, \\dotsc  , n\\}$, implies $\\psi(c_{i})\\leq_{\\mathcal{B}}\\psi(a_{i})$, and therefore $\\sigma_{\\mathcal{B}}(\\psi(c_{1}), \\dotsc  , \\psi(c_{n}))\\leq_{\\mathcal{B}}\\sigma_{\\mathcal{B}}(\\psi(a_{1}), \\dotsc  , \\psi(a_{n}))$ for $(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}$.\n\nIt follows that $\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{B}}(\\psi(c_{1}), \\dotsc  , \\psi(c_{n}))}$ is contained on $A_{\\sigma_{\\mathcal{B}}(\\psi(a_{1}), \\dotsc  , \\psi(a_{n}))}$, and therefore \n\\[\\sup \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}}A_{\\sigma_{\\mathcal{B}}(\\psi(c_{1}), \\dotsc  , \\psi(c_{n}))}\\leq_{\\mathcal{B}}\\sup A_{\\sigma_{\\mathcal{B}}(\\psi(a_{1}), \\dotsc  , \\psi(a_{n}))}=\\sigma_{\\mathcal{B}}(\\psi(a_{1}), \\dotsc  , \\psi(a_{n})).\\]\n\\end{enumerate}\n\nNow, for every atom $a$ of $\\mathcal{A}$, we have that $\\psi(a)=\\varphi(a)$, and therefore the restriction of $\\psi$ to atoms equals $\\varphi$, that is, $\\mathbb{A}(\\psi)=\\varphi$, and since $\\varphi$ was taken arbitrarily, $\\mathbb{A}$ is full.\n\\end{proof}\n\n\n\\subsection{$\\mathcal{P}_{\\mathcal{B}}$ and $\\mathbb{A}$ are adjoint}\n\nIt remains to be shown that $\\mathcal{P}_{\\mathcal{B}}$ and $\\mathbb{A}$ are adjoint: consider the bijections\n\\[\\Phi_{\\mathcal{B}, \\mathcal{A}}:Hom_{\\textbf{MAlg}(\\Sigma)}(\\mathbb{A}(\\mathcal{B}), \\mathcal{A})\\rightarrow Hom_{\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)}(\\mathcal{B}, \\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})),\\]\nfor $\\mathcal{A}$ a $\\Sigma$-multialgebra and $\\mathcal{B}$ a $(\\Sigma, \\leq)$-algebra, given by, for $\\varphi:\\mathbb{A}(\\mathcal{B})\\rightarrow\\mathcal{A}$ a homomorphism and $b$ an element of $\\mathcal{B}$,\n\\[\\Phi_{\\mathcal{B},\\mathcal{A}}(\\varphi)(b)=\\{\\varphi(c)\\ :\\  c\\in A_{b}\\}.\\]\nFirst of all, we must prove $\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)$ is truly a $(\\Sigma, \\leq)$-homomorphism.\n\n\\begin{enumerate}\n\\item If $b$ is an atom, $A_{b}=\\{b\\}$, and therefore \n\\[\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b)=\\{\\varphi(c)\\ :\\  c\\in A_{b}\\}=\\{\\varphi(b)\\},\\]\nwhich is a singleton and therefore an atom of $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$.\n\n\\item \nLet $B'$ be a non-empty subset of $\\mathcal{B}$, and we have that\n\\[\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(\\sup B')=\\{\\varphi(c)\\ :\\  c\\in A_{\\sup B'}\\}=\\{\\varphi(c)\\ :\\  c\\in \\bigcup_{b\\in B'}A_{b}\\}=\\]\n\\[\\bigcup_{b\\in B'}\\{\\varphi(c)\\ :\\  c\\in A_{b}\\}=\\bigcup_{b\\in B'}\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b)=\\sup\\{\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b)\\ :\\  b\\in B'\\},\\]\nsince $A_{\\sup B'}=\\bigcup_{b\\in B'}A_{d}$, from Lemma \\ref{Union of atoms is atoms of sup.}, and the supremum in $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$ is simply the union.\n\n\\item For $\\sigma\\in\\Sigma_{n}$ and $b_{1}, \\dotsc  , b_{n}$ elements of $\\mathcal{B}$, we have that, using Lemma \\ref{atoms of operation are union of operations over atoms},\n\\[\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n}))=\\{\\varphi(c)\\ :\\  c\\in A_{\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})}\\}=\\]\n\\[\\{\\varphi(c)\\ :\\  c\\in \\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{b_{1}}\\times\\cdots\\times A_{b_{n}}}A_{\\sigma_{\\mathcal{B}}(c_{1}, \\dotsc  , c_{n})}\\}=\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{b_{1}}\\times\\cdots\\times A_{b_{n}}}\\{\\varphi(c)\\ :\\  c\\in A_{\\sigma_{\\mathcal{B}}(c_{1}, \\dotsc  , c_{n})}\\},\\]\nand, since $c_{1}, \\dotsc  , c_{n}$ are atoms and $\\varphi$ is a homomorphism, this equals\n\\[\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{b_{1}}\\times\\cdots\\times A_{b_{n}}}\\{\\varphi(c)\\ :\\  c\\in \\sigma_{\\mathbb{A}(\\mathcal{B})}(c_{1}, \\dotsc  , c_{n})\\}\\subseteq\\]\n\\[\\bigcup_{(c_{1}, \\dotsc  , c_{n})\\in A_{b_{1}}\\times\\cdots\\times A_{b_{n}}}\\sigma_{\\mathcal{A}}(\\varphi(c_{1}), \\dotsc  , \\varphi(c_{n}))=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\{\\varphi(c)\\ :\\  c\\in A_{b_{1}}\\}\\times\\cdots\\times \\{\\varphi(c)\\ :\\  c\\in A_{b_{n}}\\}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\]\n\\[\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in \\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b_{1})\\times\\cdots\\times \\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b_{n})}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b_{1}), \\dotsc  , \\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b_{n})).\\]\n\\end{enumerate}\n\nSecond, the $\\Phi_{\\mathcal{B}, \\mathcal{A}}$ must be bijections between $Hom_{\\textbf{MAlg}(\\Sigma)}(\\mathbb{A}(\\mathcal{B}), \\mathcal{A})$ and\\\\ $Hom_{\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)}(\\mathcal{B}, \\mathcal{P}_{\\mathcal{B}}(\\mathcal{A}))$. They are certainly injective: if $\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)=\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\psi)$, for every atom $b$ we have that\n\\[\\{\\varphi(b)\\}=\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\varphi)(b)=\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\psi)(b)=\\{\\psi(b)\\},\\]\nand therefore $\\varphi=\\psi$.\n\nFor the surjectivity, take a $(\\Sigma, \\leq)$-homomorphism $\\varphi:\\mathcal{B}\\rightarrow \\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$: we then define $\\psi:\\mathbb{A}(\\mathcal{B})\\rightarrow \\mathcal{A}$ by $\\psi(b)=a$, for an atom $b$ in $\\mathcal{B}$ with $\\varphi(b)=\\{a\\}$. It is well-defined since a $(\\Sigma, \\leq)$-homomorphism takes atoms to atoms, and the atoms of $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$ are exactly the singletons.\n\nWe must show that $\\psi$ is truly a homomorphism: for $\\sigma\\in\\Sigma_{n}$ and atoms $b_{1}, \\dotsc  , b_{n}$ in $\\mathbb{A}(\\mathcal{B})$ such that $\\varphi(b_{i})=\\{a_{i}\\}$ for every $i\\in\\{1, \\dotsc  , n\\}$, we have that\n\\[\\varphi(\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n}))\\subseteq \\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(\\varphi(b_{1}), \\dotsc  , \\varphi(b_{n})),\\]\nsince $\\varphi$ is a $(\\Sigma, \\leq)$-homomorphism, and therefore\n\\[\\{\\psi(b)\\ :\\  b\\in\\sigma_{\\mathbb{A}(\\mathcal{B})}(b_{1}, \\dotsc  , b_{n})\\}=\\{\\psi(b)\\ :\\  b\\in A_{\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})}\\}=\\]\n\\[\\bigcup_{b\\in A_{\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})}}\\varphi(b)=\\sup\\{\\varphi(b)\\ :\\ b\\in A_{\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})}\\}=\\varphi(\\sup A_{\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n})})=\\varphi(\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n}))\\subseteq\\]\n\\[ \\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(\\varphi(b_{1}), \\dotsc  , \\varphi(b_{n}))=\\sigma_{\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})}(\\{a_{1}\\}, \\dotsc  , \\{a_{n}\\})=\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sigma_{\\mathcal{A}}(\\psi(b_{1}), \\dotsc  , \\psi(b_{n})).\\]\n\nNow $\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\psi)=\\varphi$ since, for any element $b$ in $\\mathcal{B}$, we have that\n\\[\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\psi)(b)=\\{\\psi(c)\\ :\\  c\\in A_{b}\\}=\\bigcup_{c\\in A_{b}}\\varphi(c)=\\sup\\{\\varphi(c)\\ :\\ c\\in A_{b}\\}=\\varphi(\\sup A_{b})=\\varphi(b),\\]\nand therefore the $\\Phi_{\\mathcal{B}, \\mathcal{A}}$ are, indeed, bijective.\n\nFinally, for $\\mathcal{A}$ and $\\mathcal{C}$ $\\Sigma$-multialgebras, $\\mathcal{B}$ and $\\mathcal{D}$ $(\\Sigma, \\leq)$-algebras, $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{C}$ a homomorphism and $\\psi:\\mathcal{D}\\rightarrow\\mathcal{B}$ a $(\\Sigma, \\leq)$-homomorphism, we must now only prove that the following diagram commutes.\n\n\\[\n\\begin{tikzcd}[row sep=5em, column sep=3em]\nHom_{\\textbf{MAlg}(\\Sigma)}(\\mathbb{A}(\\mathcal{B}), \\mathcal{A}) \\arrow{r}{\\Phi_{\\mathcal{B}, \\mathcal{A}}} \\arrow{d}{Hom(\\mathbb{A}(\\psi), \\varphi)}& Hom_{\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)}(\\mathcal{B}, \\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})) \\arrow{d}{Hom(\\psi, \\mathcal{P}_{\\mathcal{B}}(\\varphi))} \\\\%\nHom_{\\textbf{MAlg}(\\Sigma)}(\\mathbb{A}(\\mathcal{D}), \\mathcal{C}) \\arrow{r}{\\Phi_{\\mathcal{D}, \\mathcal{C}}}& Hom_{\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)}(\\mathcal{D}, \\mathcal{P}_{\\mathcal{B}}(\\mathcal{C}))\n\\end{tikzcd}\\]\n\nSo, we take a homomorphism $\\theta:\\mathbb{A}(\\mathcal{B})\\rightarrow\\mathcal{A}$ and an element $d$ of $\\mathcal{D}$. We have that\n\\[Hom(\\psi, \\mathcal{P}_{\\mathcal{B}}(\\varphi))\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\theta)=\\mathcal{P}_{\\mathcal{B}}(\\varphi)\\circ\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\theta)\\circ\\psi,\\]\nand therefore the top-right edges of the diagram give us \n\\[\\mathcal{P}_{\\mathcal{B}}(\\varphi)\\circ\\Phi_{\\mathcal{B}, \\mathcal{A}}(\\theta)\\circ\\psi(d)=\\mathcal{P}_{\\mathcal{B}}(\\varphi)(\\{\\theta(b)\\ :\\  b\\in A_{\\psi(d)}\\})=\\{\\varphi\\circ\\theta(b)\\ :\\  b\\in A_{\\psi(d)}\\}.\\]\n\nMeanwhile, the left-bottom edges give us \n\\[\\Phi_{\\mathcal{D}, \\mathcal{C}}(\\varphi\\circ \\theta\\circ \\mathbb{A}(\\psi))(d)=\\{\\varphi\\circ \\theta\\circ \\mathbb{A}(\\psi)(e)\\ :\\  e\\in A_{d}\\}=\\{\\varphi\\circ\\theta\\circ\\psi(e)\\ :\\  e\\in A_{d}\\}.\\]\n\nIf $d$ is an atom, the top-right edges give the singleton containing only $\\varphi\\circ\\theta\\circ\\psi(d)$, since in this case $A_{d}=\\{d\\}$ and, given that $\\psi$ preserves atoms, $A_{\\psi(d)}=\\{\\psi(d)\\}$; the left-bottom edges also give the singleton formed by $\\varphi\\circ\\theta\\circ\\psi(d)$, since again $A_{d}=\\{d\\}$. Since a $(\\Sigma, \\leq)$-homomorphism is determined by its action on atoms, we find that the left and right sides of the diagram are equal, and therefore the diagram commutes.\n\nAs observed before, this proves $\\mathbb{A}$ and $\\mathcal{P}_{\\mathcal{B}}$ are adjoint and, therefore, that $\\textbf{MAlg}(\\Sigma)$ and $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ are equivalent. We then have, more or less, the following functors and categories to consider.\n\n\\[ \\begin{tikzcd}\n\\textbf{MAlg}_{=}(\\Sigma)\\arrow[hook]{rr}\\arrow{dd}{\\mathcal{P}_{=}} && \\textbf{MAlg}(\\Sigma)\\arrow[looseness=3, out=30, in=330, distance=4em, \"P\"]\\arrow{ddll}{\\mathcal{P}}\\arrow[transform canvas={xshift=.7ex}]{dd}{\\mathcal{P}_{\\mathcal{B}}}\\\\\n&&\\\\\n\\textbf{Alg}(\\Sigma) && \\textbf{Alg}_{\\mathcal{B}}(\\Sigma)\\arrow[transform canvas={xshift=-.7ex}]{uu}{\\mathbb{A}}\\arrow{ll}{\\mathbb{U}}\n\\end{tikzcd}\n\\]\n\nWhile $\\mathcal{P}$ and $\\mathcal{P}_{=}$ are our failed attempts, $\\mathcal{P}_{\\mathcal{B}}$ is the successful one, and $P$ its presentation as an endofunctor of $\\textbf{MAlg}(\\Sigma)$ and part of a monad. Here, $\\mathbb{U}$ is the forgetful functor from $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ into $\\textbf{Alg}(\\Sigma)$, that ignores the order of a $(\\Sigma, \\leq)$-algebra, leaving us simply with a $\\Sigma$-algebra.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Some related results}\n\nThe result that $\\textbf{MAlg}(\\Sigma)$ and $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ are equivalent has a few consequences, and related results, we would like to stress. We start by mentioning a consequence, and move to a few related results to which, although we do not offer a proof, providing a demonstration appears straightforward.\n\nSo, take the empty signature, with no operators at all: in that case, given all multialgebras have non-empty  universes, $\\textbf{MAlg}(\\Sigma)$ becomes (or, rather, is equivalent to) the category of non-empty sets $\\textbf{Set}^{*}$\\label{Set*} (a multialgebra corresponding to its universe), with functions between them as morphisms; this is because, once one disregards the conditions demanded of a homomorphism, what remains is purely a function between the universes of the multialgebras. \n\nMeanwhile, $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ becomes the category with complete, atomic and bottomless Boolean algebras as objects (given we simply drop the operations from a $(\\Sigma, \\leq)$-algebra), with continuous, atoms-preserving functions between them as morphisms, category we will call $\\textbf{CABo}$\\label{CABo} (standing for ``complete, atomic and bottomless''). More explicitly, the equivalence is given by the pair of functors: \n\\[F:\\textbf{Set}^{*}\\rightarrow \\textbf{CABo},\\]\nwhich takes a non-empty set $X$ to the complete, atomic and bottomless Boolean algebra $\\mathcal{P}^{*}(X)$ of its non-empty sets, and a function $f:X\\rightarrow Y$ to the continuous, atoms-preserving function $Ff:\\mathcal{P}^{*}(X)\\rightarrow\\mathcal{P}^{*}(Y)$ such that, for a non-empty $A\\subseteq X$, $Ff(A)=\\{f(x)\\in Y : x\\in A\\}$; and\n\\[G:\\textbf{CABo}\\rightarrow\\textbf{Set}^{*},\\]\nwhich takes a complete, atomic and bottomless Boolean algebra to its set of minimal elements (atoms), and a continuous, atoms-preserving function $\\varphi:(A, \\leq_{\\mathcal{A}})\\rightarrow(B, \\leq_{\\mathcal{B}})$ to the function $G\\varphi$ which maps an atom $a$ in $A$ to the, still minimal, element $\\varphi(a)$ of $B$.\n\nNotice this is very closely related to the equivalence between $\\textbf{CABA}$ and $\\textbf{Set}^{op}$: the morphisms on the former are merely continuous functions, so the only extra requirement to the morphisms of $\\textbf{CABo}$ we are making is that they should preserve atoms; this, of course, allows one to exchange the opposite category of $\\textbf{Set}$ by $\\textbf{Set}$ itself (or rather $\\textbf{Set}^{*}$); the extra requirement that the maps should preserve atoms also allows one to prove a similar equivalence between an enriched $\\textbf{CABA}$ and $\\textbf{Set}$.\n\n\\subsection{Partial multialgebras}\n\nNow, a generalization of our result is to partial multialgebras\\index{Partial multialgebras}: given a signature $\\Sigma$, a partial $\\Sigma$-multialgebra is a pair $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ (with $A$ possibly empty) such that, if $\\sigma\\in\\Sigma_{n}$, $\\sigma_{\\mathcal{A}}$ is a function of the form\n\\[\\sigma_{\\mathcal{A}}:A^{n}\\rightarrow\\mathcal{P}(A)\\]\n(no longer $\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ as in the case of multialgebras); that is, a partial multialgebra is a multialgebra where operations may return the empty-set. Given partial $\\Sigma$-multialgebras $\\mathcal{A}$ and $\\mathcal{B}$, a homomorphism between them is a function $\\varphi:A\\rightarrow B$ such that, as is the case for homomorphisms for multialgebras,\n\\[\\{\\varphi(a): a\\in\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\}\\subseteq \\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n})),\\]\nfor all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$, where one or both of these sets may be empty. The class of all partial $\\Sigma$-multialgebras, with these homomorphisms between them as morphisms, becomes a category, which we shall denote by $\\textbf{PMAlg}(\\Sigma)$\\label{PMAlgSigma}.\n\nCorrespondingly, we modify the category $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$: take, as objects of $\\textbf{Alg}_{CABA}(\\Sigma)$\\label{AlgCABASigma}, triples $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{A}})$ such that:\n\\begin{enumerate}\n\\item $(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is a $\\Sigma$-algebra;\n\\item $(A, \\leq_{\\mathcal{A}})$ is a complete, atomic Boolean algebra (no longer a bottomless Boolean algebra);\n\\item for $A_{a}$ the set of atoms smaller than $a$ (equal to $\\emptyset$ if $a=0$), $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$, \n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\sup\\{\\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n}) : (b_{1}, \\dotsc  , b_{n})\\in A_{a_{1}}\\times\\cdots\\times A_{a_{n}}\\}\\]\n(where, now, the set of which we take the supremum may be empty).\n\\end{enumerate}\nThe morphisms in $\\textbf{Alg}_{CABA}(\\Sigma)$, between two of its objects $\\mathcal{A}$ and $\\mathcal{B}$, will be functions $\\varphi:A\\rightarrow B$ satisfying \n\\begin{enumerate}\n\\item for $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$, $\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\leq_{\\mathcal{B}}\\sigma_{\\mathcal{B}}(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}))$;\n\\item $\\varphi$ is continuous, \\textit{id est}, for any $S\\subseteq A$ (possibly empty), $\\varphi(\\sup S)=\\sup\\{\\varphi(a) : a\\in S\\}$;\n\\item $\\varphi$ maps atoms of $\\mathcal{A}$ to atoms of $\\mathcal{B}$.\n\\end{enumerate}\n\nNotice that the category $\\textbf{Alg}_{CABA}(\\Sigma)$ is really closely related to $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$: what we changed is that the underlying objects are not complete, atomic and bottomless Boolean algebras anymore, but rather complete, atomic Boolean algebras, while the morphisms are now required to map $0$ to $0$ (that is, be continuous on the empty set). And, with the categories $\\textbf{PMAlg}(\\Sigma)$ and $\\textbf{Alg}_{CABA}(\\Sigma)$ at hand, it becomes easy to show both of them are equivalent.\n\nIn one direction, the equivalence is given by the functor\\\\$F:\\textbf{PMAlg}(\\Sigma)\\rightarrow \\textbf{Alg}_{CABA}(\\Sigma)$ which takes: a partial multialgebra $\\mathcal{A}$ with universe $A$ to the powerset of $A$, equipped with operations given by\n\\[\\sigma_{F\\mathcal{A}}(A_{1}, \\dotsc  , A_{n})=\\bigcup_{(a_{1}, \\dotsc  , a_{n})\\in A_{1}\\times\\cdots\\times A_{n}}\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\]\nfor possibly empty $A_{1}, \\dotsc  , A_{n}\\subseteq A$; and a homomorphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ between partial $\\Sigma$-multialgebras to the function given by, for a possibly empty $A'\\subseteq A$, $F\\varphi(A')=\\{\\varphi(a)\\in B : a\\in A'\\}$.\n\nIn the other direction, the equivalence is given by $G:\\textbf{Alg}_{CABA}(\\Sigma)\\rightarrow \\textbf{PMAlg}(\\Sigma)$, taking: an object of $\\textbf{Alg}_{CABA}(\\Sigma)$ to its set of atomic elements, equipped with multioperations such that $\\sigma_{G\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$, for atoms $a_{1}$ through $a_{n}$, is the set of atoms smaller than $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})$ (now possibly empty, in the case the result of the operation is $0$); and a morphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ in $\\textbf{Alg}_{CABA}(\\Sigma)$ to its restriction to the atoms of $\\mathcal{A}$.\n\nThe equivalence between $\\textbf{PMAlg}(\\Sigma)$ and $\\textbf{Alg}_{CABA}(\\Sigma)$ is obtained directly from the equivalence between $\\textbf{MAlg}(\\Sigma)$ and $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ by merely allowing operations to return the empty-set, and correcting whatever definitions that require a set to be non-empty accordingly. \n\nThis may seem a long commentary for a result which changes very little, but there is a reason why we focus on $\\textbf{PMAlg}(\\Sigma)$: a pair $\\mathcal{M}=(\\mathcal{A}, D)$, such that $\\mathcal{A}$ is a partial $\\Sigma$-multialgebra with universe $A$ and $D$ is a subset of $A$, is called a PNmatrix, standing for a partial, non-deterministic matrix. Given a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ in the signature $\\Sigma$, we say $\\Gamma$ proves $\\varphi$ according to $\\mathcal{M}$ if, for every homomorphism $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ of partial multialgebras,\n\\[\\nu(\\Gamma)\\subseteq D\\quad\\text{implies}\\quad \\nu(\\varphi)\\in D,\\]\nwhen we write $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$.\n\nIt is easy to see that PNmatrices produce a broader semantics than that of Nmatrices: after all, every Nmatrix is a PNmatrix, but not vice-versa; so, in the effort of algebraizing logics, PNmatrices may offer new developments on logics that were, without them, uncharacterizable. PNmatrices were first developed in \\cite{Baaz}, although the presentation given here is closer to that of \\cite{CalMar}, and are indubitably very powerful; and yet, they are still not very widespread. We believe the reason for this is that they add partiality, a not so desired property for decision methods, to an already disliked by many semantical methodology, that of Nmatrices. So we extract from the logicians who have no philosophical objections against non-determinism those with no objections against partiality to find those willing to use PNmatrices.\n\nFrom our studies presented here it is clear we are supporters of Nmatrices, and of PNmatrices as well, so our intention here is not to replace them, but rather to give an alternative approach, more classically behaved, to those logicians not willing to appeal to them.\n\n\\subsection{Multihomomorphisms}\n\nFinally, consider the modified notions of homomorphism between $\\Sigma$-multialgebras $\\mathcal{A}=$\\\\$(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma})$ we briefly met in Section \\ref{Multi-valued homomorphisms}: the, perhaps, simplest multihomomorphism one can define is a function $\\varphi:A\\rightarrow\\mathcal{P}(B)\\setminus\\{\\emptyset\\}$ satisfying, for all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$,\n\n\\[\\bigcup_{a\\in \\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})}\\varphi(a)\\subseteq \\bigcup_{(b_{1}, \\dotsc  , b_{n})\\in \\varphi(a_{1})\\times\\cdots\\times\\varphi(a_{n})}\\sigma_{\\mathcal{B}}(b_{1}, \\dotsc  , b_{n});\\]\nthen, the category with $\\Sigma$-multialgebras as objects, and these multihomomorphisms as morphisms, will be denoted by $\\textbf{MMAlg}(\\Sigma)$. Notice that the composition of multihomomorphisms $\\varphi$, between $\\mathcal{A}$ and $\\mathcal{B}$, and $\\psi$, between $\\mathcal{B}$ and $\\mathcal{C}$, is not given by their composition as functions, but rather as \n\\[\\psi\\circ\\varphi(a)=\\bigcup_{b\\in \\varphi(a)}\\psi(b),\\]\nfor any element $a$ of $\\mathcal{A}$.\n\nThe idea here is quite clear: now, morphisms in our category of multialgebras return non-empty sets, much alike the multioperations in the multialgebras themselves. To find an equivalent category, we must reflect this change, what is actually rather easy: we demanded that the morphisms in $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ should preserve atoms, exactly because we wanted them all to be images, under $\\mathcal{P}_{\\mathcal{B}}$, of some homomorphism $\\varphi$ in $\\textbf{MAlg}(\\Sigma)$; and $\\mathcal{P}_{\\mathcal{B}}(\\varphi)$ preserves singletons, that is, atoms in the $(\\Sigma, \\leq)$-algebras $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$.\n\nSo, consider the category $\\textbf{MAlg}_{\\mathcal{B}}(\\Sigma)$\\label{MAlgBSigma}, with $(\\Sigma, \\leq)$-algebras as objects and, as morphisms between $(\\Sigma, \\leq)$-algebras $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{A}})$ and $\\mathcal{B}=(B, \\{\\sigma_{\\mathcal{B}}\\}_{\\sigma\\in\\Sigma}, \\leq_{\\mathcal{B}})$, any function $\\varphi:A\\rightarrow B$ such that:\n\\begin{enumerate}\n\\item for all $\\sigma\\in\\Sigma_{n}$ and $a_{1}, \\dotsc  , a_{n}\\in A$, \n\\[\\varphi(\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}))\\leq_{\\mathcal{B}}\\sigma(\\varphi(a_{1}), \\dotsc  , \\varphi(a_{n}));\\]\n\\item $\\varphi$ is continuous on non-empty sets, meaning that for any non-empty $S\\subseteq A$,\n\\[\\varphi(\\sup S)=\\sup\\{\\varphi(a) : a\\in S\\}.\\]\n\\end{enumerate}\n\nOf course, slight changes are necessaries on the functors realizing an equivalence between $\\textbf{MMAlg}(\\Sigma)$ and $\\textbf{MAlg}_{\\mathcal{B}}(\\Sigma)$. The first of them is $F:\\textbf{MMAlg}(\\Sigma)\\rightarrow\\textbf{MAlg}_{\\mathcal{B}}(\\Sigma)$, which takes a multialgebra $\\mathcal{A}$ to $\\mathcal{P}_{\\mathcal{B}}(\\mathcal{A})$, but takes a multihomomorphism between $\\mathcal{A}$ and $\\mathcal{B}$ (with universes $A$ and $B$) to the function $F\\varphi:\\mathcal{P}(A)\\setminus\\{\\emptyset\\}\\rightarrow\\mathcal{P}(B)\\setminus\\{\\emptyset\\}$ such that\n\\[F\\varphi(A')=\\bigcup_{a\\in A'}\\varphi(a),\\]\nfor a non-empty $A'\\subseteq A$. Reciprocally, we take the functor $G:\\textbf{MAlg}_{\\mathcal{B}}(\\Sigma)\\rightarrow\\textbf{MMAlg}(\\Sigma)$ mapping a $(\\Sigma, \\leq)$-algebra $\\mathcal{A}$ to the multialgebra $\\mathbb{A}(\\mathcal{A})$, but a morphism $\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ of $\\textbf{MAlg}_{\\mathcal{B}}(\\Sigma)$ to the multihomomorphism between $\\mathbb{A}(\\mathcal{A})$ and $\\mathbb{A}(\\mathcal{B})$ such that, for an atom $a$ of $\\mathcal{A}$,\n\\[G\\varphi(a)=A_{\\varphi(a)},\\]\nthat is, $G\\varphi(a)$ is the set of atoms, in $\\mathcal{B}$, below $\\varphi(a)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\\newpage\n\\thispagestyle{plain}\n\\hspace{0pt}\n\\vfill\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{.\/Image2}\n\\caption*{Specimen of \\textit{Nezara viridula} against a window, Campinas, Brazil.\\\\Photographed by Guilherme Vicentin de Toledo, all rights reserved.}\n\\end{figure}\n\\vfill\n\\hspace{0pt}\n\n\n\\part{Paraconsistent Logic}\\label{Part2}\n\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{3}\n\\chapter{Logics and Nmatrices}\\label{Chapter4}\\label{Chapter 4}\n\n\\section{Basic notions in logic}\n\n\nGiven a signature $\\Sigma$ and a countable set $\\mathcal{V}$, to which we will refer as a set of propositional variables and whose elements we will denote by $p_{i}$ for $i\\in\\mathbb{N}$, the language\\index{Language}, or propositional language, $\\mathcal{L}_{\\Sigma}$\\label{language} generated by $\\Sigma$ from $\\mathcal{V}$, which we may denote simply by $\\mathcal{L}$ if $\\Sigma$ is fixed, is the set of formulas $F(\\Sigma, \\mathcal{V})$.\n\nA consequence relation\\index{Consequence relation}\\label{vdash} $\\vdash$ on a language $\\mathcal{L}$ is a binary relation on $\\mathcal{P}(\\mathcal{L})$, and if $(\\Gamma, \\Delta)$ is in $\\vdash$, we will simply write $\\Gamma\\vdash\\Delta$. A logic\\index{Logic}\\label{logic} $\\mathscr{L}$ over a signature $\\Sigma$ is then a pair $(\\mathcal{L}, \\vdash_{\\mathscr{L}})$, where $\\vdash_{\\mathscr{L}}$ is a consequence relation on $\\mathcal{L}$ which will, at times, be denoted simply by $\\vdash$.\n\nWhen $\\Gamma\\vdash_{\\mathscr{L}}\\Delta$, we will say that $\\Gamma$ proves $\\Delta$; and for sets of formulas $\\Gamma_{1}, \\dotsc  , \\Gamma_{n}$ and $\\Delta_{1}, \\dotsc  , \\Delta_{m}$ and formulas $\\alpha_{1}, \\dotsc  , \\alpha_{k}$ and $\\beta_{1}, \\dotsc  , \\beta_{l}$, we will denote\n\\[\\Gamma_{1}\\cup\\cdots\\cup\\Gamma_{n}\\cup\\{\\alpha_{1}, \\dotsc  , \\alpha_{k}\\}\\vdash_{\\mathscr{L}}\\Delta_{1}\\cup\\cdots\\cup\\Delta_{m}\\cup\\{\\beta_{1}, \\dotsc  , \\beta_{l}\\}\\]\nsimply by \n\\[\\Gamma_{1}, \\dotsc  , \\Gamma_{n}, \\alpha_{1}, \\dotsc  , \\alpha_{k}\\vdash_{\\mathscr{L}}\\Delta_{1}, \\dotsc  , \\Delta_{m}, \\beta_{1}, \\dotsc  , \\beta_{l},\\]\nmeaning we will drop unions and curly brackets to simplify our notation. If $\\emptyset\\vdash_{\\mathscr{L}}\\varphi$, we will simply write $\\vdash_{\\mathscr{L}}\\varphi$, and call $\\varphi$ a tautology\\index{Tautology}.\n\n\\begin{definition}\nA logic $\\mathscr{L}$ is said to be a tarskian logic\\index{Logic, Tarskian} if, for any set $\\Gamma\\cup\\Delta\\cup\\{\\varphi\\}$ of formulas in the language of $\\mathscr{L}$, the following holds:\n\\begin{enumerate}\n\\item if $\\varphi\\in \\Gamma$, $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$;\n\\item if $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ and $\\Gamma\\subseteq\\Delta$, then $\\Delta\\vdash_{\\mathscr{L}}\\varphi$;\n\\item if $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ and $\\Delta\\vdash_{\\mathscr{L}}\\gamma$ for every $\\gamma\\in\\Gamma$, then $\\Delta\\vdash_{\\mathscr{L}}\\varphi$.\n\\end{enumerate}\n\\end{definition}\n\nA logic $\\mathscr{L}$ is said to be finitary\\index{Logic, Finitary} if, for all sets of formulas $\\Gamma\\cup\\{\\varphi\\}$ in its language, if $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ then there exists a finite set $\\Gamma_{0}\\subseteq\\Gamma$ such that $\\Gamma_{0}\\vdash_{\\mathscr{L}}\\varphi$.\n\nA logic $\\mathscr{L}$ over the signature $\\Sigma$ with language $\\mathcal{L}=F(\\Sigma, \\mathcal{V})$ is said to be structural\\index{Logic, Structural} when, for every $\\Gamma\\cup\\{\\varphi\\}\\subseteq\\mathcal{L}$ and every $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$, if $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ then $\\sigma(\\Gamma)\\vdash_{\\mathscr{L}}\\sigma(\\varphi)$; we call any $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow \\textbf{F}(\\Sigma, \\mathcal{V})$ a substitution.\n\n\\begin{definition}\nA set of formulas $\\Gamma$ in a logic $\\mathscr{L}$ is said to be closed\\index{Closed set of formulas} if, anytime $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$, we have that $\\varphi\\in\\Gamma$.\n\\end{definition}\n\nThe following theorem is due to Lindenbaum and \\L oz, and will be used several times in our study.\n\n\\begin{theorem}\nIf $\\mathscr{L}$ is a tarskian, finitary logic, and $\\Gamma\\cup\\{\\varphi\\}$ is a set of formulas such that $\\Gamma\\not\\vdash_{\\mathcal{L}}\\varphi$, there exists a non-trivial, closed set of formulas $\\Delta$ containing $\\Gamma$ and maximal with respect to not proving $\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nA proof can be found in \\cite{ParLog}, in Theorem $2.2.6$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Hilbert systems}\n\nPerhaps the most useful way for us to define a logic is through a set of axiom schemata plus a set of rules of inference, in what is called a Hilbert system\\index{Hilbert system}: given a language $\\mathcal{L}=F(\\Sigma, \\mathcal{V})$ over the signature $\\Sigma$, an $n$-ary rule of inference\\index{Rule of inference}, for $n\\in\\mathbb{N}$, is an element of $\\mathcal{L}^{n}\\times\\mathcal{L}$. A rule of inference $((\\alpha_{1}, \\dotsc  , \\alpha_{n}), \\alpha)$ will usually be denoted by $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$\\label{ruleofinference} or\n\\[\\frac{\\alpha_{1}, \\dotsc  , \\alpha_{n}}{\\alpha}.\\]\n An axiom scheme\\index{Axiom scheme} is any formula $\\alpha$, and its instances\\index{Axiom scheme, Instance of an} are $\\sigma(\\alpha)$, for any $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$: notice that, technically speaking, an axiom is a $0$-ary rule of inference; regardless, we will still treat then differently, given that historically this is the most common approach. More generally, given a rule of inference $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ and a homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$, $\\sigma(\\alpha_{1}), \\dotsc  , \\sigma(\\alpha_{n})|\\sigma(\\alpha)$ is an instance\\index{Rule of inference, Instance of a} of the aforementioned rule.\n\nSo, given a set of axiom schemata $\\mathfrak{A}$\\label{frakA} and a set of rules of inference $\\mathfrak{R}$\\label{frakR}, we can create a logic $\\mathscr{L}$ dependent on $\\mathfrak{A}$ and $\\mathfrak{R}$ such that, given formulas $\\Gamma\\cup\\{\\varphi\\}$ in $\\mathcal{L}$, $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ if and only if there exists a sequence of formulas $\\varphi_{1}, \\dotsc  , \\varphi_{m}$, said to be a demonstration\\index{Demonstration} of $\\varphi$ from $\\Gamma$, such that $\\varphi_{m}=\\varphi$ and $\\varphi_{i}$, for $i\\in\\{1, \\dotsc  , m\\}$, is either:\n\\begin{enumerate}\n\\item an element of $\\Gamma$, when we call it a premise\\index{Premise};\n\\item an instance of axiom $\\sigma(\\alpha)$, for an $\\alpha\\in\\mathfrak{A}$ and a $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$;\n\\item a formula $\\sigma(\\alpha)$, for a $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$ and $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ an $n$-ary rule of inference in $\\mathfrak{R}$ such that there exist $1\\leq i_{1}, \\dotsc  , i_{n}<i$ with $\\varphi_{i_{1}}=\\sigma(\\alpha_{1}), \\dotsc , \\varphi_{i_{n}}=\\sigma(\\alpha_{n})$.\n\\end{enumerate}\n\n\\begin{theorem}\nIf $\\mathscr{L}$ is the logic obtained from the set of axiom schemata $\\mathfrak{A}$ and the set of rules of inference $\\mathfrak{R}$, $\\mathscr{L}$ is a finitary, tarskian logic.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\Gamma\\cup\\Delta\\cup\\{\\varphi\\}$ be a set of formulas in $\\mathcal{L}$.\n\n\\begin{enumerate}\n\\item If $\\varphi\\in\\Gamma$, $\\varphi_{1}$ is a demonstration of $\\varphi$ from $\\Gamma$, with $\\varphi_{1}=\\varphi$: it is, in fact, a demonstration since $\\varphi_{1}$ is an element of $\\Gamma$. This way, $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$.\n\n\\item Suppose $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ and $\\Gamma\\subseteq\\Delta$: given $\\varphi_{1}, \\dotsc  , \\varphi_{m}$, a demonstration of $\\varphi$ from $\\Gamma$, we state that $\\varphi_{1}, \\dotsc  , \\varphi_{m}$ is also a demonstration of $\\varphi$ from $\\Delta$. Clearly, first of all, we have that $\\varphi_{m}=\\varphi$. Then $\\varphi_{i}$ is either:\n\\begin{enumerate}\n\\item an element of $\\Gamma$, and since $\\Gamma\\subseteq \\Delta$, an element of $\\Delta$;\n\\item an instance of axiom $\\sigma(\\alpha)$, for $\\alpha$ an axiom scheme and $\\sigma$ a substitution;\n\\item a formula $\\sigma(\\alpha)$, for $\\sigma$ a substitution and $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ a rule of inference for which there exist $1\\leq i_{1}, \\dotsc  , i_{n}<i$ with $\\varphi_{i_{1}}=\\sigma(\\alpha_{1}), \\dotsc  , \\varphi_{i_{n}}=\\sigma(\\alpha_{n})$.\n\\end{enumerate}\nThis proves $\\varphi_{1}, \\dotsc  , \\varphi_{m}$ is a demonstration of $\\varphi$ from $\\Delta$, and therefore $\\Delta\\vdash_{\\mathscr{L}}\\varphi$.\n\n\\item Suppose $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$, and let $\\varphi_{1}, \\dotsc  , \\varphi_{m}$ be a demonstration of $\\varphi$ from $\\Gamma$. We take $\\Gamma_{0}=\\Gamma\\cap\\{\\varphi_{1}, \\dotsc  , \\varphi_{m}\\}$: we state then that $\\varphi_{1}, \\dotsc  , \\varphi_{m}$ is a demonstration of $\\varphi$ from $\\Gamma_{0}\\subseteq\\Gamma$, which is a finite set.\n\nFirst of all, $\\varphi_{n}=\\varphi$. Then $\\varphi_{i}$ is either:\n\\begin{enumerate}\n\\item an element of $\\Gamma$, and then $\\varphi_{i}\\in\\Gamma\\cap\\{\\varphi_{1}, \\dotsc  , \\varphi_{n}\\}$, implying $\\varphi_{i}$ is in $\\Gamma_{0}$;\n\\item an instance $\\sigma(\\alpha)$ of the axiom scheme $\\alpha$;\n\\item a formula $\\sigma(\\alpha)$, for a rule of inference $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ and $1\\leq i_{1}, \\dotsc  , i_{n}< i$ such that $\\varphi_{i_{j}}=\\sigma(\\alpha_{j})$, for $j\\in\\{1, \\dotsc  , n\\}$.\n\\end{enumerate}\n So, to summarize, we have that $\\Gamma_{0}\\vdash_{\\mathscr{L}}\\varphi$, and therefore $\\mathscr{L}$ is finitary.\n\n\\item Finally, suppose $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ and that $\\Delta\\vdash_{\\mathscr{L}}\\gamma$ for every $\\gamma\\in\\Gamma$. By the item above, there exists a finite $\\Gamma_{0}\\subseteq\\Gamma$ such that $\\Gamma_{0}\\vdash_{\\mathscr{L}}\\varphi$, so let $\\varphi_{1}, \\dotsc  , \\varphi_{M}$ be a demonstration of $\\varphi$ from $\\Gamma_{0}$, and let $\\Gamma_{0}=\\{\\gamma_{1}, \\dotsc  , \\gamma_{l}\\}$: we will prove that we can drop one of the premises of $\\Gamma_{0}$, e.g. $\\gamma_{1}$, by adding as premises $\\Delta$, meaning\n\\[\\Delta\\cup\\{\\gamma_{1}, \\dotsc  , \\gamma_{l}\\}\\setminus\\{\\gamma_{1}\\}\\vdash_{\\mathscr{L}}\\varphi,\\]\nand by an inductive argument, we will have that $\\Delta\\vdash_{\\mathscr{L}}\\varphi$.\n\nSo, suppose $\\gamma_{1}\\neq\\varphi$ (otherwise we already have that $\\Delta\\vdash_{\\mathscr{L}}\\varphi$), and consider the subsequence $\\varphi'_{1}, \\dotsc  , \\varphi'_{m}$ of $\\varphi_{1}, \\dotsc  , \\varphi_{M}$ of formulas different from $\\gamma_{1}$; we then take a demonstration $\\psi_{1}, \\dotsc  , \\psi_{k}$ of $\\gamma_{1}$ from $\\Delta$, and state that the sequence $\\phi_{1}, \\dotsc  , \\phi_{k+m}$ equal to\n\\[\\psi_{1}, \\dotsc  , \\psi_{k}, \\varphi'_{1}, \\dotsc  , \\varphi'_{m}\\]\nis a demonstration of $\\varphi$ from $\\Delta\\cup\\Gamma_{0}\\setminus\\{\\gamma_{1}\\}$.\n\nFirst of all, since $\\varphi'_{1}, \\dotsc  , \\varphi'_{m}$ is the subsequence of $\\varphi_{1}, \\dotsc  , \\varphi_{M}$ of formulas different from $\\gamma_{1}$ and, by hypothesis, $\\gamma_{1}\\neq\\varphi$, we have that $\\varphi'_{m}=\\varphi_{M}=\\varphi$. Now, if $i\\in \\{1, \\dotsc  , k\\}$, $\\phi_{i}=\\psi_{i}$, and then:\n\\begin{enumerate}\n\\item if $\\psi_{i}$ is a premise in $\\Delta$, clearly $\\phi_{i}$ is a premise in the larger set $\\Delta\\cup\\Gamma_{0}\\setminus\\{\\gamma_{1}\\}$;\n\\item if $\\psi_{i}$ is an instance of axiom $\\sigma(\\alpha)$, clearly $\\phi_{i}$ is also an instance of $\\alpha$;\n\\item if $\\psi_{i}$ is the formula $\\sigma(\\alpha)$, for $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ an $n$-ary inference rule and $1\\leq i_{1}, \\dotsc  , i_{n}<i$ such that $\\psi_{i_{1}}=\\sigma(\\alpha_{1}), \\dotsc , \\psi_{i_{n}}=\\sigma(\\alpha_{n})$, then, since \n\\[\\psi_{i_{1}}=\\phi_{i_{1}}, \\dotsc  , \\psi_{i_{n}}=\\phi_{i_{n}},\\]\nwe have that $\\phi_{i}$ is also $\\sigma(\\alpha)$, for $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ an inference rule and $1\\leq i_{1}, \\dotsc  , i_{n}<i$ such that $\\phi_{i_{1}}=\\sigma(\\alpha_{1}), \\dotsc  , \\phi_{i_{n}}=\\sigma(\\alpha_{n})$.\n\\end{enumerate}\n\nIf $i\\in\\{k+1, \\dotsc  , k+m\\}$, $\\phi_{i}=\\varphi'_{i-k}$, and then\n\\begin{enumerate}\n\\item if $\\varphi'_{i-k}$ is a premise of the demonstration $\\varphi_{1}, \\dotsc  , \\varphi_{M}$, and therefore in $\\Gamma_{0}$, since $\\varphi'_{1}, \\dotsc  ,$\\\\$\\varphi'_{m}$ is the subsequence of elements different from $\\gamma_{1}$ we find $\\varphi'_{i-k}\\in\\Gamma_{0}\\setminus\\{\\gamma_{1}\\}$, and $\\phi_{i}$ is therefore in $\\Delta\\cup\\Gamma_{0}\\setminus\\{\\gamma_{1}\\}$;\n\\item if $\\varphi'_{i-k}$ is an instance of axiom $\\sigma(\\alpha)$, evidently $\\phi_{i}$ is also an instance of the same axiom;\n\\item if $\\varphi'_{i-k}=\\varphi_{j}$ is $\\sigma(\\alpha)$, for an inference rule $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ and $1\\leq i_{1}, .. , i_{n}<j$ such that $\\varphi_{i_{1}}=\\sigma(\\alpha_{1}), \\dotsc  , \\varphi_{i_{n}}=\\sigma(\\alpha_{n})$, there are two cases to consider for a $\\varphi_{i_{s}}$: \n\\begin{enumerate}\n\\item either $\\varphi_{i_{s}}\\neq\\gamma_{1}$, and so there exists $k+1\\leq j_{s}<k+i$ such that $\\varphi'_{j_{s}-k}=\\varphi_{i_{s}}$;\n\\item or $\\varphi_{i_{s}}=\\gamma_{1}$, when we make $j_{s}=k$ and therefore $\\phi_{j_{s}}=\\gamma_{1}$;\n\\end{enumerate}\neither way, we find that $\\phi_{i}$ is $\\sigma(\\alpha)$, for $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ an $n$-ary inference rule and $1\\leq j_{1}, \\dotsc  , j_{n}< i$ such that $\\phi_{j_{1}}=\\sigma(\\alpha_{1}), \\dotsc  , \\phi_{j_{n}}=\\sigma(\\alpha_{n})$.\n\\end{enumerate}\n\\end{enumerate}\n\n\\end{proof}\n\nAgain, let $\\mathscr{L}$ be the logic with axiom schemata $\\mathfrak{A}$ and rules of inference $\\mathfrak{R}$: beyond being a finitary, tarskian logic, we can prove that is also structural. Assume $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ and let $\\varphi_{1}, \\dotsc  , \\varphi_{m}$ be a demonstration of $\\varphi$ from $\\Gamma$ and $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma, \\mathcal{V})$ a $\\Sigma$-homomorphism.\n\nWe state then that $\\sigma(\\varphi_{1}), \\dotsc  , \\sigma(\\varphi_{m})$ is a demonstration of $\\sigma(\\varphi)$ from $\\sigma(\\Gamma)$. First of all, we have that, since $\\varphi_{m}=\\varphi$, $\\sigma(\\varphi_{m})=\\sigma(\\varphi)$. \n\\begin{enumerate}\n\\item If $\\varphi_{i}$ is a premise in $\\Gamma$, then $\\sigma(\\varphi_{i})$ is a premise in $\\sigma(\\Gamma)$.\n\\item If $\\varphi_{i}$ is an instance $\\tau(\\alpha)$ of an axiom $\\alpha$, since $\\sigma\\circ\\tau$ is still a $\\Sigma$-homomorphism, then $\\sigma(\\varphi_{i})$ is an instance $\\sigma\\circ\\tau(\\alpha)$ of the same axiom $\\alpha$.\n\\item If $\\varphi_{i}$ equals $\\tau(\\alpha)$, for a $\\Sigma$-homomorphism $\\tau$, a rule of inference $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ and $1\\leq i_{1}, \\dotsc  , i_{n}<i$ such that $\\varphi_{i_{1}}=\\tau(\\alpha_{1}), \\dotsc  , \\varphi_{i_{n}}=\\tau(\\alpha_{n})$, then $\\sigma\\circ\\tau$ is still a $\\Sigma$-homomorphism, and $\\sigma(\\varphi_{i})$ equals $\\sigma\\circ\\tau(\\alpha)$ for a rule of inference $\\alpha_{1}, \\dotsc  , \\alpha_{n}|\\alpha$ and $1\\leq i_{1}, \\dotsc  , i_{n}<i$ such that $\\sigma(\\varphi_{i_{1}})=\\sigma\\circ\\tau(\\alpha_{1}), \\dotsc  , \\sigma(\\varphi_{i_{n}})=\\sigma\\circ\\tau(\\alpha_{n})$.\n\\end{enumerate}\n\nOne important thing to notice is that, when presenting an axiom or rule of inference, is common to use a not so precise notation: suppose our signature has a binary symbol $\\uparrow$, and that $p_{1}\\uparrow p_{2}$ is an axiom; we will usually write it as $\\alpha\\uparrow\\beta$, using $\\alpha$ and $\\beta$ as meta-variables for formulas, although, in principle, this notation is incorrect. The same will be done for rules of inference, where, to name one example, a rule such as\n\\[\\frac{p_{1}, p_{1}\\uparrow p_{2}}{{\\sim}  p_{2}}\\]\nwill be denoted by $\\alpha, \\alpha\\uparrow\\beta|{\\sim} \\beta$.\n\n\n\\subsection{Paraconsistent logics and $\\textbf{LFI}'s$}\n\nWe will understand as classical propositional logic\\index{Logic, Classical propositional} the logic we will denote by $\\textbf{CPL}$\\label{CPL} over the signature $\\Sigma^{\\textbf{CPL}}$\\label{SigmaCPL}, such that $\\Sigma^{\\textbf{CPL}}_{0}=\\{\\bot, \\top\\}$, $\\Sigma^{\\textbf{CPL}}_{1}=\\{{\\sim} \\}$, $\\Sigma^{\\textbf{CPL}}_{2}=\\{\\vee, \\wedge, \\rightarrow\\}$ and $\\Sigma^{\\textbf{CPL}}_{n}=\\emptyset$ for $n>2$, with axiom schemata\n\\begin{enumerate}\n\\item[\\textbf{Ax\\: 1}] $\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$;\n\\item[\\textbf{Ax\\: 2}] $\\big(\\alpha\\rightarrow (\\beta\\rightarrow \\gamma)\\big)\\rightarrow\\big((\\alpha\\rightarrow\\beta)\\rightarrow(\\alpha\\rightarrow\\gamma)\\big)$;\n\\item[\\textbf{Ax\\: 3}] $\\alpha\\rightarrow\\big(\\beta\\rightarrow(\\alpha\\wedge\\beta)\\big)$;\n\\item[\\textbf{Ax\\: 4}] $(\\alpha\\wedge\\beta)\\rightarrow \\alpha$;\n\\item[\\textbf{Ax\\: 5}] $(\\alpha\\wedge\\beta)\\rightarrow \\beta$;\n\\item[\\textbf{Ax\\: 6}] $\\alpha\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 7}] $\\beta\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 8}] $(\\alpha\\rightarrow\\gamma)\\rightarrow\\Big((\\beta\\rightarrow\\gamma)\\rightarrow \\big((\\alpha\\vee\\beta)\\rightarrow\\gamma\\big)\\Big)$;\n\\item[\\textbf{Ax\\: 9}] $(\\alpha\\rightarrow\\beta)\\rightarrow\\big((\\alpha\\rightarrow{\\sim} \\beta)\\rightarrow{\\sim} \\alpha\\big)$;\n\\item[\\textbf{Ax\\: 10}] $\\alpha\\rightarrow({\\sim} \\alpha\\rightarrow\\beta)$;\n\\item[\\textbf{Ax\\: 11}] $\\alpha\\vee{\\sim} \\alpha$;\n\\item[\\textbf{Ax\\: 12}] $\\bot\\rightarrow\\alpha$;\n\\item[\\textbf{Ax\\: 13}] $\\alpha\\rightarrow\\top$;\n\\end{enumerate}\nand following the inference rule of Modus Ponens\n\\[\\frac{\\alpha\\quad\\alpha\\rightarrow\\beta}{\\beta}.\\]\n\n\nAs opposed to $\\textbf{CPL}$, a tarskian logic $\\mathscr{L}$ will be said to be paraconsistent\\index{Logic, Paraconsistent} when it possesses an unary symbol \"$\\neg $\", that we shall refer to as a negation, such that there exist formulas $\\alpha$ and $\\beta$ of $\\mathscr{L}$ satisfying $\\alpha, \\neg\\alpha\\not\\vdash_{\\mathscr{L}}\\beta$. A good, intuitive way to look at paraconsistent logics is through the paradigm introduced by da Costa that the formulas of such a logic can, at times, be divided into well-behaved and badly-behaved (nowadays more commonly referred to as inconsistent) ones. Think of a scientist performing an experiment: from her school years, said scientist knows that opposite poles of a magnet attract each other; so, if during the experiment two poles she thought to be opposite are instead repealing each other, she can be certain that she was wrong, and the two poles are instead the same, both north or both south. Her reasoning works because the sentence ``opposite poles attract'' is well-behaved (that is, it can not coexist with its negation) and true, while what she had apparently observed was that ``opposite poles repeal'', a well-behaved although false sentence.\n\nNow, suppose that our scientist has discovered that all monopoles (conjectured magnetic particles that, instead of having both north and south poles, have only one pole, hence the name) have exactly the same pole and is now testing two hypotheses: ``all monopoles are north poles'' and ``all monopoles are south poles''. Again, we have contradictory sentences, as was the case with ``opposite poles attract'' and ``opposite poles repeal'', but this time our scientist can not reach the conclusion that, at some prior step, she made a mistake: why is that?\\footnote{Notice that we are presenting this logical problem in a somewhat convoluted way in order to avoid explicit explosivity of the theory at hand: scientists, and most mathematicians, appear to prefer thinking about these topics in terms of non-contradiction (what is eventually equivalent).} Simply put: none of the hypotheses to be tested, ``all monopoles are north poles'' and ``all monopoles are south poles'', is well-behaved, meaning that we can not discard their negations as necessarily false.\n\nWhat we are doing is dividing those sentences found in science between true-or-false sentences (such as those involving attracting or repealing poles) and hypothetical sentences (such as those involving all monopoles). But this is not exclusive to scientists: mathematicians also have true-or-false sentences and conjectural sentences: after all, some mathematicians believe that, e.g., the Riemann hypothesis is true, while others believe it to be false, and that doesn't make mathematics as a whole trivial. In daily life, conjectural and hypothetical sentences may be replaced with rumors, that can also be contradictory without making logical reasoning unfeasible. da Costa's hierarchy, which deals with these distinctions plus higher degrees of consistency, will be studied algebraically in Chapters \\ref{Chapter5} and \\ref{Chapter6}. \n\nIf our logic $\\mathscr{L}$ also possesses a binary symbol \"$\\rightarrow$\" satisfying the deduction meta-theorem, i.e., $\\Gamma, \\psi\\vdash_{\\mathscr{L}}\\varphi$ if and only if $\\Gamma\\vdash_{\\mathscr{L}}\\psi\\rightarrow\\varphi$, then the condition that there exist formulas $\\alpha$ and $\\beta$ such that $\\alpha, \\neg\\alpha\\not\\vdash_{\\mathscr{L}}\\beta$ is equivalent to the existence of formulas $\\alpha$ and $\\beta$ such that $\\not\\vdash_{\\mathscr{L}}\\alpha\\rightarrow(\\neg\\alpha\\rightarrow\\beta)$, which is know as the failure of the explosion law\\index{Explosion law}; if this result were to be true for any $\\alpha$ and $\\beta$, that would mean the explosion law would be valid, which indeed happens in $\\textbf{CPL}$ as one can see by its $\\textbf{Ax\\: 10}$.\n\nIf a formula $\\psi$ has all its propositional variables among the set $\\{p_{1}, \\dotsc  , p_{n}\\}$, we may write $\\psi(p_{1}, \\dotsc  , p_{n})$\\label{varformula}: to define logics of formal inconsistency, we shall need a set of formulas $\\bigcirc(p)$, each of them dependent exactly on the propositional variable $p$, that is, each one of them contains $p$ and solely $p$ as a variable.\n\nAnd if we apply to a formula $\\psi(p_{1}, \\dotsc  , p_{n})$ a homomorphism $\\sigma$ sending $p_{i}$ to the formula $\\alpha_{i}$, for $i\\in\\{1, \\dotsc  , n\\}$, we shall write $\\psi(\\alpha_{1}, \\dotsc  , \\alpha_{n})$ for $\\sigma(\\psi)$: this way, $\\bigcirc(\\alpha)$ will be the set of formulas obtained from $\\bigcirc(p)$ after we apply to each of its elements the homomorphism taking $p$ to $\\alpha$, that is,\n\\[\\bigcirc(\\alpha)=\\{\\psi(\\alpha)\\ :\\  \\psi(p)\\in\\bigcirc(p)\\}.\\]\n\n\\begin{definition}\\label{Definition of LFI}\nA tarskian, finitary and structural logic $\\mathscr{L}$ containing a negation and a set of formulas $\\bigcirc(p)\\neq\\emptyset$ depending exactly on the propositional variable $p$ is a logic of formal inconsistency ($\\textbf{LFI}$)\\index{Logic of formal inconsistency}\\label{LFI} when:\n\\begin{enumerate}\n\\item there exist formulas $\\phi$ and $\\theta$ in $\\mathscr{L}$ such that $\\phi, \\neg\\phi\\not\\vdash_{\\mathscr{L}}\\theta$;\n\\item there exist formulas $\\alpha$ and $\\beta$ in $\\mathscr{L}$ such that\n\\begin{enumerate}\n\\item $\\bigcirc(\\alpha), \\alpha\\not\\vdash_{\\mathscr{L}}\\beta$ and\n\\item $\\bigcirc(\\alpha), \\neg\\alpha\\not\\vdash_{\\mathscr{L}}\\beta$;\n\\end{enumerate}\n\\item for all formulas $\\varphi$ and $\\psi$ in $\\mathscr{L}$,\n\\[\\bigcirc(\\varphi), \\varphi, \\neg\\varphi\\vdash_{\\mathscr{L}}\\psi.\\]\n\\end{enumerate}\n\\end{definition}\n\nThe logic $\\mathscr{L}$ will be called a weak $\\textbf{LFI}$\\index{Weak $\\textbf{LFI}$} if the second condition of Definition \\ref{Definition of LFI} is replaced by there existing formulas $\\alpha_{1}$ and $\\beta_{1}$ such that\n\\[\\bigcirc(\\alpha_{1}), \\alpha_{1}\\not\\vdash_{\\mathscr{L}}\\beta_{1}\\]\nand $\\alpha_{2}$ and $\\beta_{2}$, possibly different from respectively $\\alpha_{1}$ and $\\beta_{1}$, such that\n\\[\\bigcirc(\\alpha_{2}), \\neg\\alpha_{2}\\not\\vdash_{\\mathscr{L}}\\beta_{2}.\\]\n\nThe logic $\\mathscr{L}$ will be called a strong $\\textbf{LFI}$\\index{Strong $\\textbf{LFI}$} if the first and second conditions of Definition \\ref{Definition of LFI} are replaced by there existing a single pair of formulas $\\alpha$ and $\\beta$ satisfying simultaneously\n\\begin{enumerate}\n\\item $\\alpha, \\neg\\alpha\\not\\vdash_{\\mathscr{L}}\\beta$,\n\\item $\\bigcirc(\\alpha), \\alpha\\not\\vdash_{\\mathscr{L}}\\beta$ and\n\\item $\\bigcirc(\\alpha), \\neg\\alpha\\not\\vdash_{\\mathscr{L}}\\beta$.\n\\end{enumerate}\nAll these definitions are the Definitions $2.1.7$, $2.1.8$ and $2.1.9$ of \\cite{ParLog}.\n\nWhen the set $\\bigcirc(p)$ contains a single formula, this formula will be denoted by $\\circ p$\\label{circ}, being the consistency of $p$. More often than not, we will want \"$\\circ$\" to be a primitive connective, as well as consistency a primitive notion.\n\nThe $\\textbf{LFI}$ we will most frequently work with is one that, beyond having a primitive consistency, emulates only the positive fragment of classical propositional logic and excluded middle: this way, $\\textbf{mbC}$\\label{mbC} is often regarded as the simplest logic of formal inconsistency. It has the signature we will denote by $\\Sigma_{\\textbf{LFI}}$\\label{SigmaLFI}, with $(\\Sigma_{\\textbf{LFI}})_{0}=\\emptyset$, $(\\Sigma_{\\textbf{LFI}})_{1}=\\{\\neg , \\circ\\}$, $(\\Sigma_{\\textbf{LFI}})_{2}=\\{\\vee, \\wedge, \\rightarrow\\}$ and $(\\Sigma_{\\textbf{LFI}})_{n}=\\emptyset$ for $n>2$; it has as axiom schemata\n\\begin{enumerate}\n\\item[\\textbf{Ax\\: 1}] $\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$;\n\\item[\\textbf{Ax\\: 2}] $\\big(\\alpha\\rightarrow (\\beta\\rightarrow \\gamma)\\big)\\rightarrow\\big((\\alpha\\rightarrow\\beta)\\rightarrow(\\alpha\\rightarrow\\gamma)\\big)$;\n\\item[\\textbf{Ax\\: 3}] $\\alpha\\rightarrow\\big(\\beta\\rightarrow(\\alpha\\wedge\\beta)\\big)$;\n\\item[\\textbf{Ax\\: 4}] $(\\alpha\\wedge\\beta)\\rightarrow \\alpha$;\n\\item[\\textbf{Ax\\: 5}] $(\\alpha\\wedge\\beta)\\rightarrow \\beta$;\n\\item[\\textbf{Ax\\: 6}] $\\alpha\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 7}] $\\beta\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 8}] $(\\alpha\\rightarrow\\gamma)\\rightarrow\\Big((\\beta\\rightarrow\\gamma)\\rightarrow \\big((\\alpha\\vee\\beta)\\rightarrow\\gamma\\big)\\Big)$;\n\\item[$\\textbf{Ax\\: 9}^{*}$] $(\\alpha\\rightarrow \\beta)\\vee\\alpha$;\n\\item[$\\textbf{Ax\\: 11}^{*}$] $\\alpha\\vee\\neg \\alpha$,\n\\end{enumerate}\nplus \n\\[\\tag{\\textbf{bc1}}\\circ\\alpha\\rightarrow(\\alpha\\rightarrow(\\neg \\alpha\\rightarrow \\beta)),\\]\nand as inference rules that of Modus Ponens,\n\\[\\frac{\\alpha\\quad\\alpha\\rightarrow\\beta}{\\beta}.\\]\n\nOther logics of formal incompatibility we will make use of are:\n\\begin{enumerate}\n\\item $\\textbf{mbCciw}$\\label{mbCciw}, obtained from the Hilbert system for $\\textbf{mbC}$ by adding the axiom schema\\label{ciw}\n\\[\\tag{\\textbf{ciw}} \\circ\\alpha\\vee(\\alpha\\wedge\\neg \\alpha);\\]\n\\item $\\textbf{mbCci}$\\label{mbCci}, obtained from $\\textbf{mbC}$ by adding\\label{ci}\n\\[\\tag{\\textbf{ci}}\\neg \\circ\\alpha\\rightarrow(\\alpha\\wedge\\neg \\alpha);\\]\n\\item $\\textbf{mbCcl}$\\label{mbCcl}, obtained from $\\textbf{mbC}$ by adding\\label{cl}\n\\[\\tag{\\textbf{cl}}\\neg(\\alpha\\wedge\\neg \\alpha)\\rightarrow\\circ\\alpha.\\]\n\\end{enumerate}\n\n\n\n\n\\section{Matrices and generalizations}\n\nA logical matrix\\index{Matrix, Logical} over a signature $\\Sigma$ is a pair $\\mathcal{M}=(\\mathcal{A}, D)$ such that $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is a $\\Sigma$-algebra and $D\\subseteq A$ is said to be the set of designated elements\\index{Designated elements} of the matrix. Given formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma$, we say $\\Gamma$ semantically proves $\\varphi$ according to $\\mathcal{M}$ (and write $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$\\label{semant.proves}) if, for every homomorphism $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ such that $\\nu(\\gamma)\\in D$, for every $\\gamma\\in\\Gamma$, one has that $\\nu(\\varphi)\\in D$.\n\nA logic $\\mathcal{L}$, over the signature $\\Sigma$, is said to be characterized\\index{Characterized} by $\\mathcal{M}$ if, for every set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over $\\Sigma$, $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$. \n\nQuite analogously, given a class $\\mathbb{M}$ of matrices $\\mathcal{M}$ over the same signature, we write $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$\\label{vDashM} if $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$ for every $\\mathcal{M}\\in\\mathbb{M}$, and a logic $\\mathcal{L}$ is said to be characterized by $\\mathbb{M}$ when $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$.\n\nIt is a well know result by W\\'ojcicki\\index{W\\'ojciki} that every tarskian logic can be characterized by a suitable class of logical matrices, see \\cite{Woj}. And, although one could see such a result as settling the matter of logical matrices, many times the suitable class of logical matrices obtained for a logic is not efficient, as in, to name one example, its algebras can be too large or complex, or the class may be infinite. So alternatives have been offered to classes of logical matrices, as in W\\'ojcicki \\cite{Woj2}.\n\nAnother approach is the one we will call that of restricted matrices\\index{Matrix, Restricted}, or Rmatrices\\index{Rmatrix}, (although that was not its original nomenclature, Piochi\\index{Piochi} named then $\\mathcal{E}$-matrices instead), see \\cite{Piochi} or, for an approach focusing more on structurality of the related closure operators, \\cite{Piochi2} and \\cite{Piochi3}. A restricted matrix over a signature $\\Sigma$ is a triple $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ such that:\n\\begin{enumerate}\n\\item $\\mathcal{A}=(A,\\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ is a $\\Sigma$-algebra;\n\\item $D$ is a subset of $A$;\n\\item $\\mathcal{F}$ is a set of homomorphisms from $\\textbf{F}(\\Sigma, \\mathcal{V})$ to $\\mathcal{A}$.\n\\end{enumerate}\n\nThe set $\\mathcal{F}$ will be called the set of restrictions\\index{Restrictions}. Given formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma$, we say $\\Gamma$ proves $\\varphi$ according to a restricted matrix $\\mathcal{M}$ (also over $\\Sigma$) and write $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$ if, for every homomorphism $\\nu\\in\\mathcal{F}$, $\\nu(\\gamma)\\in D$, for every $\\gamma\\in\\Gamma$, implies $\\nu(\\varphi)\\in D$; we say a logic $\\mathcal{L}$ is characterized by a restricted matrix $\\mathcal{M}$ when $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$, and it is possible to prove that every tarskian logic can be characterized by a, potentially infinite, restricted matrix (see Piochi's \\cite{Piochi} and \\cite{Piochi2}).\n\nAnd yet, given sometimes this is the most efficient approach, we may define $\\vDash_{\\mathbb{M}}$ for $\\mathbb{M}$ a class of restricted matrices in much the same way we did for a class of matrices. \n\nMany of these generalizations have one simple objective, which is to offer a reasonable decision method for a logic; of course, methods that depend on infinite matrices or infinite classes of matrices are not really decision methods, and so finite matrices, or finite sets of finite matrices, are the preferable outcomes of the algebraization process of a logic. Of course, such outcomes are not always possible, as many results on uncharacterizability of logics show: probably the earliest one is G{\\\"o}del's\\index{G{\\\"o}del} proof that propositional, intuitionistic logic is not characterizable by a single, finite logical matrix, found in \\cite{Godel}.\n\nInspired by G{\\\"o}del's proof, Dugundji (\\cite{Dugundji})\\index{Dugundji} proved that no system lying between the modal logics $\\textbf{S1}$ and $\\textbf{S5}$ admits a single, finite logic matrix which characterizes it as well, and in many places we may refer to results regarding uncharacterizability of logics as ``Dugundji-like'' theorems. As we enter the domain of paraconsistent logics, these results abound, given the intrinsic complexity of many of those systems: one example would be Avron's\\index{Avron} proof that many logics, including da Costa's $C_{1}$, do not possess a characterizing finite Nmatrix, or even a characterizing finite set of finite Nmatrices (\\cite{Avron, Avron3}).\n\nMost of the work shown in Sections \\ref{Restricted Nmatrices}, \\ref{A brief history of RNmatrices}, \\ref{Structurality} and \\ref{Examples: characterizing some logics with RNmatrices} was submitted to an online repository in \\cite{CostaRNmatrix}, and then finally published in \\cite{TwoDecisionProcedures}.\n\n\n\n\n\\subsection{Restricted Nmatrices}\\label{Restricted Nmatrices}\n\nAs we explained before, although the problem of finding semantics for a given logic is somewhat solved in the case the logic at hand is tarskian, the corresponding class of matrices or even restricted matrix associated to the logic may not be sufficiently efficient, and in the realm of paraconsistency and, furthermore, in the presence of incompatibility, which we will further study ahead, non-deterministic matrices have been proven fruitful. The first approach to non-deterministic matrices is found in the work of Rescher\\index{Rescher} (see \\cite{Rescher}) and Ivlev\\index{Ivlev} (see \\cite{Lev}, \\cite{Lev2}, \\cite{Lev3} and \\cite{Lev4}), although our reasoning will be closer to that of \\cite{Avron}.\n\n\\begin{definition}\nGiven a signature $\\Sigma$, a non-deterministic matrix over $\\Sigma$, or Nmatrix\\index{Nmatrix}\\index{Matrix, Non-deterministic}, is a pair $\\mathcal{M}=(\\mathcal{A}, D)$ such that $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\Sigma})$ is a $\\Sigma-$multialgebra and $D$ is a subset of $A$, said to be its set of designated elements. \n\\end{definition}\n\nOne may check definition $6.3.1$ of \\cite{ParLog} and \\cite{Avron} for equivalent definitions, with slightly different emphases. \n\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma$ and an Nmatrix $\\mathcal{M}$, we say that $\\Gamma$ proves $\\varphi$ according to $\\mathcal{M}$ if, for every homomorphism of multialgebras $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow \\mathcal{A}$ such that $\\nu(\\gamma)\\in D$ for every $\\gamma\\in\\Gamma$, one has $\\nu(\\varphi)\\in D$, when we then write $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$: notice how close this definition is of that for $\\vDash_{\\mathcal{M}}$ when $\\mathcal{M}$ is simply a matrix. \n\nOne, perhaps very important, observation is that Nmatrices, as defined, are slightly redundant, being enough to add a symbol to our signature in order to work only their underlying multialgebra: consider an Nmatrix $\\mathcal{M}=(\\mathcal{A}, D)$ over the signature $\\Sigma$, add a symbol $\\top$ to $\\Sigma_{0}$ therefore producing the signature $\\Sigma^{\\top}$\\label{Sigmatop} and define the $\\Sigma^{\\top}$-multialgebra $\\mathcal{A}^{\\top}$ such that\n\\[\\sigma_{\\mathcal{A}^{\\top}}=\\sigma_{\\mathcal{A}}, \\quad\\text{for every $\\sigma$ in $\\Sigma$},\\]\nand $\\top_{\\mathcal{A}^{\\top}}=D$. Then, given formulas $\\Gamma\\cup\\{\\varphi\\}$ in the signature $\\Sigma$, we say that $\\Gamma$ proves $\\varphi$ according to $\\mathcal{A}^{\\top}$, and write $\\Gamma\\vDash_{\\mathcal{A}^{\\top}}\\varphi$, if for every homomorphism of $\\Sigma$-multialgebras $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}^{\\top}$,\n\\[\\nu(\\Gamma)\\subseteq \\top_{\\mathcal{A}^{\\top}}\\quad\\text{implies}\\quad \\nu(\\varphi)\\in \\top_{\\mathcal{A}^{\\top}},\\]\nwhat is possible given that $\\Sigma\\subseteq \\Sigma^{\\top}$ implies $\\Gamma\\cup\\{\\varphi\\}\\subseteq F(\\Sigma, \\mathcal{V})\\subseteq F(\\Sigma^{\\top}, \\mathcal{V})$.\nOf course, such a simplification is not possible when dealing with matrices, unless the set of designated values is a singleton.\n\nGiven a class $\\mathbb{M}$ of Nmatrices over the same signature $\\Sigma$, we say $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$ if, for every $\\mathcal{M}\\in \\mathbb{M}$, $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$. Given every matrix is an Nmatrix, we have that every tarskian logic may be characterized by a class of Nmatrices.\n\n\\begin{definition}\nA restricted non-deterministic matrix, or restricted Nmatrix or RNmatrix\\index{RNmatrix}\\index{Matrix, Restricted non-deterministic}, over a signature $\\Sigma$ is a triple \n\\[\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})\\]\nsuch that:\n\\begin{enumerate}\n\\item $(\\mathcal{A}, D)$ is a non-deterministic matrix over $\\Sigma$;\n\\item $\\mathcal{F}$ is a subset of the set of all homomorphisms from $\\textbf{F}(\\Sigma, \\mathcal{V})$ to $\\mathcal{A}$.\n\\end{enumerate}\n\\end{definition}\n\n\nWe define $\\vDash_{\\mathcal{M}}$ as in the case that $\\mathcal{M}$ is a restricted matrix. Notice that every restricted matrix is a restricted Nmatrix, and then every tarskian logic may be characterized by a restricted Nmatrix, but we can achieve a more powerful result, similar to the one commonly known as Suszko`s Thesis, found in \\cite{Suszko}; but, unlike Suszko, we do not focus on bivaluations, and we do not wish to advocate that all logics are two-valued.\n\n\\begin{theorem}\\label{2-valued RNmatrix}\nEvery tarskian logic is characterizable by a two-valued RNmatrix.\\footnote{By an $n$-valued RNmatrix we understand an RNmatrix $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ where the universe of $\\mathcal{A}$ has $n$ elements.}\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathfrak{L}=(\\mathcal{L}, \\vdash)$ be a tarskian logic over the signature $\\Sigma$. Consider then the $\\Sigma$-multialgebra $\\textbf{2}(\\Sigma)$\\label{2Sigma} with universe $\\{0,1\\}$ and, for an $n$-ary $\\sigma\\in\\Sigma$, operations defined by\n\\[\\sigma_{\\textbf{2}(\\Sigma)}(x_{1}, \\dotsc  , x_{n})=\\{0,1\\}, \\forall x_{1}, \\dotsc  , x_{n}\\in \\{0,1\\}.\\]\nWe then define the set $\\mathcal{F}_{\\mathfrak{L}}$ of valuations $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{2}(\\Sigma)$ such that there exists a closed set of formulas $\\Gamma$ over $\\Sigma$ for which $\\nu(\\gamma)=1$ if, and only if, $\\gamma\\in\\Gamma$; notice that all functions from $\\textbf{F}(\\Sigma, \\mathcal{V})$ to $\\textbf{2}(\\Sigma)$ are homomorphisms, and therefore no further restrictions are necessary.\n\nWe then define the RNmatrix \n\\[\\textbf{2}(\\mathfrak{L})=(\\textbf{2}(\\Sigma), \\{1\\}, \\mathcal{F}_{\\mathfrak{L}}),\\]\n\\label{2L}and we state that $\\Gamma\\vdash\\varphi$ if, and only if, $\\Gamma\\vDash_{\\textbf{2}(\\mathfrak{L})}\\varphi$. \n\nRegarding the first direction, if $\\Gamma\\vdash\\varphi$, for any valuation $\\nu\\in\\mathcal{F}_{\\mathfrak{L}}$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$ there must exist a closed set $\\Delta$ of formulas over $\\Sigma$ such that $\\nu(\\delta)=1$ if, and only if, $\\delta\\in\\Delta$, and since $\\nu(\\Gamma)\\subseteq\\{1\\}$ we have $\\Gamma\\subseteq \\Delta$; now, given $\\Gamma\\vdash\\varphi$ and $\\Gamma\\subseteq \\Delta$, it follows that $\\Delta\\vdash\\varphi$ and, since $\\Delta$ is closed, $\\varphi\\in \\Delta$, meaning $\\nu(\\varphi)=1$ and, therefore, $\\Gamma\\vDash_{\\textbf{2}(\\mathfrak{L})}\\varphi$.\n\nFor the second direction, assume $\\Gamma\\vDash_{\\textbf{2}(\\mathfrak{L})}\\varphi$: if $\\Gamma\\not\\vdash\\varphi$, by Lindenbaum-\\L oz there exists a non-trivial extension $\\Delta$ of $\\Gamma$ such that $\\varphi\\notin\\Delta$; if $\\nu$ is the valuation of $\\mathcal{F}_{\\mathfrak{L}}$ such that $\\nu(\\delta)=1$ if, and only if, $\\delta\\in\\Delta$, this means that $\\nu(\\Delta)\\subseteq\\{1\\}$, and therefore $\\nu(\\Gamma)\\subseteq\\{1\\}$ since $\\Gamma\\subseteq\\Delta$, and $\\nu(\\varphi)=0$, contradicting the fact that $\\Gamma\\vDash_{\\textbf{2}(\\mathfrak{L})}\\varphi$. This proves $\\Gamma\\vDash_{\\textbf{2}(\\mathfrak{L})}\\varphi$ implies $\\Gamma\\vdash\\varphi$, and the theorem is proved.\n\\end{proof}\n\nFor a restricted Nmatrix $\\mathcal{M}$ we will want to consider, for a subset $\\Gamma$ of $F(\\Sigma, \\mathcal{V})$, the closure $K_{\\mathcal{M}}(\\Gamma)$\\label{KMGamma}, that is, the set of formulas $\\varphi\\in F(\\Sigma, \\mathcal{V})$ such that $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$.\n\n\\begin{definition}\nGiven a signature $\\Sigma$, an operator $K:\\mathcal{P}(F(\\Sigma, \\mathcal{V}))\\rightarrow \\mathcal{P}(F(\\Sigma, \\mathcal{V}))$ is said to be a tarskian\\index{Operator, Tarskian} (or closure) one if, for all $\\Gamma, \\Theta\\subseteq F(\\Sigma, \\mathcal{V})$, it satisfies:\n\\begin{enumerate}\n\\item $\\Gamma\\subseteq K(\\Gamma)$;\n\\item if $\\Theta\\subseteq \\Gamma$, $K(\\Theta)\\subseteq K(\\Gamma)$;\n\\item if $\\Theta=K(\\Gamma)$, $K(\\Theta)=\\Theta$.\n\\end{enumerate}\n\\end{definition}\n\nNotice how this generalizes the notion of a logic being tarskian: the consequence relation in a tarskian logic is a tarskian operator; for such a reason, one may often call the pair $(\\textbf{F}(\\Sigma, \\mathcal{V}), K)$, for $K$ a tarskian operator on $F(\\Sigma, \\mathcal{V})$, a sentential logic itself. Even more, we may define a logic $\\mathcal{L}$ over the signature $\\Sigma$ by $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ if, and only if, $\\varphi\\in K(\\Gamma)$, and it is clear how $\\mathcal{L}$ is tarskian: to every tarskian operator there corresponds a tarskian logic and vice-versa.\n\n\\begin{proposition}\nGiven a restricted Nmatrix $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$, the operator $\\Gamma\\mapsto K_{\\mathcal{M}}(\\Gamma)$ is a tarskian one.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Take $\\varphi\\in \\Gamma$, and suppose $\\nu\\in\\mathcal{F}$ is such that $\\nu(\\gamma)\\in D$ for all $\\gamma\\in\\Gamma$; then $\\nu(\\varphi)\\in D$ and therefore $\\varphi\\in K_{\\mathcal{M}}(\\Gamma)$, so that $\\Gamma\\subseteq K_{\\mathcal{M}}(\\Gamma)$.\n\n\\item Suppose $\\Theta\\subseteq\\Gamma$ and that $\\varphi\\in K_{\\mathcal{M}}(\\Theta)$: then, if $\\nu\\in\\mathcal{F}$ satisfies that $\\nu(\\theta)\\in D$, for every $\\theta\\in\\Theta$, one has $\\nu(\\varphi)\\in D$.\n\nNow, take a restricted homomorphism $\\nu\\in\\mathcal{F}$ such that $\\nu(\\gamma)\\in D$, for every $\\gamma\\in\\Gamma$: since $\\Theta\\subseteq \\Gamma$, in this case $\\nu(\\theta)\\in D$, $\\forall \\theta\\in \\Theta$, and therefore $\\nu(\\varphi)\\in D$, implying that $\\varphi\\in K_{\\mathcal{M}}(\\Gamma)$. It follows that $K_{\\mathcal{M}}(\\Theta)\\subseteq K_{\\mathcal{M}}(\\Gamma)$.\n\n\\item By the first item above, $\\Theta\\subseteq K_{\\mathcal{M}}(\\Theta)$, so it only remains to be shown that $K_{\\mathcal{M}}(\\Theta)\\subseteq \\Theta$. Given one $\\varphi\\in K_{\\mathcal{M}}(\\Theta)$, for any $\\nu\\in\\mathcal{F}$ such that $\\nu(\\theta)\\in D$, for all $\\theta\\in\\Theta$, $\\nu(\\varphi)\\in D$.\n\nNow suppose $\\nu\\in\\mathcal{F}$ satisfies that $\\nu(\\gamma)\\in D$, for every $\\gamma\\in\\Gamma$: then, since $\\Theta=K_{\\mathcal{M}}(\\Gamma)$, $\\nu(\\theta)\\in D$ for all $\\theta\\in\\Theta$, and therefore $\\nu(\\varphi)\\in D$. It follows that $\\varphi\\in K_{\\mathcal{M}}(\\Gamma)=\\Theta$ and $K_{\\mathcal{M}}(\\Theta)=\\Theta$.\n\\end{enumerate}\n\\end{proof}\n\nGiven a class $\\mathbb{M}$ of RNmatrices, we can then consider the closure of a set $\\Gamma\\subseteq F(\\Sigma, X)$ under $\\mathbb{M}$, that is, $\\varphi\\in K_{\\mathbb{M}}(\\Gamma)$\\label{KMMGamma} if and only if $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$. Clearly \n\\[K_{\\mathbb{M}}(\\Gamma)=\\bigcap_{\\mathcal{M}\\in\\mathbb{M}}K_{\\mathcal{M}}(\\Gamma).\\]\n\n\\begin{lemma}\nIf $K_{\\lambda}$ is a tarskian operator for every $\\lambda\\in\\Lambda$, $K$ defined as \n\\[K(\\Gamma)=\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Gamma)\\]\nis also tarskian.\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Since, for every $\\lambda\\in\\Lambda$ we have that $\\Gamma\\subseteq K_{\\lambda}(\\Gamma)$, we have that $\\Gamma\\subseteq \\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Gamma)=K(\\Gamma)$.\n\n\\item If $\\Theta\\subseteq \\Gamma$, we have $K_{\\lambda}(\\Theta)\\subseteq K_{\\lambda}(\\Gamma)$ for every $\\lambda\\in\\Lambda$, so \n\\[K(\\Theta)=\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Theta)\\subseteq\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Gamma)=K(\\Gamma).\\]\n\n\\item Finally, if $\\Theta=K(\\Gamma)$, then $\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Gamma)=\\Theta$, so that $\\Theta\\subseteq K_{\\lambda}(\\Gamma)$ for every $\\lambda\\in\\Lambda$: clearly $\\Theta\\subseteq K(\\Theta)$, so it remains for us to show that $K(\\Theta)\\subseteq \\Theta$.\n\nLet us denote $K_{\\lambda}(\\Gamma)$ by $\\Theta_{\\lambda}$, and since $\\Theta\\subseteq K_{\\lambda}(\\Gamma)$, $K_{\\lambda}(\\Theta)\\subseteq K_{\\lambda}(\\Theta_{\\lambda})=\\Theta_{\\lambda}$, for every $\\lambda\\in\\Lambda$; notice that $\\Theta=\\bigcap_{\\lambda\\in\\Lambda}\\Theta_{\\lambda}$.\n\nThen, \n\\[K(\\Theta)=\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Theta)\\subseteq \\bigcap_{\\lambda\\in\\Lambda}\\Theta_{\\lambda}=\\Theta,\\]\nwhat finishes the proof.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theorem}\nGiven a class $\\mathbb{M}$ of RNmatrices, $K_{\\mathbb{M}}$ is a tarskian operator.\n\\end{theorem}\n\nIn other words, classes of restricted Nmatrices can, at most, describe tarskian logics, and from Theorem \\ref{2-valued RNmatrix} they indeed characterize all of these. So no non-tarskian logic can be characterized by either RNmatrices or their classes,\\footnote{Although slight generalizations of RNmatrices could possibly change this.} but the fact is we do not see that as a real problem: we are not looking exclusively for expressive power in our semantics, being efficiency a more desirable property. That is, the point of restricted Nmatrices will not be what they can express, but how easily will be to define and use them. Most importantly, however, is that RNmatrices will be able to provide decision methods through what are, essentially, truth-tables, where none were available before: one example, we will stress repeatedly, is that of $C_{1}$; although decidable, this system is not only not characterizable by finite matrices, but neither by finite sets of finite matrices, finite Nmatrices or finite sets of finite Nmatrices (\\cite{Avron}). And, despite all of this, this logic, and in fact all of da Costa's hierarchy, may be characterized by finite RNmatrices, what we will achieve in Chapter \\ref{Chapter5}.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{A brief history of RNmatrices}\\label{A brief history of RNmatrices}\n\nWe first developed RNmatrices by studying semantics for the logic $\\textbf{mbC}$: when analyzing Fidel structures for that very system, we hoped to simplify the ($3$-valued) Nmatrix for $\\textbf{mbC}$ to a matrix, by slightly altering its signature. By replacing the unary connective, standing for consistency, for a binary connective ($\\uparrow$) for incompatibility, and therefore creating the logics of incompatibility we study in Chapters \\ref{Chapter7}, \\ref{Chapter8} and \\ref{Chapter9}, we created a system (within $\\textbf{nbI}^{-}$, that is, $\\textbf{nbI}$ without the commutativity of $\\uparrow$) equivalent to $\\textbf{mbC}$, with simpler semantics, but which was also, unfortunately, not characterizable by finite matrices.\n\nWell, a very natural system related to $\\textbf{nbI}^{-}$ is $\\textbf{bI}$, which adds the commutativity of the binary incompatibility connective and subtracts the negation; rather unfortunate was then our discovery that this logic is not characterizable by, not only finite matrices, but rather finite Nmatrices as well. But we had a semantic for it almost ready, simply by adapting the decision method (a two-valued Nmatrix) for $\\textbf{bI}^{-}$: it was enough to demand that every valuation to be taken into consideration should satisfy $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$. \n\nOf course, within a semantics of matrices, or even Nmatrices, this is not permissible; but, inspired by Piochi's work on $\\mathcal{E}$-matrices (which do restrict valuations) and our previous knowledge of Nmatrices, we have coined what we begun to call restricted non-deterministic matrices, or RNmatrices. We quickly grew to realize that RNmatrices are actually recurrent in the history of non-classical logics, although not defined as such but rather as a specific solution to a difficult system: examples abound, including \\index{Bivaluation}bivaluations, Fidel structures, Kearn's $4$-valued semantics for modal logics, static semantics, PNmatrices, among others.\n\nTo our great surprise, we found out that RNmatrices semantics have quite recently started to resurface: Pawlowski and Urbaniak, in \\cite{Pawlowski, PawlowskUrbaniak}, had noticed, shortly before us, previous uses of the semantics of RNmatrces (although their nomenclature is, naturally, different), specially in the areas of informal provability and modal logic. We add to the topic a more comprehensive analysis of the literature and a more in depth study of the expressiveness of these semantics, in addition to the first explicit use of RNmatrices to paraconsistent logics.\n\nAnd it is important, here, to explain what we mean by the expressiveness of RNmatrix semantics: usually, the expressiveness of semantics refers to which logics can these semantics characterize, a concept that is distinctively easy to understand when our semantics are matricial in nature. However, according to Theorem \\ref{2-valued RNmatrix}, RNmatrices are, in a way, as expressive as possible, meaning that any tarskian logic can be characterized by a finite RNmatrix. Hence, we are actually more concerned about, first of all, the expressiveness of decidable RNmatrices, \\textit{i. e.} RNmatrices where the problem of identifying which valuations are restricted valuations is decidable, see Section \\ref{Decision methods} of Chapter \\ref{Chapter5} for more details; second, the expressiveness of other, related semantics, such as PNmatrices of Example \\ref{example of PNMatrices} below, which we know to be no more expressive than RNmatrices (that is, all logics characterized by PNmatrices can also be characterized by RNmatrices), but are not sure whether they are as expressive as RNmatrices (meaning, if PNmatrices characterize all tarskian logics, as RNmatrices do). \n\n\n\n\n\n\n\n\n\\begin{example}\\label{Example Kearns}\nAs we have already discussed, Dugundji proved that the logics between the modal systems $\\textbf{S1}$ and $\\textbf{S5}$ can not be characterized by single, finite matrices; to circumvent that, J. Kearns\\index{Kearns} (\\cite{Kearns}) proposed a (four-valued) Nmatrix semantics for the modal $\\textbf{T}$, $\\textbf{S4}$ and $\\textbf{S5}$ for which just some specific valuations can be considered, what clearly characterizes these as RNmatrices semantics. Kearns idea proved itself to be very popular, and generalizations can be found in, \\textit{exempli gratia}, \\cite{CCP} (and \\cite{CCPErr}) and \\cite{OmoriSkurt}.\n\nUsing a more modern approach to Kearns technique, as in \\cite{CCP}, we proceed as follows:\\label{Semantics, Kearns} consider the universe $\\{T, t, f, F\\}$; operations are defined as necessary for each logic $\\textbf{L}\\in\\{\\textbf{T}, \\textbf{S4}, \\textbf{S5}\\}$. Consider the set $Val^{\\textbf{L}}$ of all valuations for $\\textbf{L}$; we define $Val_{k}^{\\textbf{L}}\\subseteq Val^{\\textbf{L}}$ by induction as $Val_{0}^{\\textbf{L}}=Val^{\\textbf{L}}$ and, for $k\\in\\mathbb{N}$,\\[Val_{k+1}^{\\textbf{L}}=\\{\\nu\\in Val_{k}^{\\textbf{L}} : \\text{for every formula $\\alpha$, $Val_{k}^{\\textbf{L}}(\\alpha)\\subseteq D$ implies $\\nu(\\alpha)=T$}\\},\\]\nwhere $Val_{k}^{\\textbf{L}}(\\alpha)=\\{\\mu(\\alpha) : \\mu\\in Val_{k}^{\\textbf{L}}\\}$. We then define the restricted valuations as $\\mathcal{F}_{\\textbf{L}}=\\bigcap_{k\\in\\mathbb{N}}Val_{k}^{\\textbf{L}}$, and Kearns proved that $\\vdash_{\\textbf{L}}\\alpha$ if, and only if, $\\nu(\\alpha)=T$ for every $\\nu\\in\\mathcal{F}_{\\textbf{L}}$, making his semantics for $\\textbf{L}$ correspond to the RNmatrix $\\mathcal{K}_{\\textbf{L}}=(\\mathcal{A}_{\\textbf{L}}, \\{T\\}, \\mathcal{F}_{\\textbf{L}})$, for $\\mathcal{A}_{\\textbf{L}}$ the multialgebra for $\\textbf{L}$, with universe $\\{T, t, f, F\\}$, whose precise definition we omitted.\n\nNow, we can also prove $\\mathcal{K}_{\\textbf{L}}$\\label{KL} is structural: through an induction on $k$, one shows that, for every valuation $\\nu\\in Val_{k}^{\\textbf{L}}$ and every substitution $\\rho$, $\\nu\\circ\\rho\\in Val_{k}^{\\textbf{L}}$, being the case $k=0$ trivially true; so, suppose the result holds for a certain $k\\in\\mathbb{N}$. Given $\\nu\\in Val_{k+1}^{\\textbf{L}}$ and a substitution $\\rho$, by definition of $Val_{k+1}^{\\textbf{L}}$ we have $\\nu\\in Val_{k}^{\\textbf{L}}$ and, by induction hypothesis, $\\nu\\circ\\rho\\in Val_{k}^{\\textbf{L}}$. Let $\\alpha$ be a formula satisfying $Val_{k}^{\\textbf{L}}(\\alpha)\\subseteq D$: for every $\\mu\\in Val_{k}^{\\textbf{L}}$, $\\mu\\circ\\rho\\in Val_{k}^{\\textbf{L}}$ (again by induction hypothesis), meaning that $\\mu\\circ\\rho(\\alpha)=\\mu(\\rho(\\alpha))\\in D$, and therefore $Val_{k}^{\\textbf{L}}(\\rho(\\alpha))\\subseteq D$.\n\nThis implies, since $\\nu\\in Val_{k+1}^{\\textbf{L}}$, that $\\nu\\circ\\rho(\\alpha)=\\nu(\\rho(\\alpha))=T$, and henceforth $\\nu\\circ\\rho\\in Val_{k+1}^{\\textbf{L}}$. Obviously this proves that $\\nu\\in Val_{k}^{\\textbf{L}}$ implies $\\nu\\circ\\rho\\in Val_{k}^{\\textbf{L}}$ for all $k\\in\\mathbb{N}$: so, if $\\nu\\in \\mathcal{F}_{\\textbf{L}}$ and $\\rho$ is a substitution, for every $k\\in \\mathbb{N}$, $\\nu\\in Val_{k}^{\\textbf{L}}$ implies $\\nu\\circ\\rho\\in Val_{k}^{\\textbf{L}}$ (again, for every $k\\in\\mathbb{N}$), and so $\\nu\\circ\\rho\\in \\bigcap_{k\\in\\mathbb{N}}Val_{k}^{\\textbf{L}}=\\mathcal{F}_{\\textbf{L}}$.\n\\end{example}\n\n\\begin{example}\nThe first proof of the decidability of da Costa's calculi $C_{n}$ (see Chapter \\ref{Chapter5} for a definition of these logics) is due to Fidel, which in 1977 created a new class of structures, both algebraic and relational, to provide precisely such a demonstration in \\cite{Fidel3}: these objects are now known as \\textit{Fidel structures}\\index{Fidel structure}. Essentially, a \\index{Fidel structure for $C_{n}$}Fidel structure for $C_{n}$, which in this presentation are equipped with a connective $(n)$, is a triple $\\mathcal{N}=(\\mathcal{A}, \\{N_{a}\\}_{a\\in A}, \\{N_{a}^{(n)}\\}_{a\\in A})$ such that:\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is a Boolean algebra with universe $A$;\n\\item for every $a\\in A$, $N_{a}$\\label{Na} and $N_{a}^{(n)}$ are subsets of $A$ (and therefore unary relations indexed by $A$) satisfying certain desired properties.\n\\end{enumerate}\nA valuation over a Fidel structure $\\mathcal{N}$ for $C_{n}$ is a function $\\nu$ from the formulas of $C_{n}$ to $A$ satisfying, among other conditions, that $\\nu(\\alpha\\#\\beta)=\\nu(\\alpha)\\#\\nu(\\beta)$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, \n\\[\\nu(\\neg\\alpha)\\in N_{\\nu(\\alpha)}\\quad\\text{and}\\quad\\nu(\\alpha^{(n)})\\in N_{\\nu(\\alpha)}^{(n)}.\\]\nIf, for every element $a$ of $\\mathcal{A}$, we define $\\neg a=N_{a}$ and $n(a)=N_{a}^{(n)}$, we have enriched $\\mathcal{A}$ to become a multialgebra, which we denote by $\\mathcal{A}_{\\mathcal{N}}^{n}$; it is not difficult to see, then, that the consequence operator induced by the Fidel structure $\\mathcal{N}$ equals the one produced by the RNmatrix $\\mathcal{M}_{\\mathcal{N}}^{n}=(\\mathcal{A}_{\\mathcal{N}}^{n}, D, \\mathcal{F}_{\\mathcal{N}}^{n})$, for $D=\\{1\\}$ and $\\mathcal{F}_{\\mathcal{N}}^{n}$ the set of valuations for $C_{n}$ over $\\mathcal{N}$.\n\nThus, we understand Fidel structures as one of the earliest, and best understood, applications of RNmatrices; more important, however, is its success in surviving the test of time, being a methodology that is still relevantly applied today. See, for an example, the characterizations of $\\textbf{LFI}$'s such as $\\textbf{mbC}$, $\\textbf{mbCcl}$ and $\\textbf{CILA}$ in \\cite{ParLog} through Fidel structures, all of which may be recast as RNmatrices, one may add.\n\nNow, to be more precise, \\cite{Fidel3} uses a Fidel structure called $\\textbf{C}$ over the two-element Boolean algebra $\\textbf{2}$ as a decision procedure for all of $C_{n}$: that way, for any $n\\geq 1$, the multialgebras $\\mathcal{A}_{\\textbf{C}}^{n}$ are the same and satisfy $\\neg 0=n(0)=\\{1\\}$ and $\\neg1=n(1)=\\{0,1\\}$; this forces most of the semantical power of these structures to lie in their valuations, making of $\\mathcal{F}_{\\textbf{C}}^{n}$ rather complicated sets (look, for an example of a similar nature, to $\\textbf{2}(\\mathfrak{L})$ in Theorem \\ref{2-valued RNmatrix}). In a certain sense, when we present our own RNmatrices for $C_{n}$ in Chapter \\ref{Chapter5} we make a compromise: we allow for the underlying multialgebras to have larger universes (with $n+2$ elements for $C_{n}$), but in return we have far simpler valuations.\n\\end{example}\n\n\n\\begin{example}\nIn the work of Avron and Konikowska \\cite{AK:05}, valuations over Nmatrices induce semantics which they called dynamic semantics\\index{Semantics, Dynamic} over Nmatrices; as opposed to these, they have also considered semantics which restrict the usual valuations, what they have baptized as static semantics\\index{Semantics, Static}. Essentially, given an Nmatrix $\\mathcal{M}=(\\mathcal{A}, D)$, its static semantics is given by $\\mathcal{M}^{s}=(\\mathcal{A}, D, \\mathcal{F}^{s}_{\\mathcal{M}})$\\label{Ms}, for $\\mathcal{F}^{s}_{\\mathcal{M}}$ the set of valuations $\\nu$ over $\\mathcal{M}$ satisfying that: for an $n$-ary connective $\\sigma$, and formulas $\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n})$ and $\\sigma(\\beta_{1}, \\dotsc  , \\beta_{n})$,  $\\nu(\\alpha_{i})=\\nu(\\beta_{i})$, for every $i\\in\\{1, \\dotsc  , n\\}$, implies\n\\[\\nu(\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}))=\\nu(\\sigma(\\beta_{1}, \\dotsc  , \\beta_{n})).\\]\nIt is easy to prove that the RNmatrices $\\mathcal{M}^{s}$ are all structural.\n\\end{example}\n\n\\begin{example}\\label{example of PNMatrices}\nPNmatrices\\index{PNmatrix}\\index{Matrix, Partial non-deterministic}, whose first appearance in the literature was in \\cite{Baaz}, generalize Nmatrices by replacing multialgebras for partial multialgebras (that is, algebras of relations). Our brief presentation here, however, is closer to that of \\cite{CalMar}: a partial $\\Sigma$-multialgebra is a pair $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$ such that, if $\\sigma\\in\\Sigma_{n}$, $\\sigma_{\\mathcal{A}}$ is a function of the form $\\sigma_{\\mathcal{A}}:A^{n}\\rightarrow\\mathcal{P}(A)$ (no longer $\\mathcal{P}(A)\\setminus\\{\\emptyset\\}$ as in the case of multialgebras). A valuation for a partial $\\Sigma$-multialgebra $\\mathcal{A}$, as described, is then a map $\\nu: F(\\Sigma, \\mathcal{V})\\rightarrow A$ such that\n\\[\\nu(\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{n}))\\in\\sigma_{\\mathcal{A}}(\\nu(\\alpha_{1}), \\dotsc  , \\nu(\\alpha_{n})),\\]\nfor every $\\sigma\\in\\Sigma_{n}$ and formulas $\\alpha_{1}, \\dotsc  , \\alpha_{n}$; notice that, if $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=\\emptyset$, then there cannot exist a valuation $\\nu$ and formulas $\\alpha_{1}$ trough $\\alpha_{n}$ such that $\\nu(\\alpha_{i})=a_{i}$, for $1\\leq i\\leq n$, given $\\nu$ is supposed to be a total function. Finally, given a partial $\\Sigma$-multialgebra $\\mathcal{A}$ and a subset $D$ of its universe, $\\mathcal{M}=(\\mathcal{A}, D)$ is a PNmatrix: given formulas $\\Gamma\\cup\\{\\varphi\\}$ on $\\Sigma$, we say $\\Gamma$ proves $\\varphi$ according to $\\mathcal{M}$, and write $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$, whenever, for every valuation $\\nu$ for $\\mathcal{A}$, $\\nu(\\Gamma)\\subseteq D$ implies $\\nu(\\varphi)\\in D$.\n\nWhat we can prove, however, is that the semantics induced by PNmatrices can be also induced by RNmatrices, making of the latter a more expressive technique. Given a PNmatrix $\\mathcal{M}=(\\mathcal{A}, D)$, over $\\Sigma$, we consider the $\\Sigma$-multialgebra $\\mathcal{A}^{\\emptyset}=(A\\cup\\{o\\}, \\{\\sigma_{\\mathcal{A}^{\\emptyset}}\\}_{\\sigma\\in\\Sigma})$, for an element $o\\notin A$, such that\n\\[\\sigma_{\\mathcal{A}^{\\emptyset}}(a_{1}, \\dotsc  , a_{n})=\n\\begin{cases*}\n\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n}) & if $a_{1}, \\dotsc  , a_{n}\\in A$ and $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\neq\\emptyset$,\\\\\n\\{o\\} & otherwise,\n\\end{cases*}\\]\nfor all $a_{1}, \\dotsc  , a_{n}\\in A\\cup\\{o\\}$. By defining\n\\[\\mathcal{F}=\\{\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}^{\\emptyset}\\ :\\ o\\notin\\nu(F(\\Sigma, \\mathcal{V}))\\},\\]\nwhere $\\nu(F(\\Sigma, \\mathcal{V}))=\\{\\nu(\\alpha) : \\alpha\\in F(\\Sigma, \\mathcal{V})\\}$, one sees that the valuations for $\\mathcal{A}$ correspond, more or less, to $\\mathcal{F}$, that is, the set of valuations for $\\mathcal{A}^{\\emptyset}$ without $o$ in their range; then $\\mathcal{M}^{\\emptyset}=(\\mathcal{A}^{\\emptyset}, D, \\mathcal{F})$\\label{Mo} has the same deduction operator as $\\mathcal{A}$.\n\\end{example}\n\n\\begin{example}\\label{PTS}\nPossible-translations semantics were first defined in \\cite{PTS-first}, but here we use an approach similar to that found in \\cite{PTS-defined}, the difference being that we only focus on zeroth order logics: take signatures $\\Sigma$ and $\\Sigma^{*}$ and its respective languages, $\\mathcal{L}_{\\Sigma}$ and $\\mathcal{L}_{\\Sigma^{*}}$; a translation between logics $\\mathscr{L}=(\\mathcal{L}_{\\Sigma}, \\vdash_{\\mathscr{L}})$ and $\\mathscr{L}^{*}=(\\mathcal{L}_{\\Sigma^{*}}, \\vdash_{\\mathscr{L}^{*}})$ is any function $t:\\mathcal{L}_{\\Sigma}\\rightarrow \\mathcal{L}_{\\Sigma^{*}}$ such that, for any set of formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\mathcal{L}_{\\Sigma}$,\n\\[\\text{$\\Gamma\\vdash_{\\mathscr{L}}\\varphi$\\quad implies\\quad $t(\\Gamma)\\vdash_{\\mathscr{L}^{*}}t(\\varphi)$,}\\]\nwhere $t(\\Gamma)=\\{t(\\gamma)\\ :\\ \\gamma\\in\\Gamma\\}$. Fixed a system $\\mathscr{L}$ over $\\Sigma$, a possible-translations semantics\\index{Possible-translations semantics} for $\\mathscr{L}$ is then a pair $\\mathcal{PT}=(\\textbf{Log}, \\textbf{Tr})$ where $\\textbf{Log}=\\{\\mathscr{L}_{i}\\}_{i\\in I}$ are logics over possibly distinct signatures $\\Sigma^{i}$, and $\\textbf{Tr}=\\{t_{i}\\}_{i\\in I}$ are translations from $\\mathscr{L}$ to $\\mathscr{L}_{i}$. The possible-translations semantics leads to a consequence operator $\\Vdash_{\\mathcal{PT}}$ on the formulas of the signature $\\Sigma$ whenever we define, for a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over $\\Sigma$,\n\\[\\text{$\\Gamma\\Vdash_{\\mathcal{PT}}\\varphi$\\quad if and only if\\quad $t_{i}(\\Gamma_{i})\\vdash_{\\mathscr{L}_{i}}t_{i}(\\varphi)$,\\quad for all $i\\in I$.}\\]\nOf course, $\\mathscr{L}$ is characterized by $\\mathcal{PT}$ when $\\Gamma\\vdash_{\\mathscr{L}}\\varphi$ iff $\\Gamma\\Vdash_{\\mathcal{PT}}\\varphi$; intuitively, a logic is characterized by a given possible-translations semantics whenever $\\mathscr{L}$ can be defined by specific properties of the $\\mathscr{L}_{i}$, which the translations $t_{i}$ combine together to obtain $\\Vdash_{\\mathcal{PT}}$. \n\nOf course, RNmatrices cannot possibly be as expressive as even this weaker definition of possible-translations semantics, given that they can only deal with one signature at a time,\\footnote{It seems, however, possible to combine RNmatrices over different signatures as long as one has at their disposal adequate maps between signatures, sometimes also known as translations.} but a connection between the two semantics is self-evident: both of them shift the expressive power of semantics, from structures to maps themselves; specifically, translations in the case of possible-translations semantics, and restricted valuations in the case of RNmatrices.\n\nHowever, consider a specific case of possible-translations semantics: suppose all logics of $\\textbf{Log}$ have the same signature $\\Sigma$ as $\\mathscr{L}$, and that all of them are characterized by RNmatrices $\\mathcal{M}_{i}$ or, what is equivalent, that all of them are tarskian (and, therefore, so is $\\mathscr{L}$); then $\\mathscr{L}$ is characterized by a class of of RNmatrices in a straightforward way. In fact, if $\\mathcal{M}_{i}=(\\mathcal{A}_{i}, D_{i}, \\mathcal{F}_{i})$, then \n\\[\\text{$\\Gamma\\vdash_{\\mathscr{L}_{i}}\\varphi$\\quad iff \\quad $\\nu(\\Gamma)\\subseteq D_{i}$\\quad implies\\quad $\\nu(\\varphi)\\in D_{i}$\\quad for every $\\nu\\in\\mathcal{F}_{i}$;}\\]\nby defining $\\mathcal{F}_{i}^{t}=\\{\\nu\\circ t_{i}\\ :\\ \\nu\\in\\mathcal{F}_{i}\\}$ and $\\mathcal{M}_{i}^{t}=(\\mathcal{A}_{i}, D_{i}, \\mathcal{F}_{i}^{t})$, it is easy to see that $t_{i}(\\Gamma)\\vdash_{\\mathscr{L}_{i}}t_{i}(\\varphi)$ if, and only if, $\\Gamma\\vDash_{\\mathcal{M}_{i}^{t}}\\varphi$. By making $\\mathbb{M}=\\{\\mathcal{M}_{i}^{t}\\}_{i\\in I}$, we see that $\\Gamma\\Vdash_{\\mathcal{PT}}\\varphi$ is equivalent to $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$, and so $\\mathscr{L}$ is characterized by $\\mathbb{M}$.\n\n\n\n\n\n\n\\end{example}\n\n\n\n\n\\subsection{Structurality}\\label{Structurality}\n\n\\begin{definition}\nFor any subsemigroup $\\mathcal{E}$ of the semigroup $\\textit{End}(F(\\Sigma, \\mathcal{V}))$ of endomorphisms on $F(\\Sigma, \\mathcal{V})$, we say an operator $K:\\mathcal{P}(F(\\Sigma, \\mathcal{V}))\\rightarrow\\mathcal{P}(F(\\Sigma, \\mathcal{V}))$ is $\\mathcal{E}$-structural\\index{Structural, $\\mathcal{E}$-} when \n\\[\\{\\sigma(\\varphi)\\ :\\  \\varphi\\in K(\\Gamma)\\}\\subseteq K(\\{\\sigma(\\varphi)\\ :\\  \\varphi\\in\\Gamma\\})\\]\nfor every $\\sigma\\in\\mathcal{E}$, or as we shall write it, $\\sigma K(\\Gamma)\\subseteq K(\\sigma\\Gamma)$; the operator is said to be structural when it is $\\textit{End}(F(\\Sigma, \\mathcal{V}))$-structural. \n\\end{definition}\n\nStructurality is rather important when dealing with logics arising from a Hilbert calculus given that instances of axiom schemata and rules of inference are obtained by application of endomorphisms; so, when studying a deduction operator, it is important to understand how structural it is. While most of the systems we study here are structural, not only $\\mathcal{E}$-structural for a proper subsemigroup $\\mathcal{E}$, developing a general theory of structurality for RNmatrices, as Piochi did for Rmatrices in \\cite{Piochi}, \\cite{Piochi2} and \\cite{Piochi3}, can have useful applications on non-structural logics.\n\n\\begin{proposition}\nIf $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ is an RNmatrix such that, for every $\\nu\\in\\mathcal{F}$ and $\\sigma\\in\\mathcal{E}$, $\\nu\\circ\\sigma\\in\\mathcal{F}$, then $K_{\\mathcal{M}}$ is $\\mathcal{E}$-structural.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\Gamma$ be a subset of $F(\\Sigma, \\mathcal{V})$ and $\\varphi\\in K_{\\mathcal{M}}(\\Gamma)$: we must prove that, for a given $\\sigma\\in \\mathcal{E}$, $\\sigma(\\varphi)$ is in $K_{\\mathcal{M}}(\\sigma\\Gamma)$, or what is equivalent, that $\\sigma\\Gamma\\vDash_{\\mathcal{M}}\\sigma(\\varphi)$. So, let $\\nu\\in\\mathcal{F}$ be a valuation satisfying $\\nu(\\sigma(\\gamma))=\\nu\\circ\\sigma(\\gamma)\\in D$, $\\forall\\gamma\\in\\Gamma$, and we must prove that $\\nu(\\sigma(\\varphi))=\\nu\\circ\\sigma(\\varphi)\\in D$.\n\nBy our hypothesis, $\\nu\\circ\\sigma\\in\\mathcal{F}$, and since $\\nu\\circ\\sigma(\\gamma)\\in D$ for every $\\gamma\\in\\Gamma$ and $\\varphi\\in K_{\\mathcal{M}}(\\Gamma)$ (meaning $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$), we find that $\\nu\\circ\\sigma(\\varphi)\\in D$, what ends the proof.\n\\end{proof}\n\n\\begin{lemma}\nIf all operators $K_{\\lambda}$, for $\\lambda\\in\\Lambda$, are $\\mathcal{E}$-structural, then so it is $K$ defined by \n\\[K(\\Gamma)=\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\Gamma),\\]\nfor every $\\Gamma\\subseteq F(\\Sigma, \\mathcal{V})$.\n\\end{lemma}\n\n\\begin{proof}\nTake $\\sigma\\in\\mathcal{E}$ and $\\varphi\\in K(\\Gamma)$: we have that $\\varphi\\in K_{\\lambda}(\\Gamma)$ for every $\\lambda\\in\\Lambda$, and then $\\sigma(\\varphi)\\in K_{\\lambda}(\\sigma\\Gamma)$, since each $K_{\\lambda}$ is structural.\n\nIt follows that $\\sigma(\\varphi)\\in\\bigcap_{\\lambda\\in\\Lambda}K_{\\lambda}(\\sigma\\Gamma)=K(\\sigma\\Gamma)$, and so $\\sigma K(\\Gamma)\\subseteq K(\\sigma\\Gamma)$.\n\\end{proof}\n\n\\begin{theorem}\nGiven a class $\\mathbb{M}$ of RNmatrices $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ such that, for every $\\nu\\in\\mathcal{F}$ and $\\sigma\\in\\mathcal{E}$, $\\nu\\circ\\sigma\\in\\mathcal{F}$, $K_{\\mathbb{M}}$ is a $\\mathcal{E}$-structural operator.\n\\end{theorem}\n\n\nNow, for any tarskian logic $\\mathfrak{L}$, we remember for an instant the two-valued RNmatrix $\\textbf{2}(\\mathfrak{L})$ which characterizes $\\mathfrak{L}$ from Theorem \\ref{2-valued RNmatrix}: notice that, if $\\mathfrak{L}$ is $\\mathcal{E}$-structural, so is the deduction operator induced by $\\textbf{2}(\\mathfrak{L})$. To see that, take an endomorphism $\\sigma$ in $\\mathcal{E}$; if $\\Gamma\\vDash_{\\textbf{2}(\\mathfrak{L})}\\varphi$, $\\Gamma\\vdash\\varphi$, and from the fact that $\\mathfrak{L}$ is $\\mathcal{E}$-structural, $\\{\\sigma(\\gamma)\\ :\\  \\gamma\\in\\Gamma\\}\\vdash\\sigma(\\varphi)$; this means, of course, that \n\\[\\{\\sigma(\\gamma)\\ :\\ \\gamma\\in\\Gamma\\}\\vDash_{\\textbf{2}(\\mathfrak{L})}\\sigma(\\varphi),\\]\nwhat proves the result. Of course, if $\\mathfrak{L}$ is structural, so is the operator of $\\textbf{2}(\\mathfrak{L})$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Examples: characterizing some logics with RNmatrices}\\label{Examples: characterizing some logics with RNmatrices}\n\nWe would like to know that our approach using restricted Nmatrices has something to offer that other approaches don't. It is a classical result by Avron, found in \\cite{Avron}, that no finite Nmatrix can characterize the logics between $\\mbCcl$ and $\\CILA$, both included: the problem arises specifically from axiom $\\cl$. This is an important Dugundji-like uncharacterizability theorem, that limits greatly the expressive power of Nmatrices. We will show, by offering examples of such finite RNmatrices, that no such restriction exists for finite RNmatrices over those logics.\n\n\\subsection{$\\mbCcl$}\\label{RNmatrix for mbCcl}\n\nThis logic, together with the closely related $\\mbCci$, were first defined, under these names, in \\cite{ParLog}; however, the system $\\textbf{B}[\\{\\textbf{i1}, \\textbf{i2}\\}]$\\label{Bi1i2}, equivalent to $\\mbCci$, appeared already in \\cite{Avron2}, while $\\textbf{Bi}$\\label{Bi} and $\\textbf{Bl}$\\label{Bl}, equivalent to respectively $\\mbCci$ and $\\mbCcl$, had also been defined in \\cite{Avron}.\n\nConsider the $\\Sigma_{\\textbf{LFI}}$-multialgebra $\\mathcal{A}_{\\mbCciw}$ with universe $\\{F, t, T\\}$ and operations given by the tables below.\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\vee$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$t$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Disjunction}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\wedge$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{F\\}$ & $\\{F\\}$ & $\\{F\\}$ \\\\ \\hline\n$t$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Conjunction}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\neg$ \\\\ \\hline\n$F$ & $\\{t, T\\}$\\\\ \\hline\n$t$ & $\\{t, T\\}$\\\\ \\hline\n$T$ & $\\{F\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for $\\neg$}\n\\end{minipage}\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\rightarrow$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$t$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Implication}\n\\end{minipage}\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\circ$ \\\\ \\hline\n$F$ & $\\{t, T\\}$\\\\ \\hline\n$t$ & $\\{F\\}$\\\\ \\hline\n$T$ & $\\{t,T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for $\\circ$}\n\\end{minipage}\n\\end{figure}\n\nWhen making $D=\\{t, T\\}$, is it is shown in Corollary $6.5.5$ of \\cite{ParLog} that the Nmatrix $\\mathcal{M}_{\\mbCciw}=(\\mathcal{A}_{\\mbCciw}, D)$\\label{MmbCciw} is adequate for $\\mbCciw$, that is, for any set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\textbf{LFI}}$ we have $\\Gamma\\vdash_{\\mbCciw}\\varphi$ if, and only if, $\\Gamma\\vDash_{\\mathcal{M}_{\\mbCciw}}\\varphi$.\n\nNow consider the restricted Nmatrix \\label{MmbCcl}\n\\[\\mathcal{M}_{\\mbCcl}=(\\mathcal{A}_{\\mbCciw}, D, \\mathcal{F}_{\\mbCcl})\\]\nsuch that $\\mathcal{F}_{\\mbCcl}$ is the set of homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\mathcal{A}_{\\mbCcl}$ satisfying that, if $\\nu(\\alpha)=t$, then $\\nu(\\alpha\\wedge\\neg\\alpha)=T$. Clearly such an RNmatrix is structural, since for any endomorphism $\\sigma$ of $\\textbf{F}(\\Sigma_{\\textbf{LFI}},\\mathcal{V})$ we have that if $\\nu\\circ\\sigma(\\alpha)=t$, then $\\nu(\\sigma(\\alpha))=t$, meaning $\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))=T$ and, therefore, $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)=T$.\n\nIt is easy to see $\\mathcal{M}_{\\mbCcl}$ models the axiom schemata and rules of inference of\\\\ $\\mbCciw$, but it is also true that, given an instance $\\psi=\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\circ\\alpha$ of $\\cl$, we have $\\vDash_{\\mathcal{M}_{\\mbCcl}}\\psi$: assume that, for $\\nu\\in\\mathcal{F}_{\\mbCcl}$, $\\nu(\\psi)\\notin D$, meaning that $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))\\in D$ and $\\nu(\\circ\\alpha)=F$, which in turn implies $\\nu(\\alpha)=t$.\n\nSince $\\nu\\in\\mathcal{F}_{\\mbCcl}$, $\\nu(\\alpha)=t$ implies $\\nu(\\alpha\\wedge\\neg\\alpha)=T$, and therefore $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=F$, reaching a contradiction. We have, then, that $\\nu(\\psi)\\in D$, for any $\\nu\\in\\mathcal{F}_{\\mbCcl}$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\mbCcl$, if $\\Gamma\\vdash_{\\mbCcl}\\varphi$ then $\\Gamma\\vDash_{\\mathcal{M}_{\\mbCcl}}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nIf $\\Gamma\\vdash_{\\mbCcl}\\varphi$, there exists a demonstration $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ of $\\varphi$ from $\\Gamma$, with $\\alpha_{n}=\\varphi$.\n\nLet $\\nu\\in\\mathcal{F}_{\\mbCcl}$ be a valuation satisfying that that $\\nu(\\Gamma)\\subseteq D$: we want to prove that, in this case, $\\nu(\\varphi)\\in D$; so we prove, by induction, that $\\alpha_{1}$ through $\\alpha_{n}$ have image in $D$ under $\\nu$, and therefore $\\nu(\\varphi)=\\nu(\\alpha_{n})=1$.\n\nThe formula $\\alpha_{1}$ is either an instance of an axiom, when $\\nu(\\alpha_{1})\\in D$ since all instances of axioms have image in $D$ through any elements of $\\mathcal{F}_{\\mbCcl}$, or $\\alpha_{1}$ is a premise, that is, an element of $\\Gamma$, and since $\\nu(\\Gamma)\\subseteq D$ we have that $\\nu(\\alpha_{1})\\in D$.\n\nSuppose then that $\\nu(\\alpha_{1}), \\dotsc  , \\nu(\\alpha_{i-1})\\in D$, and we have three cases to consider:\n\\begin{enumerate}\n\\item if $\\alpha_{i}$ is an instance of an axiom, as mentioned above $\\nu(\\alpha_{i})\\in D$;\n\\item if $\\alpha_{i}$ is a premise, $\\nu(\\alpha_{i})\\in D$ since $\\alpha_{i}\\in\\Gamma$ and $\\nu(\\Gamma)\\subseteq D$;\n\\item if there are $\\alpha_{j}$ and $\\alpha_{k}$ with $j,k<i$ such that $\\alpha_{j}=\\alpha_{k}\\rightarrow\\alpha_{i}$ or $\\alpha_{k}=\\alpha_{j}\\rightarrow\\alpha_{i}$, since $\\nu(\\alpha_{j}), \\nu(\\alpha_{k})\\in D$ we find in both cases that $\\nu(\\alpha_{i})\\in D$, what ends the proof.\n\\end{enumerate}\n\\end{proof}\n\nReciprocally, consider that a map $\\nu:F(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\{0,1\\}$ is said to be a \\index{Bivaluation for $\\mbCciw$}bivaluation for $\\mbCciw$ when it satisfies\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\vee\\beta)=1$ if and only if $\\nu(\\alpha)=1$ or $\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\wedge\\beta)=1$ if and only if $\\nu(\\alpha)=\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\rightarrow\\beta)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha)=0$ implies $\\nu(\\neg\\alpha)=1$;\n\\item $\\nu(\\circ\\alpha)=1$ if and only if $\\nu(\\alpha)\\neq\\nu(\\neg\\alpha)$;\n\\end{enumerate}\n\nA \\index{Bivaluation for $\\mbCcl$}bivaluation for $\\mbCcl$ is simply a bivaluation for $\\mbCciw$ such that, in addition to the previous conditions,\n\\[\\nu(\\circ\\alpha)=0\\quad\\text{implies}\\quad\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=0.\\]\n\n If we define, for a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\textbf{LFI}}$, that $\\Gamma\\vDash_{\\mbCcl}\\varphi$ whenever, for every bivaluation for $\\mbCcl$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$, $\\nu(\\varphi)=1$, it is proved in Theorem $3.3.28$ of \\cite{ParLog} that $\\Gamma\\vdash_{\\mbCcl}\\varphi$ if and only if $\\Gamma\\vDash_{\\mbCcl}\\varphi$.\n\nWe want to show $\\Gamma\\vDash_{\\mathcal{M}_{\\mbCcl}}\\varphi$ implies $\\Gamma\\vdash_{\\mbCcl}\\varphi$, so we will show instead that, given a bivaluation $\\nu$ for $\\mbCcl$, there exists a homomorphism $\\overline{\\nu}:\\textbf{F}(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\mathcal{A}_{\\mbCciw}$ which lies in $\\mathcal{F}_{\\mbCcl}$ and satisfies that $\\nu(\\alpha)=1$ if and only if $\\overline{\\nu}(\\alpha)\\in D$. \n\nThis way, when we assume $\\Gamma\\vDash_{\\mathcal{M}_{\\mbCcl}}\\varphi$, if $\\nu(\\Gamma)\\subseteq\\{1\\}$ we have $\\overline{\\nu}(\\Gamma)\\subseteq D$, and therefore $\\overline{\\nu}(\\varphi)\\in D$, what means that $\\nu(\\varphi)=1$, thus proving $\\Gamma\\vDash_{\\mbCcl}\\varphi$ or, what is equivalent, $\\Gamma\\vdash_{\\mbCcl}\\varphi$.\n\nSo, suppose $\\Gamma\\vDash_{\\mathcal{M}_{\\mbCcl}}\\varphi$ and let $\\nu:F(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\{0,1\\}$ be a bivaluation for $\\mbCcl$: we then consider the map $\\overline{\\nu}:F(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\{F,t,T\\}$ such that:\n\\begin{enumerate}\n\\item $\\overline{\\nu}(\\alpha)=F$ if and only if $\\nu(\\alpha)=0$ and $\\nu(\\neg\\alpha)=1$;\n\\item $\\overline{\\nu}(\\alpha)=t$ if and only if $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=1$;\n\\item $\\overline{\\nu}(\\alpha)=T$ if and only if $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=0$.\n\\end{enumerate}\n\nNotice $\\overline{\\nu}$ is well defined since $\\nu(\\alpha)=0$ implies $\\nu(\\neg\\alpha)=1$, and we therefore can not have $\\nu(\\alpha)=\\nu(\\neg\\alpha)=0$.\nThe proof of the following results is long and tedious, but rather straightforward, so we will prefer to wait to show similar proofs in the case of $\\CILA$, in Section \\ref{RNmatrix for CILA}.\n\n\\begin{theorem}\n$\\overline{\\nu}$ is a $\\Sigma_{\\textbf{LFI}}$-homomorphism between $\\textbf{F}(\\Sigma_{\\textbf{LFI}}, \\mathcal{V})$ and $\\mathcal{A}_{\\mbCcl}$.\n\\end{theorem}\n\n\\begin{theorem}\n$\\overline{\\nu}$ is in $\\mathcal{F}_{\\mbCcl}$.\n\\end{theorem}\n\n\n\n\n\n\\subsection{$\\CILA$}\\label{RNmatrix for CILA}\n\n\\label{CILA}This logic was defined in \\cite{CM}, and already in this study it was shown to be equivalent to da Costa's $C_{1}$; it is obtained over the signature $\\Sigma_{\\textbf{LFI}}$ by adding to the Hilbert calculus of $\\textbf{mbC}$ the axiom schemata\\label{cf}\n\\[\\tag{\\textbf{ci}}\\neg\\circ\\alpha\\rightarrow(\\alpha\\wedge\\neg\\alpha);\\]\n\\[\\tag{\\textbf{cl}}\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\circ\\alpha;\\]\n\\[\\tag{$\\textbf{ca}_{\\wedge}$}(\\circ\\alpha\\wedge\\circ\\beta)\\rightarrow\\circ(\\alpha\\wedge\\beta);\\]\n\\[\\tag{$\\textbf{ca}_{\\vee}$}(\\circ\\alpha\\wedge\\circ\\beta)\\rightarrow\\circ(\\alpha\\vee\\beta);\\]\n\\[\\tag{$\\textbf{ca}_{\\rightarrow}$}(\\circ\\alpha\\wedge\\circ\\beta)\\rightarrow\\circ(\\alpha\\rightarrow\\beta);\\]\n\\[\\tag{\\textbf{cf}}\\neg\\neg\\alpha\\rightarrow\\alpha.\\]\n\nConsider the $\\Sigma_{\\textbf{LFI}}$-multialgebra $\\mathcal{A}_{\\textbf{Cila}}$ given by the tables below.\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\vee$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{T\\}$ \\\\ \\hline\n$t$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{T\\}$ & $\\{t, T\\}$ & $\\{T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Disjunction}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\wedge$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{F\\}$ & $\\{F\\}$ & $\\{F\\}$ \\\\ \\hline\n$t$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Conjunction}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\neg$ \\\\ \\hline\n$F$ & $\\{T\\}$\\\\ \\hline\n$t$ & $\\{t, T\\}$\\\\ \\hline\n$T$ & $\\{F\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Negation}\n\\end{minipage}\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\rightarrow$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{T\\}$ & $\\{t, T\\}$ & $\\{T\\}$ \\\\ \\hline\n$t$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Implication}\n\\end{minipage}\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\circ$ \\\\ \\hline\n$F$ & $\\{T\\}$\\\\ \\hline\n$t$ & $\\{F\\}$\\\\ \\hline\n$T$ & $\\{T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Consistency}\n\\end{minipage}\n\\end{figure}\n\nWe then define the restricted Nmatrix $\\mathcal{M}_{\\textbf{Cila}}=(\\mathcal{A}_{\\textbf{Cila}}, D, \\mathcal{F}_{\\textbf{Cila}})$\\label{MCILA} such that $D=\\{t, T\\}$ and $\\mathcal{F}_{\\textbf{Cila}}$ is the set of homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\textbf{LFI}}, \\mathcal{V})\\rightarrow\\mathcal{A}_{\\CILA}$ with the property that, if $\\nu(\\alpha)=t$, then $\\nu(\\alpha\\wedge\\neg\\alpha)=T$.\n\n\nTo prove the soundness of our RNmatrix, we start by defining the logic $\\Ci$\\label{Ci}, obtained from $\\mbC$ by addition of the axiom schemata $\\ci$ and $\\cf$: by Theorem $6.5.24$ of \\cite{ParLog}, $\\Ci$ is characterized by the Nmatrix $\\mathcal{M}_{\\Ci}=(\\mathcal{A}_{\\Ci}, D)$, with $\\mathcal{A}_{\\Ci}$ the $\\Sigma_{\\textbf{LFI}}$-multialgebra given by the tables below.\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\vee$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$t$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Disjunction}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\wedge$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{F\\}$ & $\\{F\\}$ & $\\{F\\}$ \\\\ \\hline\n$t$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Conjunction}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\neg$ \\\\ \\hline\n$F$ & $\\{T\\}$\\\\ \\hline\n$t$ & $\\{t, T\\}$\\\\ \\hline\n$T$ & $\\{F\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Negation}\n\\end{minipage}\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\rightarrow$ & $F$ & $t$ & $T$ \\\\ \\hline\n$F$ & $\\{t, T\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$t$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n$T$ & $\\{F\\}$ & $\\{t, T\\}$ & $\\{t, T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Implication}\n\\end{minipage}\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\circ$ \\\\ \\hline\n$F$ & $\\{T\\}$\\\\ \\hline\n$t$ & $\\{F\\}$\\\\ \\hline\n$T$ & $\\{T\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Consistency}\n\\end{minipage}\n\\end{figure}\n\nSince $\\mathcal{A}_{\\CILA}$ is a submultialgebra of $\\mathcal{A}_{\\Ci}$, it is clear $\\mathcal{M}_{\\CILA}$ models Modus Ponens and those axiom schemata of $\\Ci$: it remains to be shown that this RNmatrix also models the axioms concerning propagation of consistency and $\\cl$.\n\n\\begin{enumerate}\n\\item So, take an instance $\\psi=(\\circ\\alpha\\wedge\\circ\\beta)\\rightarrow\\circ(\\alpha\\#\\beta)$ of $\\ca_{\\#}$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$: we have that, for a $\\nu\\in\\mathcal{F}_{\\CILA}$, $\\nu(\\psi)\\notin D$ if and only if $\\nu(\\circ\\alpha\\wedge\\circ\\beta)\\in D$ and $\\nu(\\circ(\\alpha\\#\\beta))=F$, which imply $\\nu(\\circ\\alpha), \\nu(\\circ\\beta)\\in D$ and $\\nu(\\alpha\\#\\beta)=t$. \n\nSince $\\circ$ always gives classical values (that is, either $F$ or $T$), we have $\\nu(\\circ\\alpha)=\\nu(\\circ\\beta)=T$, and therefore $\\nu(\\alpha), \\nu(\\beta)\\in \\{F, T\\}$, where we finally reach the desired contradiction: if both $\\nu(\\alpha)$ and $\\nu(\\beta)$ are classically valued, so is $\\nu(\\alpha\\#\\beta)$ from the tables for $\\mathcal{A}_{\\CILA}$, and therefore we must have $\\nu(\\psi)\\in D$.\n\n\\item Take an instance $\\psi=\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\circ\\alpha$ of $\\cl$, and assume that, for $\\nu\\in\\mathcal{F}_{\\CILA}$, $\\nu(\\psi)\\notin D$, meaning that $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))\\in D$ and $\\nu(\\circ\\alpha)=F$, which implies $\\nu(\\alpha)=t$.\n\nHowever, since $\\nu\\in\\mathcal{F}_{\\CILA}$, $\\nu(\\alpha)=t$ in turn implies $\\nu(\\alpha\\wedge\\neg\\alpha)=T$, and therefore $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=F$, a contradiction. We have, then, that $\\nu(\\psi)\\in D$.\n\\end{enumerate}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\CILA$, if $\\Gamma\\vdash_{\\CILA}\\varphi$ then $\\Gamma\\vDash_{\\mathcal{M}_{\\CILA}}\\varphi$.\n\\end{theorem}\n\nTo prove the completeness of our RNmatrix, we will detour through bivaluations.\n\n\\begin{definition}\nA \\index{Bivaluation for $\\CILA$}bivaluation for $\\CILA$ is a bivaluation for $\\mbCcl$ satisfying additionally:\n\\begin{enumerate}\n\\item $\\nu(\\alpha)=0$ implies $\\nu(\\neg\\neg\\alpha)=0$;\n\\item $\\nu(\\circ\\alpha)=\\nu(\\circ\\beta)=1$ implies $\\nu(\\circ(\\alpha\\#\\beta))=1$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$;\n\\item $\\nu(\\neg\\circ\\alpha)=1$ implies $\\nu(\\circ\\alpha)=0$.\n\\end{enumerate}\n\nWe say $\\Gamma$ semantically proves $\\varphi$ in $\\CILA$ if, for every bivaluation $\\nu$ for $\\CILA$ such that $\\nu(\\gamma)=1$, for every $\\gamma\\in\\Gamma$, one has that $\\nu(\\varphi)=1$; in this case we write $\\Gamma\\vDash_{\\CILA}\\varphi$.\n\\end{definition}\n\nAs it is discussed in \\cite{ParLog} and \\cite{Handbook}, $\\Gamma\\vdash_{\\CILA}\\varphi$ if and only if $\\Gamma\\vDash_{\\CILA}\\varphi$. We now want to prove that, if $\\Gamma\\vDash_{\\mathcal{M}_{\\CILA}}\\varphi$, then $\\Gamma\\vdash_{\\CILA}\\varphi$, so what we will do instead is prove that, if $\\Gamma\\vDash_{\\mathcal{M}_{\\CILA}}\\varphi$, then $\\Gamma\\vDash_{\\CILA}\\varphi$. Again, we suppose $\\Gamma\\vDash_{\\mathcal{M}_{\\CILA}}\\varphi$ and let $\\nu:F(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\{0,1\\}$ be a bivaluation for $\\CILA$, to then consider the map $\\overline{\\nu}:F(\\Sigma_{\\textbf{LFI}},\\mathcal{V})\\rightarrow\\{F,t,T\\}$ such that:\n\\begin{enumerate}\n\\item $\\overline{\\nu}(\\alpha)=F$ if and only if $\\nu(\\alpha)=0$ and $\\nu(\\neg\\alpha)=1$;\n\\item $\\overline{\\nu}(\\alpha)=t$ if and only if $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=1$;\n\\item $\\overline{\\nu}(\\alpha)=T$ if and only if $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=0$.\n\\end{enumerate}\n\n\n\\begin{theorem}\n$\\overline{\\nu}$ is a $\\Sigma_{\\textbf{LFI}}$-homomorphism between $\\textbf{F}(\\Sigma_{\\textbf{LFI}}, \\mathcal{V})$ and $\\mathcal{A}_{\\CILA}$.\n\\end{theorem}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item If $\\overline{\\nu}(\\alpha)=t$, $\\nu(\\alpha)=\\nu(\\neg\\alpha)=1$ and therefore $\\nu(\\alpha\\vee\\beta)=1$, meaning \n\\[\\overline{\\nu}(\\alpha\\vee\\beta)\\in\\{t, T\\}=\\overline{\\nu}(\\alpha)\\vee\\overline{\\nu}(\\beta).\\]\nWe can do the same if $\\overline{\\nu}(\\beta)=t$.\n\nIf $\\overline{\\nu}(\\alpha)=T$ and $\\overline{\\nu}(\\beta)=F$, we have $\\nu(\\alpha)=\\nu(\\neg\\beta)=1$ and $\\nu(\\neg\\alpha)=\\nu(\\beta)=0$, which imply $\\nu(\\circ\\alpha)=\\nu(\\circ\\beta)=1$ and therefore $\\nu(\\circ(\\alpha\\vee\\beta))=1$; since $\\nu(\\alpha\\vee\\beta)=1$, $\\nu(\\neg(\\alpha\\vee\\beta))=0$, and therefore $\\overline{\\nu}(\\alpha\\vee\\beta)=T\\in \\overline{\\nu}(\\alpha)\\vee\\overline{\\nu}(\\beta)$, the same happening if $\\overline{\\nu}(\\alpha)=F$ and $\\overline{\\nu}(\\beta)=T$.\n\nIf $\\overline{\\nu}(\\alpha)=F$ and $\\overline{\\nu}(\\beta)=F$, we have that $\\nu(\\alpha\\vee\\beta)=0$ and, since $\\nu(\\circ(\\alpha\\vee\\beta))=1$, one has $\\nu(\\neg(\\alpha\\vee\\beta))=1$; so $\\overline{\\nu}(\\alpha\\vee\\beta)=F\\in \\overline{\\nu}(\\alpha)\\vee\\overline{\\nu}(\\beta)$. If $\\overline{\\nu}(\\alpha)=T$ and $\\overline{\\nu}(\\beta)=T$, $\\nu(\\alpha\\vee\\beta)=1$ and $\\nu(\\circ(\\alpha\\vee\\beta))=1$, that is, $\\nu(\\neg(\\alpha\\vee\\beta))=0$: with this, $\\overline{\\nu}(\\alpha\\vee\\beta)=T\\in \\overline{\\nu}(\\alpha)\\vee\\overline{\\nu}(\\beta)$.\n\nThis finishes proving that, for any values of $\\overline{\\nu}(\\alpha)$ and $\\overline{\\nu}(\\beta)$, $\\overline{\\nu}(\\alpha\\vee\\beta)\\in \\overline{\\nu}(\\alpha)\\vee\\overline{\\nu}(\\beta)$.\n\n\\item If $\\overline{\\nu}(\\alpha)=F$, $\\nu(\\alpha\\wedge\\beta)=0$ and therefore $\\overline{\\nu}(\\alpha\\wedge\\beta)=F\\in\\overline{\\nu}(\\alpha)\\wedge\\overline{\\nu}(\\beta)$. The same can be done if $\\overline{\\nu}(\\beta)=F$.\n\nIf $\\overline{\\nu}(\\alpha)=t$ and $\\overline{\\nu}(\\beta)$ is either $t$ or $T$, we have $\\nu(\\alpha)=\\nu(\\beta)=1$ and therefore $\\nu(\\alpha\\wedge\\beta)=1$, meaning \n\\[\\overline{\\nu}(\\alpha\\wedge\\beta)\\in \\{t, T\\}=\\overline{\\nu}(\\alpha)\\wedge\\overline{\\nu}(\\beta).\\]\nThe same can be done if the conditions are reversed, $\\overline{\\nu}(\\alpha)$ is either $t$ and $T$, and $\\overline{\\nu}(\\beta)=t$.\n\nFinally, if $\\overline{\\nu}(\\alpha)=\\overline{\\nu}(\\beta)=T$, $\\nu(\\alpha\\wedge\\beta)=1$ and, since $\\nu(\\circ(\\alpha\\wedge\\beta))=1$, $\\nu(\\neg(\\alpha\\wedge\\beta))=0$, what means $\\overline{\\nu}(\\alpha\\wedge\\beta)=T\\in \\overline{\\nu}(\\alpha)\\wedge\\overline{\\nu}(\\beta)$.\n\n\\item First of all, suppose $\\overline{\\nu}(\\beta)=t$, what signifies $\\nu(\\beta)=1$ and allows to find $\\nu(\\alpha\\rightarrow\\beta)=1$, \\textit{id est}, $\\overline{\\nu}(\\alpha\\rightarrow\\beta)\\in\\{t, T\\}=\\overline{\\nu}(\\alpha)\\rightarrow\\overline{\\nu}(\\beta)$.\n\nIf $\\overline{\\nu}(\\alpha)=F$ and $\\overline{\\nu}(\\beta)$ is either $F$ or $T$, we clearly have $\\nu(\\alpha\\rightarrow\\beta)=1$; and since $\\nu(\\circ\\alpha)=\\nu(\\circ\\beta)=1$, $\\nu(\\circ(\\alpha\\rightarrow\\beta))=1$, meaning $\\nu(\\neg(\\alpha\\rightarrow\\beta))=0$ and therefore $\\overline{\\nu}(\\alpha\\rightarrow\\beta)=T\\in \\overline{\\nu}(\\alpha)\\rightarrow\\overline{\\nu}(\\beta)$.\n\nIf $\\overline{\\nu}(\\beta)=F$ and $\\overline{\\nu}(\\alpha)\\in \\{t, T\\}$, $\\nu(\\alpha)=1$ but $\\nu(\\beta)=0$, implying that $\\nu(\\alpha\\rightarrow\\beta)=0$, $\\overline{\\nu}(\\alpha\\rightarrow\\beta)=F\\in \\overline{\\nu}(\\alpha)\\rightarrow\\overline{\\nu}(\\beta)$.\n\nIf $\\overline{\\nu}(\\alpha)=t$ and $\\overline{\\nu}(\\beta)=T$, $\\nu(\\alpha)=1$ and $\\nu(\\beta)=1$, what means that $\\nu(\\alpha\\rightarrow\\beta)=1$ and \n\\[\\overline{\\nu}(\\alpha\\rightarrow\\beta)\\in\\{t, T\\}=\\overline{\\nu}(\\alpha)\\rightarrow\\overline{\\nu}(\\beta).\\]\nIf $\\overline{\\nu}(\\alpha)=T$ and $\\overline{\\nu}(\\beta)=T$, $\\nu(\\alpha\\rightarrow\\beta)=1$ and $\\nu(\\circ\\alpha)=\\nu(\\circ\\beta)=1$, from what we derive that $\\nu(\\circ(\\alpha\\rightarrow\\beta))=1$ and $\\nu(\\neg(\\alpha\\rightarrow\\beta))=0$, that is, $\\overline{\\nu}(\\alpha\\rightarrow\\beta)=T\\in \\overline{\\nu}(\\alpha)\\rightarrow\\overline{\\nu}(\\beta)$.\n\n\\item If $\\overline{\\nu}(\\alpha)=F$, $\\nu(\\alpha)=0$, $\\nu(\\neg\\alpha)=1$ and, therefore, $\\nu(\\neg\\neg\\alpha)=0$, meaning $\\overline{\\nu}(\\neg\\alpha)=T\\in \\neg\\overline{\\nu}(\\alpha)$.\n\nIf $\\overline{\\nu}(\\alpha)=t$, $\\nu(\\neg\\alpha)=1$, and it follows that $\\overline{\\nu}(\\neg\\alpha)\\in\\{t, T\\}=\\neg\\overline{\\nu}(\\alpha)$.\n\nIf $\\overline{\\nu}(\\alpha)=T$, one has $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=0$ and, therefore, $\\nu(\\neg\\neg\\alpha)=1$, implying $\\overline{\\nu}(\\neg\\alpha)=F\\in\\neg\\overline{\\nu}(\\alpha)$.\n\n\\item If $\\overline{\\nu}(\\alpha)$ equals $F$ or $T$, $\\nu(\\circ\\alpha)=1$ and $\\nu(\\neg\\circ\\alpha)=0$, since otherwise we would be forced to have $\\nu(\\circ\\alpha)=0$; this means $\\overline{\\nu}(\\circ\\alpha)=T\\in\\circ\\overline{\\nu}(\\alpha)$.\n\nIf $\\overline{\\nu}(\\alpha)=t$, $\\nu(\\circ\\alpha)=0$ and therefore $\\overline{\\nu}(\\circ\\alpha)=F\\in \\circ\\overline{\\nu}(\\alpha)$.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theorem}\n$\\overline{\\nu}$ is in $\\mathcal{F}_{\\CILA}$.\n\\end{theorem}\n\n\\begin{proof}\nIf $\\overline{\\nu}(\\alpha)=t$, $\\nu(\\alpha)=\\nu(\\neg\\alpha)=1$ and therefore $\\nu(\\alpha\\wedge\\neg\\alpha)=1$; meanwhile, $\\nu(\\alpha)=\\nu(\\neg\\alpha)=1$ imply $\\nu(\\circ\\alpha)=0$ and therefore $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=0$, meaning $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)=T$.\n\\end{proof}\n\nFrom the facts that $\\overline{\\nu}$ is in $\\mathcal{F}_{\\CILA}$, $\\Gamma\\vDash_{\\mathcal{M}_{\\CILA}}\\varphi$ and $\\nu(\\gamma)=1$ for every $\\gamma\\in\\Gamma$, implying $\\overline{\\nu}(\\gamma)\\in D$ for every $\\gamma\\in\\Gamma$, we obtain that $\\overline{\\nu}(\\varphi)\\in D$, meaning $\\nu(\\varphi)=1$. With this, $\\Gamma\\vDash_{\\CILA}\\varphi$, and we have that $\\Gamma\\vdash_{\\CILA}\\varphi$, what finishes proving $\\mathcal{M}_{\\CILA}$ characterizes $\\CILA$.\n\n\n\\begin{example}\nWith the described RNmatrix for $\\CILA$, we have obtained a simple, elegant and rather efficient decision procedure for $\\CILA$ and, given their equivalence, $C_{1}$ as well.\\footnote{For how this decision procedure works in practice, and the proof that it actually works, look at Section \\ref{Row-branching,row-eliminating} ahead.} So, let us stop for a moment to actually apply the method. Consider\n\\[\\neg(\\alpha\\vee\\beta)\\rightarrow(\\neg\\alpha\\wedge\\neg\\beta),\\]\nwhich is not a theorem of $\\CILA$ (nor of $C_{1}$). Consider the homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\textbf{LFI}}, \\mathcal{V})\\rightarrow \\mathcal{A}_{\\CILA}$, lying in $\\mathcal{F}_{\\CILA}$, such that \n\\[\\nu(\\alpha)=t,\\quad\\nu(\\beta)=T,\\quad\\nu(\\alpha\\vee\\beta)=t\\quad\\text{and}\\quad\\nu(\\neg(\\alpha\\vee\\beta))=t.\\]\nThen we have $\\nu(\\neg\\beta)=F$, what means $\\nu(\\neg\\alpha\\wedge\\neg\\beta)=F$ (regardless of the value of $\\neg\\alpha$) and therefore $\\nu(\\neg(\\alpha\\vee\\beta)\\rightarrow(\\neg\\alpha\\wedge\\neg\\beta))=F$, what indeed shows that the formula is not a theorem.\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{4}\n\\chapter{RNmatrices for da Costa's hierarchy}\\label{Chapter5}\\label{Chapter 5}\n\nThe most important result of this chapter will be a full treatment of da Costa's hierarchy. In 1963, Newton C. A. da Costa\\index{da Costa} presented his \\textit{Tese de C{\\'a}tedra} (the Brazilian equivalent of  an habilitation thesis) under the title ``Sistemas Formais Inconsistentes'' (\\textit{inconsistent formal systems} in Portuguese, \\cite{Costa3}). This work is one of the founding studies in paraconsistent logic, together with Stanislaw Ja{\\'s}kowski discussive (or discursive) logic $\\textbf{D2}$, based on suggestions made to him by Jan Lukaziewicz (\\cite{Jaskowski, Jaskowski2}). da Costa's systems $C_{n}$, however, war far better scrutinized by him than $\\textbf{D2}$ by Ja{\\'s}kowski, what may explain their far greater success in the long run.\n\nOne of da Costa's innovations was the separation of the formulas for a given $C_{n}$ in two: those that, indeed, do not trivialize an argument when presented together with their negation (and are, in a sense, ``badly-behaved''), and those that do, usually called ``well-behaved'' or ``classically-behaved''. To facilitate this division, he introduced a connective $\\circ_{n}$, for each $C_{n}$, which asserts the well-behavior of a formula when analyzed in light of the explosion law: this way, in $C_{n}$, $\\{\\alpha, \\neg\\alpha\\}$ does not necessarily have a trivial deductive closure (meaning its negation is not explosive), while $\\{\\alpha, \\neg\\alpha, \\circ_{n}\\alpha\\}$ does have it, implying\n\\[\\alpha, \\neg\\alpha, \\circ_{n}\\alpha\\vdash_{C_{n}}\\beta\\]\nfor any formulas $\\alpha$ and $\\beta$ of $C_{n}$. This approach was further generalized, later on, by Carnielli\\index{Carnielli} and Marcos \\cite{CM}, through the notion of logics of formal inconsistency ($\\textbf{LFI}$'s) which adds the connective $\\circ$, the consistency operator, as a primitive one.\n\nMany logicians will agree that da Costa's hierarchy $C_{n}$ is astoundingly interesting, but very few will deny that it is also of exceedingly difficult treatment: by Avron's aforementioned results, $C_{1}$, which is equivalent in a different signature to $\\CILA$, can not be characterized by a finite Nmatrix; even more, it is not characterizable by a finite set of finite Nmatrices, and is not algebraizable by \\index{Blok and Pigozzi}Blok and Pigozzi's methodology (\\cite{Avron, Mortensen80, Mortensen89, Lewin}). It is, fortunately, decidable, and decision methods vary from bivaluations (\\cite{Loparic}) to Fidel structures (\\cite{Fidel3}). \n\nWe here provide other decision procedures, now based on RNmatrices, for the entire hierarchy, which we believe to be easier to apply, given the very algebraic nature of RNmatrices, than many other such procedures found in the literature; in particular, they seem to suggest that the $n$th logic of da Costa should be seem as an $n+2$-valued logic, something other approaches cannot do. In Chapter \\ref{Chapter6} we proceed to generalize these RNmatrices for $C_{n}$ to a more general, and categorical, semantics through restricted swap structures (\\cite{ParLog}).\n\nMost of the work we present in this chapter can be found submitted to an online repository in \\cite{CostaRNmatrix}, and published in \\cite{TwoDecisionProcedures}.\n\n\n\\section{Definition of da Costa's hierarchy}\n\nConsider the signature $\\Sigma_{\\textbf{C}}$\\label{SigmaC} with $(\\Sigma_{\\textbf{C}})_{0}=\\emptyset$, $(\\Sigma_{\\textbf{C}})_{1}=\\{\\neg\\}$, $(\\Sigma_{\\textbf{C}})_{2}=\\{\\vee, \\wedge, \\rightarrow\\}$ and $(\\Sigma_{\\textbf{C}})_{n}=\\emptyset$ for $n>2$.\n\nFor simplicity, we define $\\alpha^{0}=\\alpha$\\label{alpha^} and\n\\[\\alpha^{n+1}=\\neg(\\alpha^{n}\\wedge\\neg(\\alpha^{n}))\\]\nfor $n\\in\\mathbb{N}$; we also define $\\alpha^{(0)}=\\alpha$, $\\alpha^{(1)}=\\alpha^{1}$\\label{alpha^()} and \n\\[\\alpha^{(n+1)}=\\alpha^{(n)}\\wedge\\alpha^{n+1}\\]\nfor $n\\in\\mathbb{N}\\setminus\\{0\\}$. Again for simplicity, $\\alpha^{1}$ may be denoted by $\\alpha^{\\circ}$\\label{alphacirc}, since this formula will play a role equivalent to $\\circ\\alpha$'s role in, for example, $\\mbC$.\n\nThe Hilbert calculus for the positive fragment of intuitionistic logic\\index{Logic, Intuitionistic} is composed of the following axiom schemata\n\\begin{enumerate}\n\\item[\\textbf{Ax\\: 1}] $\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$;\n\\item[\\textbf{Ax\\: 2}] $\\big(\\alpha\\rightarrow (\\beta\\rightarrow \\gamma)\\big)\\rightarrow\\big((\\alpha\\rightarrow\\beta)\\rightarrow(\\alpha\\rightarrow\\gamma)\\big)$;\n\\item[\\textbf{Ax\\: 3}] $\\alpha\\rightarrow\\big(\\beta\\rightarrow(\\alpha\\wedge\\beta)\\big)$;\n\\item[\\textbf{Ax\\: 4}] $(\\alpha\\wedge\\beta)\\rightarrow \\alpha$;\n\\item[\\textbf{Ax\\: 5}] $(\\alpha\\wedge\\beta)\\rightarrow \\beta$;\n\\item[\\textbf{Ax\\: 6}] $\\alpha\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 7}] $\\beta\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 8}] $(\\alpha\\rightarrow\\gamma)\\rightarrow\\Big((\\beta\\rightarrow\\gamma)\\rightarrow \\big((\\alpha\\vee\\beta)\\rightarrow\\gamma\\big)\\Big)$;\n\\end{enumerate}\nplus Modus Ponens as inference rule\n\\[\\frac{\\alpha\\quad\\alpha\\rightarrow\\beta}{\\beta}.\\]\n\nWe define the da costa's system $C_{\\omega}$\\label{Comega}, over $\\Sigma_{\\textbf{C}}$, by adding to the Hilbert calculus for the positive fragment of intuitionistic logic the axiom schemata\n\\begin{enumerate}\n\\item[\\textbf{Ax\\: 10}] $\\alpha\\vee\\neg\\alpha$;\n\\item[\\textbf{cf}] $\\neg\\neg\\alpha\\rightarrow\\alpha$.\n\\end{enumerate}\n\n\\begin{definition}\nFor $n\\in\\mathbb{N}\\setminus\\{0\\}$, we define the da Costa's\\index{Hierarchy, da Costa's} system $C_{n}$\\label{Cn}, over the signature $\\Sigma_{\\textbf{C}}$, as the logic obtained from $C_{\\omega}$ by addition of the axiom schemata\\label{bcn}\\label{phashn}\n\\[\\tag{$\\textbf{bc}_{n}$}\\alpha^{(n)}\\rightarrow\\big(\\alpha\\rightarrow(\\neg\\alpha\\rightarrow\\beta)\\big);\\]\n\\[\\tag{$\\textbf{p}\\vee_{n}$}(\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\vee\\beta)^{(n)};\\]\n\\[\\tag{$\\textbf{p}\\wedge_{n}$}(\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\wedge\\beta)^{(n)};\\]\n\\[\\tag{$\\textbf{p}\\rightarrow_{n}$}(\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\rightarrow\\beta)^{(n)}.\\]\n\\end{definition}\n\nThese systems were originally developed by Newton da Costa, in his seminal work \\cite{Costa3}; originally, the axiom $\\textbf{bc}_{n}$ was presented as\n\\[\\alpha^{(n)}\\rightarrow\\Big((\\beta\\rightarrow\\alpha)\\rightarrow\\big((\\beta\\rightarrow\\neg\\alpha)\\rightarrow\\neg\\beta\\big)\\Big),\\]\nwhich deals with non-contradiction instead of explosivity, but it is clear how both approaches are equivalent. For further comments on this finer distinction, refer back to \\cite{ParLog}.\n\n\\begin{definition}\\label{bival-def}\nA bivaluation for $C_{n}$, also known as a $C_{n}$-bivaluation\\index{Bivaluation for $C_{n}$}, is a function\\\\ $\\mathsf{b}:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\{0,1\\}$ satisfying:\n\\begin{enumerate}\n\\item[$(B1)$] $\\mathsf{b}(\\alpha\\wedge\\beta)=1$ if and only if $\\mathsf{b}(\\alpha)=1$ and $\\mathsf{b}(\\beta)=1$;\n\\item[$(B2)$] $\\mathsf{b}(\\alpha\\vee\\beta)=1$ if and only if $\\mathsf{b}(\\alpha)=1$ or $\\mathsf{b}(\\beta)=1$;\n\\item[$(B3)$] $\\mathsf{b}(\\alpha\\rightarrow\\beta)=1$ if and only if $\\mathsf{b}(\\alpha)=0$ or $\\mathsf{b}(\\beta)=1$;\n\\item[$(B4)$] $\\mathsf{b}(\\alpha)=0$ implies $\\mathsf{b}(\\neg\\alpha)=1$;\n\\item[$(B5)$] $\\mathsf{b}(\\neg\\neg\\alpha)=1$ implies $\\mathsf{b}(\\alpha)=1$;\n\\item[$(B6)_{n}$] $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}(\\neg(\\alpha^{n-1}))$ if and only if $\\mathsf{b}(\\alpha^{n})=0$;\n\\item[$(B7)$] $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)$ if and only if $\\mathsf{b}(\\neg(\\alpha^{1}))=1$;\n\\item[$(B8)$] for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta)\\neq\\mathsf{b}(\\neg\\beta)$ imply, together, that $\\mathsf{b}(\\alpha\\#\\beta)\\neq\\mathsf{b}(\\neg(\\alpha\\#\\beta))$.\n\\end{enumerate}\n\\end{definition}\n\nThe previous notion of bivaluations for $C_{n}$ has its origin in \\cite{Loparic}. If we denote the fact that, for every $C_{n}$-bivaluation $\\mathsf{b}$, $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$ implies $\\mathsf{b}(\\varphi)=1$ by $\\Gamma\\vDash_{C_{n}}\\varphi$, it is proved in the same paper that \n\\[\\Gamma\\vdash_{C_{n}}\\varphi\\quad\\text{if and only if}\\quad\\Gamma\\vDash_{C_{n}}\\varphi,\\]\nfor any set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\textbf{C}}$.\n\nWe shall provide, in the next sections, a semantical approach to da Costa's hierarchy trough finite restricted Nmatrices.\n\n\\section{$C_{2}$}\n\nWe will start from the simpler case that is $C_{2}$. For $\\mathsf{b}$ a $C_{2}$-bivaluation, $(B6)_{2}$ implies that $\\mathsf{b}(\\alpha^{1})=\\mathsf{b}(\\neg(\\alpha^{1}))$ if and only if $\\mathsf{b}(\\alpha^{2})=0$; so $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=\\mathsf{b}(\\alpha^{1})=1$ implies, from $(B7)$, that $\\mathsf{b}(\\alpha^{2})=0$. Furthermore, $\\mathsf{b}(\\alpha)=0$ leads us to $\\mathsf{b}(\\neg\\neg\\alpha)=0$ from $(B5)$, and so the following four scenarios are possible.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|c|r|}\n\\hline\n$\\alpha$ & $\\neg\\alpha$ & $\\alpha\\wedge\\neg\\alpha$ & $\\alpha^{1}$ & $\\neg(\\alpha^{1})$ & $\\alpha^{1}\\wedge\\neg(\\alpha^{1})$ & $\\alpha^{2}$ & $\\alpha^{(2)}$\\\\ \\hline\n\\multirow{3}{*}{$1$} & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$1$} & $1$ & $1$ & $1$ & $0$ & $0$\\\\\\cline{4-8}\n& & & $0$ & $1$ & $0$ & $1$ & $0$\\\\\\cline{2-8}\n& $0$ & $0$ & $1$ & $0$ & $0$ & $1$ & $1$\\\\\\hline\n$0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\caption*{Table for the scenarios on $C_{2}$}\n\\end{figure}\n\nNow, for a formula $\\alpha$ and a $C_{2}$-bivaluation, we would like to consider the triple $(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}))$ in $\\{0,1\\}^{3}$; from the previous table, we see that there are only $4$ possibilities, that we list below:\n\\[T_{2}=(1,0,1),\\quad t^{2}_{0}=(1,1,0),\\quad t^{2}_{1}=(1,1,1,)\\quad\\text{and}\\quad F_{2}=(0,1,1).\\]\nWe call the set of those elements $B_{2}$, and it is clear that \n\\[B_{2}=\\{z\\in \\{0,1\\}^{3}\\ :\\  z_{1}\\vee z_{2}=1\\quad\\text{and}\\quad (z_{1}\\wedge z_{2})\\vee z_{3}=1\\},\\]\nwhere $z_{i}$ will denote the $i$th coordinate of an element $z\\in\\{0,1\\}^{3}$. We then define the $\\Sigma_{\\textbf{C}}$-multialgebra $\\mathcal{A}_{C_{2}}$ with universe $B_{2}$ and operations given by\n\\[\\tilde{\\neg}z=\\{w\\in B_{2}\\ :\\  w_{1}=z_{2}\\quad\\text{and}\\quad w_{2}\\leq z_{1}\\}\\]\nand\n\\[\\text{for}\\quad \\#\\in\\{\\vee, \\wedge, \\rightarrow\\},\\quad z\\tilde{\\#}w=\\begin{cases*}\n\\{u\\in B_{2}\\ :\\  u_{1}=z_{1}\\# w_{1}\\}\\cap Boo_{2} & if $z, w\\in Boo_{2}$\\\\\n\\{u\\in B_{2}\\ :\\  u_{1}=z_{1}\\# w_{1}\\} & otherwise\n\\end{cases*},\\]\nwhere: we will denote an operation $\\sigma_{\\mathcal{A}_{C_{2}}}$ on $\\mathcal{A}_{C_{2}}$ simply by $\\tilde{\\sigma}$; the operations $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$ are defined as usual in $\\{0,1\\}$; and $Boo_{2}=\\{F_{2}, T_{2}\\}$ is the set of classically-behaving elements.\n\nOne could argue that a more natural definition would be that $z\\tilde{\\#}w$ always equals $\\{u\\in B_{2}\\ :\\  u_{1}=z_{1}\\# w_{1}\\}$, but that would be problematic. Condition $(B8)$ for being a bivaluation for $C_{n}$ states that $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta\\neq\\mathsf{b}(\\neg\\beta)$ imply, together, that $\\mathsf{b}(\\alpha\\#\\beta)\\neq\\mathsf{b}(\\neg(\\alpha\\#\\beta))$; and one easily sees that the elements $(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}))$ of $B_{2}$ satisfying $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ are precisely $F_{2}$ and $T_{2}$. This all means that $(B8)$ translates to demanding that, if $x,y\\in Boo_{2}$, then one must necessarily have $x\\tilde{\\#}y\\in Boo_{2}$. Notice that this also justifies the nomenclature of ``classical'' for the elements $F_{2}$ and $T_{2}$: they correspond to formulas such that their negation, and the formula itself, are not simultaneously true.\n\nIf we denote the set $\\{t_{1}^{2}, t_{0}^{2}, T_{2}\\}$ by $D_{2}$, the tables for $\\mathcal{A}_{C_{2}}$ are as below. We will also use the notation $\\{t_{0}^{2}, t_{1}^{2}\\}=I_{2}$ when necessary.\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n$\\vee$ & $F_{2}$ & $t_{1}^{2}$ & $t_{0}^{2}$ & $T_{2}$ \\\\ \\hline\n$F_{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $\\{T_{2}\\}$ \\\\ \\hline\n$t_{1}^{2}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ \\\\ \\hline\n$t_{0}^{2}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ \\\\ \\hline\n$T_{2}$ & $\\{T_{2}\\}$ & $D_{2}$ & $D_{2}$ & $\\{T_{2}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Disjunction}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n$\\wedge$ & $F_{2}$ & $t_{1}^{2}$ & $t_{0}^{2}$ & $T_{2}$ \\\\ \\hline\n$F_{2}$ & $\\{F_{2}\\}$ & $\\{F_{2}\\}$ & $\\{F_{2}\\}$ & $\\{F_{2}\\}$ \\\\ \\hline\n$t_{1}^{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ \\\\ \\hline\n$t_{0}^{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ \\\\ \\hline\n$T_{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $\\{T_{2}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Conjunction}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\neg$ \\\\ \\hline\n$F_{2}$ & $\\{T_{2}\\}$\\\\ \\hline\n$t_{1}^{2}$ & $D_{2}$\\\\ \\hline\n$t_{0}^{2}$ & $D_{2}$\\\\ \\hline\n$T_{2}$ & $\\{F_{2}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for negation}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{|l|c|c|c|r|}\n\\hline\n$\\rightarrow$ & $F_{2}$ & $t_{1}^{2}$ & $t_{0}^{2}$ & $T_{2}$ \\\\ \\hline\n$F_{2}$ & $\\{T_{2}\\}$ & $D_{2}$ & $D_{2}$ & $\\{T_{2}\\}$ \\\\ \\hline\n$t_{1}^{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ \\\\ \\hline\n$t_{0}^{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $D_{2}$ \\\\ \\hline\n$T_{2}$ & $\\{F_{2}\\}$ & $D_{2}$ & $D_{2}$ & $\\{T_{2}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Implication}\n\\end{minipage}\n\\end{figure}\n\nIf we consider the Nmatrix $\\mathcal{M}_{C_{2}}=(\\mathcal{A}_{C_{2}}, D_{2})$, we cannot hope to characterize $C_{2}$ with it given that $C_{2}$ is not characterizable by a single finite Nmatrix (\\cite{Avron}). What we will do instead, and which will be a successful endeavor, is to restrict the set of valuations for this Nmatrix, creating an RNmatrix, in order to characterize $C_{2}$.\n\n\\begin{definition}\\label{FC2}\nLet $\\mathcal{F}_{C_{2}}$ be the set of homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow \\mathcal{A}_{C_{2}}$ (which are called valuations over $\\mathcal{A}_{C_{2}}$) such that:\n\\begin{enumerate}\n\\item if $\\nu(\\alpha)=t_{0}^{2}$, then $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{2}$;\n\\item if $\\nu(\\alpha)=t_{1}^{2}$, then $\\nu(\\alpha\\wedge\\neg\\alpha)\\in I_{2}$ and $\\nu(\\alpha^{\\circ})=t_{0}^{2}$.\n\\end{enumerate}\nWe will denote the restricted Nmatrix $(\\mathcal{A}_{C_{2}}, D_{2}, \\mathcal{F}_{C_{2}})$ by $\\mathcal{RM}_{C_{2}}$.\n\\end{definition}\n\nSuppose $\\nu\\in\\mathcal{F}_{C_{2}}$: notice that, if $\\nu(\\alpha)=t_{0}^{2}$, then $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{2}$ and so $\\nu(\\alpha^{\\circ})=F_{2}$; if $\\nu(\\alpha)=t_{1}^{2}$, $\\nu(\\alpha^{\\circ})=t_{0}^{2}$ and therefore $\\nu(\\alpha^{\\circ\\circ})=F_{2}$. Let $\\bot_{\\alpha}$ denote $(\\alpha\\wedge\\neg\\alpha)\\wedge\\alpha^{(2)}$ and ${\\sim}\\alpha$ denote $\\alpha\\rightarrow\\bot_{\\alpha}$ (the strong negation definable in $C_{2}$), and we arrive at the following table, where an asterisk signifies that a certain value would be different if the table were constructed using the Nmatrix $(\\mathcal{A}_{C_{2}}, D_{2})$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|r|}\n\\hline\n$\\alpha$ & $\\neg\\alpha$ & $\\alpha\\wedge\\neg\\alpha$ & $\\alpha^{\\circ}$ & $\\neg(\\alpha^{\\circ})$ & $\\alpha^{\\circ}\\wedge\\neg(\\alpha^{\\circ})$ & $\\alpha^{\\circ\\circ}$ & $\\alpha^{(2)}$ & $\\bot_{\\alpha}$ & ${\\sim}\\alpha$ \\\\ \\hline\n$T_{2}$ & $F_{2}$ & $F_{2}$ & $T_{2}$ & $F_{2}$ & $F_{2}$ & $T_{2}$ & $T_{2}$ & $F_{2}$ & $F_{2}$\\\\ \\hline\n$t_{0}^{2}$ & $D_{2}$ & $T_{2}^{*}$ & $F_{2}$ & $T_{2}$ & $F_{2}$ & $T_{2}$ & $F_{2}$ & $F_{2}$ & $F_{2}$\\\\ \\hline\n$t_{1}^{2}$ & $D_{2}$ & $I_{2}^{*}$ & $t_{0}^{2 *}$ & $D_{2}$ & $T_{2}^{*}$ & $F_{2}$ & $F_{2}$ & $F_{2}$ & $F_{2}$ \\\\ \\hline\n$F_{2}$ & $T_{2}$ & $F_{2}$ & $T_{2}$ & $F_{2}$ & $F_{2}$ & $T_{2}$ & $T_{2}$ & $F_{2}$ & $T_{2}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for the scenarios in $\\mathcal{RM}_{C_{2}}$}\n\\end{figure}\n\n\\begin{proposition}\nFor a homomorphism $\\nu$ in $\\mathcal{F}_{C_{2}}$ and an endomorphism $\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$, $\\nu\\circ\\sigma\\in \\mathcal{F}_{C_{2}}$.\n\\end{proposition}\n\n\\begin{proof}\nOf course $\\nu\\circ\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{2}}$ remains a homomorphism.\n\\begin{enumerate}\n\\item If $\\nu\\circ\\sigma(\\alpha)=t_{0}^{2}$, given $\\nu$ is in $\\mathcal{F}_{C_{2}}$ we derive that $\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))=T_{2}$; since $\\sigma$ is an endomorphism of $\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$, $\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha)=\\sigma(\\alpha\\wedge\\neg\\alpha)$, and so $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)=T_{2}$.\n\n\\item If $\\nu\\circ\\sigma(\\alpha)=t_{1}^{2}$, since $\\nu\\in\\mathcal{F}_{C_{2}}$ we get $\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))\\in I_{2}$ and \n\\[\\nu(\\sigma(\\alpha)^{\\circ})=\\nu\\Big(\\neg(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))\\Big)=t_{0}^{2};\\]\ngiven $\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha)=\\sigma(\\alpha\\wedge\\neg\\alpha)$ and $\\neg(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))=\\sigma(\\neg(\\alpha\\wedge\\neg\\alpha))$, we obtain that $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)\\in I_{2}$ and $\\nu\\circ\\sigma(\\neg(\\alpha\\wedge\\neg\\alpha))=\\nu\\circ\\sigma(\\alpha^{\\circ})=t_{0}^{2}$, what finishes the proof.\n\\end{enumerate}\n\\end{proof}\n\nThe previous proposition implies $\\mathcal{RM}_{C_{2}}$ is structural. \n\nThe following technical lemmas are necessary in order to prove the desired Theorem \\ref{Char C2}.\n\n\\begin{lemma}\\label{Sound C2}\nFor $\\nu$ a valuation in $\\mathcal{F}_{C_{2}}$, the mapping $\\mathsf{b}: F(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow \\{0,1\\}$, such that $\\mathsf{b}(\\alpha)=1$ if and only if $\\nu(\\alpha)\\in D_{2}$, is a $C_{2}$-bivaluation.\n\\end{lemma}\n\n\\begin{proof}\nLet us see that $\\mathsf{b}$ satisfies the clauses of Definition \\ref{bival-def} for $n=2$. Clauses $(B1)$ through $(B3)$ are clearly satisfied: since \n\\[\\mathsf{b}(\\alpha \\# \\beta) = \\nu(\\alpha \\# \\beta)_1 = \\nu(\\alpha)_1 \\# \\nu(\\beta)_1\\]\nfor $\\# \\in \\{\\vee, \\wedge, \\rightarrow\\}$. \n\nOn the other hand, $\\mathsf{b}(\\neg\\alpha) = \\nu(\\neg\\alpha)_1 = \\nu(\\alpha)_2$, hence $\\mathsf{b}(\\alpha) \\vee \\mathsf{b}(\\neg\\alpha)=1$, by definition of $B_{2}$. From this, clause $(B2)$ is satisfied. Since $\\mathsf{b}(\\neg\\neg\\alpha)=\\nu(\\neg\\neg\\alpha)_1=\\nu(\\neg\\alpha)_2\\leq \\nu(\\alpha)_1 = \\mathsf{b}(\\alpha)$ it follows that $\\mathsf{b}$ satisfies $(B5)$. \n\nConcerning $(B6)_2$, suppose that $\\mathsf{b}(\\alpha^\\circ)=\\mathsf{b}(\\neg(\\alpha^\\circ))=1$: this means that $\\nu(\\alpha^\\circ)_1=\\nu(\\neg(\\alpha^\\circ))_1=\\nu(\\alpha^\\circ)_2=1$. That is, $\\nu(\\alpha^\\circ) \\in \\{t^2_0,t^2_1\\}$. From the possible scenarios in $\\mathcal{RM}_{C_{2}}$ it follows that $\\nu(\\alpha^\\circ) = t^2_1$ and so $\\nu(\\alpha^2)=F$, what implies that $\\mathsf{b}(\\alpha^2)=\\nu(\\alpha^2)_1=0$. Conversely, $\\mathsf{b}(\\alpha^2)=\\nu(\\alpha^2)_1=0$ means that $\\nu(\\alpha^2)=F$, which implies that $\\nu(\\alpha^\\circ) = t^2_1$, and so $\\mathsf{b}(\\alpha^\\circ)=\\mathsf{b}(\\neg(\\alpha^\\circ))=1$. From this, clause $(B6)_2$ is satisfied. \n\nNow,  $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)$ if and only if $\\nu(\\alpha)_1 = \\nu(\\alpha)_2=1$, what in turn happens if and only if $\\nu(\\alpha) \\in \\{t^2_0,t^2_1\\}$. This last condition is equivalent to the fact that $\\nu(\\neg(\\alpha^\\circ)) \\in D_2$, by the possible scenarios in $\\mathcal{RM}_{C_{2}}$, which is equivalent to $\\mathsf{b}(\\neg(\\alpha^\\circ))=1$. Then, $(B7)$ is satisfied. \n\nFinally, suppose that $\\mathsf{b}(\\alpha) \\neq \\mathsf{b}(\\neg\\alpha)$ and $v(\\beta) \\neq \\mathsf{b}(\\neg\\beta)$. This means that $\\nu(\\alpha),\\nu(\\beta) \\in \\{T_2,F_2\\}$ and so $\\nu(\\alpha\\# \\beta) \\in \\{T_2,F_2\\}$, by definition of  $\\mathcal{A}_{C_2}$. Hence clause $(B8)$ is fulfilled, and the proof is complete.\n\\end{proof}\n\n\\begin{lemma}\\label{Comp C2}\nFor $\\mathsf{b}$ a $C_{2}$-bivaluation, the mapping $\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow \\mathcal{A}_{C_{2}}$, such that $\\nu(\\alpha)=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{\\circ}))$, is a valuation which lies in $\\mathcal{F}_{C_{2}}$ and satisfies that $\\mathsf{b}(\\alpha)=1$ if, and only if, $\\nu(\\alpha)\\in D_{2}$.\n\\end{lemma}\n\n\\begin{proof}\nBy definition, $\\nu(\\neg\\alpha)=(\\mathsf{b}(\\neg\\alpha),\\mathsf{b}(\\neg\\neg\\alpha),\\mathsf{b}((\\neg\\alpha)^\\circ))$, and clearly this belongs to $\\tilde{\\neg} \\nu(\\alpha)$ according to the definition of $\\tilde{\\neg}$, by the property $(B5)$ of $\\mathsf{b}$. Analogously, by definition of $\\tilde{\\#}$ it follows that $\\nu(\\alpha\\# \\beta) \\in \\nu(\\alpha) \\tilde{\\#} \\nu(\\beta)$, with use of the properties $(B1)$ through $(B3)$ of $\\mathsf{b}$. This shows that $\\nu$ is a valuation for $\\mathcal{A}_{C_2}$. It remains to be shown that $\\nu$ satisfies conditions $1$ and $2$ of Definition \\ref{FC2}.\n\nRegarding the first of these conditions, assume that $\\nu(\\alpha) = t^2_0$. This means that $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=1$ and $\\mathsf{b}(\\alpha^\\circ)=0$. Let $\\beta=\\alpha\\wedge\\neg\\alpha$ (thus $\\neg\\beta=\\alpha^\\circ$). By $(B1)$, $\\mathsf{b}(\\beta)=1$. Since, by hypothesis, $\\mathsf{b}(\\beta) \\neq \\mathsf{b}(\\neg\\beta)$, it follows that $\\mathsf{b}(\\neg(\\beta^\\circ))=0$, by~$( B7)$, and so $\\mathsf{b}(\\beta^\\circ)=1$, by~$(B4)$. This shows that \n\\[\\nu(\\alpha\\wedge\\neg\\alpha) = (1,0,1)=T_2,\\]\nas we wished to prove. \n\nRegarding the second condition, suppose that $\\nu(\\alpha) = t^2_1$. Then, $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=\\mathsf{b}(\\alpha^\\circ)=1$. Consider again $\\beta=\\alpha \\land \\neg\\alpha$. From this, $\\mathsf{b}(\\beta)=\\mathsf{b}(\\neg\\beta)=1$ and so \n\\[\\nu(\\alpha\\wedge\\neg\\alpha) = (\\mathsf{b}(\\beta), \\mathsf{b}(\\neg\\beta), \\mathsf{b}(\\beta^\\circ)) \\in \\{t^2_1,t^2_1\\}.\\]\nSince $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)$ it follows by $(B7)$ that  $\\mathsf{b}(\\neg(\\alpha^\\circ))=1$. Hence, since $\\mathsf{b}(\\alpha^1)=\\mathsf{b}(\\neg(\\alpha^1))$, we infer that $\\mathsf{b}(\\alpha^2)=0$, by $(B6)_2$. That is, $\\nu(\\alpha^\\circ) = (1,1,0)=t^2_0$. This, of course, finishes proving that $\\nu$ is in $\\mathcal{F}_{C_{2}}$.\n\\end{proof}\n\n\\begin{theorem}\\label{Char C2}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $C_{2}$, $\\Gamma\\vdash_{C_{2}}\\varphi$ if, and only if, $\\Gamma\\vDash_{\\mathcal{RM}_{C_{2}}}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose first that $\\Gamma \\vdash_{C_{2}} \\varphi$, and let $\\nu$ be  a valuation in $\\mathcal{F}_{C_2}$ such that $\\nu(\\gamma) \\in D_{2}$ for every $\\gamma \\in \\Gamma$. By Lemma \\ref{Sound C2}, $\\mathsf{b}$ defined as $\\mathsf{b}(\\alpha)=\\nu(\\alpha)_1$, for every $\\alpha$, is a $C_2$-bivaluation such that, by definition, $\\mathsf{b}(\\gamma) =1$ for every $\\gamma \\in \\Gamma$. By  hypothesis and by soundness of $C_2$ with respect to bivaluations, it follows that $\\mathsf{b}(\\varphi)=1$. That is, $\\nu(\\varphi) \\in D_2$. This shows that $\\Gamma\\vDash_{\\mathcal{RM}_{C_{2}}}\\varphi$.\n\nNow assume that $\\Gamma\\vDash_{\\mathcal{RM}_{C_{2}}}\\varphi$, and let $\\mathsf{b}$ be a $C_2$-bivaluation such that $\\mathsf{b}(\\gamma) =1$ for every $\\gamma \\in \\Gamma$. By Lemma \\ref{Comp C2}, the function $\\nu$ defined as $\\nu(\\alpha)=(\\mathsf{b}(\\alpha),\\mathsf{b}(\\neg\\alpha),\\mathsf{b}(\\alpha^\\circ))$, for every $\\alpha$, is a valuation in $\\mathcal{F}_{C_2}$ such that, by definition, $\\nu(\\gamma) \\in D_2$ for every $\\gamma \\in \\Gamma$. By hypothesis, $\\nu(\\varphi) \\in D_2$, which means that $\\mathsf{b}(\\varphi)=1$. By completeness of $C_2$  with respect to bivaluations, it follows that $\\Gamma \\vdash_{C_2} \\varphi$.\n\\end{proof}\n\n\n\n\n\\section{The general case}\\label{The general case}\n\n\\begin{lemma}\\label{Values of i-consistency}\nGiven a $C_{n}$-bivaluation $\\mathsf{b}$, if $\\mathsf{b}(\\alpha^{i})=1$ and $\\mathsf{b}(\\neg(\\alpha^{i}))=0$ for some $i\\in\\mathbb{N}$, then $\\mathsf{b}(\\alpha^{j})=1$ for all $j\\geq i$.\n\\end{lemma}\n\n\\begin{proof}\nWe prove that, for all $j\\geq i$, $\\mathsf{b}(\\alpha^{j})=1$ and $\\mathsf{b}(\\neg(\\alpha^{j}))=0$ by induction, being the base case done. So, suppose $\\mathsf{b}(\\alpha^{j})=1$ and $\\mathsf{b}(\\neg(\\alpha^{j}))=0$: we have $\\mathsf{b}(\\alpha^{j}\\wedge\\neg(\\alpha^{j}))=0$ from $(B1)$, and $\\mathsf{b}(\\alpha^{j+1})=\\mathsf{b}(\\neg(\\alpha^{j}\\wedge\\neg(\\alpha^{j})))=1$ from $(B4)$; and from $(B7)$, $\\mathsf{b}(\\neg(\\alpha^{j+1}))=\\mathsf{b}(\\neg((\\alpha^{j})^{1}))=0$, since $\\mathsf{b}(\\alpha^{j})\\neq\\mathsf{b}(\\neg(\\alpha^{j}))$, what finishes the proof.\n\\end{proof}\n\n\\begin{proposition}\nIf $\\mathsf{b}$ is a $C_{n}$-bivaluation, at most one of the elements $\\mathsf{b}(\\alpha)$, $\\mathsf{b}(\\neg\\alpha)$, $\\mathsf{b}(\\alpha^{1}), \\dotsc  ,\n$\\\\$\\mathsf{b}(\\alpha^{n-1})$ equals $0$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $\\mathsf{b}(\\alpha)=0$: from condition $(B4)$, this means $\\mathsf{b}(\\neg\\alpha)=1$. Then $\\mathsf{b}(\\alpha\\wedge\\neg\\alpha)=0$ (from $(B1)$), meaning $\\mathsf{b}(\\alpha^{1})=\\mathsf{b}(\\neg(\\alpha\\wedge\\neg\\alpha))=1$, again from $(B4)$; but notice, furthermore, that, from the converse of $(B5)$, $\\mathsf{b}(\\alpha\\wedge\\neg\\alpha)=0$ implies $\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\neg\\neg(\\alpha\\wedge\\neg\\alpha))=0$. From Lemma \\ref{Values of i-consistency}, we find $\\mathsf{b}(\\alpha^{2})=\\cdots=\\mathsf{b}(\\alpha^{n-1})=1$, what finishes proving that, if $\\mathsf{b}(\\alpha)=0$, then $\\mathsf{b}(\\neg\\alpha)=\\mathsf{b}(\\alpha^{1})=\\cdots=\\mathsf{b}(\\alpha^{n-1})=1$.\n\nNow, suppose $\\mathsf{b}(\\neg\\alpha)=0$: we already have $\\mathsf{b}(\\alpha)=1$, since otherwise one could derive $\\mathsf{b}(\\neg\\alpha)=1$ from the previous remarks. So $\\mathsf{b}(\\alpha\\wedge\\neg\\alpha)=0$ and therefore $\\mathsf{b}(\\alpha^{1})=1$. Again by the converse of $(B5)$, one finds $\\mathsf{b}(\\neg(\\alpha^{1}))=0$, and again from Lemma \\ref{Values of i-consistency}, we obtain $\\mathsf{b}(\\alpha^{1})=\\cdots=\\mathsf{b}(\\alpha^{n-1})=1$.\n\nFinally, suppose that for some $1\\leq i<j< n-1$ one has $\\mathsf{b}(\\alpha^{i})=\\mathsf{b}(\\alpha^{j})=0$: we find $\\mathsf{b}(\\alpha^{i}\\wedge\\neg(\\alpha^{i}))=0$, $\\mathsf{b}(\\alpha^{i+1})=1$ and, from $(B5)$'s converse, $\\mathsf{b}(\\neg(\\alpha^{i+1}))=0$; once again using Lemma \\ref{Values of i-consistency}, we obtain $\\mathsf{b}(\\alpha^{i+1})=\\cdots=\\mathsf{b}(\\alpha^{n-1})=1$, contradicting $\\mathsf{b}(\\alpha^{j})=0$. The obvious conclusion is that at most one of $\\mathsf{b}(\\alpha^{1})$ trough $\\mathsf{b}(\\alpha^{n-1})$ can equal $0$, in which case one also has $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=1$.\n\\end{proof}\n\nWe consider $(n+1)$-tuples $z=(z_{1}, \\dotsc  , z_{n+1})$ in $\\textbf{2}^{n+1}$, corresponding to $\\mathsf{b}(\\alpha)$, $\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\alpha^{1})$ trough $\\mathsf{b}(\\alpha^{n-1})$ under a given $C_{n}$-bivaluation $\\mathsf{b}$, which we will call snapshots\\index{Snapshots} for $C_{n}$; given the restrictions established by the proposition, there are precisely $n+2$ of such tuples:\\label{Tn}\\label{Fn}\n\\[T_{n}=(1, 0, 1, \\dotsc  , 1), \\quad t_{0}^{n}=(1, 1, 0, 1, \\dotsc  , 1),\\quad\\cdots\\quad,\\quad t_{n-2}^{n}=(1, \\dotsc  , 1, 0), \\quad t_{n-1}^{n}=(1, \\dotsc  , 1)\\]\n\\[\\text{and}\\quad F_{n}=(0, 1, \\dotsc  , 1).\\]\nThe set $B_{n}$\\label{Bn} of all snapshots for $C_{n}$ is clearly the set of all elements of $\\{0,1\\}^{n+1}$ with at most one coordinate equal to $0$, and may be defined trough the following equality.\n\\[B_{n}=\\{z\\in \\textbf{2}^{n+1}\\ :\\  (\\bigwedge_{i=1}^{k}z_{i})\\vee z^{k+1}=1,\\quad\\text{for every $1\\leq k\\leq n$}\\}.\\]\n\nThe set of designated elements, that will soon be part of appropriate restricted Nmatrices for $C_{n}$ with universe $B_{n}$, is $D_{n}=B_{n}\\setminus\\{F_{n}\\}$\\label{Dn}. Other useful sets are: \n\\begin{enumerate}\n\\item the set of undesignated elements, $U_{n}=\\{F_{n}\\}$;\n\\item the set of Boolean elements\\index{Boolean elements}, $Boo_{n}=\\{F_{n}, T_{n}\\}$, also equal to $\\{z\\in B_{n}\\ :\\  z_{1}\\wedge z_{2}=0\\}$\\label{Boon}; \n\\item and the set of inconsistent elements, $I_{n}=B_{n}\\setminus Boo_{n}$.\n\\end{enumerate}\n\nNotice that an element $z$ of $B_{n}$ is in $Boo_{n}$ if, and only if, it equals $(a, {\\sim}a, 1, \\dotsc  , 1)$, for some element $a$ of $\\textbf{2}$ and ${\\sim}a$ its complement in $\\textbf{2}$; this is easy to see given that $z_{1}\\vee z_{2}=1$, from the definition of $B_{n}$, and $z_{1}\\wedge z_{2}=0$, from the definition of $Boo_{n}$.\n\nNow, we will define the multialgebra $\\mathcal{A}_{C_{n}}$\\label{ACn}, which happens to be a swap structure\\index{Swap structure} in the sense of \\cite{ParLog}; unlike the swap structures in this reference, however, we will need to restrict the valuations taken into consideration, making of the semantics with underlying $\\mathcal{A}_{C_{n}}$ an RNmatrix.\n\n\\begin{definition}\\label{Define operations in Cn}\nWe define the multioperations on the $\\Sigma_{\\textbf{C}}$-multialgebra $\\mathcal{A}_{C_{n}}$\\label{ACn}, with universe $B_{n}$, by means of the following equations:\n\\[\\tilde{\\neg}z=\\{w\\in B_{n}\\ :\\  w_{1}=z_{2}\\quad\\text{and}\\quad w_{2}\\leq z_{1}\\}\\]\nand\n\\[\\text{for}\\quad \\#\\in\\{\\vee, \\wedge, \\rightarrow\\},\\quad z\\tilde{\\#}w=\\begin{cases*}\n\\{u\\in B_{n}\\ :\\  u_{1}=z_{1}\\# w_{1}\\}\\cap Boo_{n} & if $z, w\\in Boo_{n}$\\\\\n\\{u\\in B_{n}\\ :\\  u_{1}=z_{1}\\# w_{1}\\} & otherwise\n\\end{cases*}.\\]\n\\end{definition}\n\nThe operations on this multialgebra may be illustrated, concisely, by the following tables.\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\vee$ & $F_{n}$ & $I_{n}$ & $T_{n}$ \\\\ \\hline\n$F_{n}$ & $\\{F_{n}\\}$ &  $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n$I_{n}$ & $D_{n}$ & $D_{n}$ & $D_{n}$ \\\\ \\hline\n$T_{n}$ & $\\{T_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Disjunction}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\wedge$ & $F_{n}$ & $I_{n}$ & $T_{n}$ \\\\ \\hline\n$F_{n}$ & $\\{F_{n}\\}$ & $\\{F_{n}\\}$ & $\\{F_{n}\\}$ \\\\ \\hline\n$I_{n}$ & $\\{F_{n}\\}$  & $D_{n}$ & $D_{n}$ \\\\ \\hline\n$T_{n}$ & $\\{F_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Conjunction}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n & $\\neg$ \\\\ \\hline\n$F_{n}$ & $\\{T_{n}\\}$\\\\ \\hline\n$I_{n}$ & $D_{n}$\\\\ \\hline\n$T_{n}$ & $\\{F_{n}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for negation}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\rightarrow$ & $F_{n}$ & $I_{n}$ & $T_{n}$ \\\\ \\hline\n$F_{n}$ & $\\{T_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n$I_{n}$ & $\\{F_{n}\\}$ & $D_{n}$ & $D_{n}$ \\\\ \\hline\n$T_{n}$ & $\\{F_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Implication}\n\\end{minipage}\n\\end{figure}\n\n\\begin{definition}\nLet $\\mathcal{F}_{C_{n}}$\\label{FCn} be the set of valuations $\\nu$ over $\\mathcal{A}_{C_{n}}$ satisfying that:\n\\begin{enumerate}\n\\item $\\nu(\\alpha)=t_{0}^{n}$ implies $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{n}$;\n\\item for every $2\\leq k\\leq n$, $\\nu(\\alpha)=t_{k-1}^{n}$ implies $\\nu(\\alpha\\wedge\\neg\\alpha)\\in I_{n}$ and $\\nu(\\alpha^{1})=t_{k-2}^{n}$.\n\\end{enumerate}\nWe denote the RNmatrix $(\\mathcal{A}_{C_{n}}, D_{n}, \\mathcal{F}_{C_{n}})$ by $\\mathcal{RM}_{C_{n}}$\\label{RMCn}.\n\\end{definition}\n\n\\begin{proposition}\nFor a homomorphism $\\nu$ in $\\mathcal{F}_{C_{n}}$ and an endomorphism $\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$, $\\nu\\circ\\sigma\\in \\mathcal{F}_{C_{n}}$.\n\\end{proposition}\n\n\\begin{proof}\nGiven the composition of homomorphisms returns homomorphisms, $\\nu\\circ\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}$ is certainly still a homomorphism.\n\n\\begin{enumerate}\n\\item Suppose $\\nu\\circ\\sigma(\\alpha)=t_{0}^{n}$: this means $\\nu(\\sigma(\\alpha))=t_{0}^{n}$, and since $\\nu\\in\\mathcal{F}_{C_{n}}$, $\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))=T_{n}$. Given $\\alpha$ is an endomorphism of $\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$, \n\\[\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha)=\\sigma(\\alpha\\wedge\\neg\\alpha),\\]\nand so $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)=\\nu(\\sigma(\\alpha\\wedge\\neg\\alpha))=T_{n}$.\n\n\\item For a $2\\leq k\\leq n$, suppose $\\nu\\circ\\sigma(\\alpha)=t_{k-1}^{n}$. That is, $\\nu(\\sigma(\\alpha))=t_{k-1}^{n}$, and since $\\nu$ is in $\\mathcal{F}_{C_{n}}$, we have that $\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))\\in I_{n}$ and $\\nu(\\sigma(\\alpha)^{1})=\\nu(\\neg(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha)))=t_{k-2}^{n}$. Again, given $\\sigma$ is an endomorphism of $\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$,\n\\[\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha)=\\sigma(\\alpha\\wedge\\neg\\alpha)\\quad\\text{and}\\quad \\neg(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))=\\sigma(\\neg(\\alpha\\wedge\\neg\\alpha)),\\]\nwhat implies $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)=\\nu(\\sigma(\\alpha\\wedge\\neg\\alpha))\\in I_{n}$ and $\\nu\\circ\\sigma(\\alpha^{1}))=\\nu(\\sigma(\\alpha^{1}))=t_{k-2}^{n}$.\n\\end{enumerate}\n\\end{proof}\n\nThe previous result proves $\\mathcal{RM}_{C_{n}}$ is structural.\n\nNotice that, for $\\nu\\in\\mathcal{F}_{C_{n}}$, $\\nu(\\alpha)=t_{0}^{n}$ implies $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{n}$ and, therefore, $\\nu(\\alpha^{1})=F_{n}$; if $\\nu(\\alpha)=t_{1}^{n}$, $\\nu(\\alpha^{1})=t_{0}^{n}$ and, by the previous remark, $\\nu(\\alpha^{2})=F_{n}$. Proceeding inductively, we discover that if $\\nu(\\alpha)=t_{i}^{n}$, $\\nu(\\alpha^{j})\\in D_{n}$, for $1\\leq j\\leq i$, and $\\nu(\\alpha^{i+1})=F_{n}$. With the aid of these observations, we obtain the following possible scenarios in $\\mathcal{RM}_{C_{n}}$, where we will write $X^{*}$ to mean that the value $X$ is obtained from the fact a valuation is in $\\mathcal{F}_{C_{n}}$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|r|}\n\\hline\n$\\alpha$ & $\\alpha\\wedge\\neg\\alpha$ & $\\alpha^{1}$ & $\\alpha^{1}\\wedge\\neg(\\alpha^{1})$ & $\\alpha^{2}$ & $\\alpha^{2}\\wedge\\neg(\\alpha^{2})$ & $\\cdots$ & $\\alpha^{n-1}$ & $\\alpha^{n-1}\\wedge\\neg(\\alpha^{n-1})$ & $\\alpha^{n}$ & $\\alpha^{(n)}$\\\\ \\hline\n$T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ & $\\cdots$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $T_{n}$ \\\\ \\hline\n$t_{0}^{n}$ & $T_{n}^{*}$ & $F_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ & $\\cdots$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ \\\\ \\hline\n$t_{1}^{n}$ & $I_{n}^{*}$ & $t_{0}^{n *}$ & $T_{n}^{*}$ & $F_{n}$ & $F_{n}$ & $\\cdots$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ \\\\ \\hline\n$t_{2}^{n}$ & $I_{n}^{*}$ & $t_{1}^{n *}$ & $I_{n}^{*}$ & $t_{0}^{n *}$ & $T_{n}^{*}$ & $\\cdots$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ \\\\ \\hline\n$\\vdots$ & $\\vdots$ & $\\vdots$ & $\\vdots$ & $\\vdots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\vdots$ & $\\vdots$ & $\\vdots$ \\\\ \\hline\n$t_{n-3}^{n}$ & $I_{n}^{*}$ & $t_{n-4}^{n *}$ & $I_{n}^{*}$ & $t_{n-5}^{n *}$ & $I_{n}^{*}$ & $\\cdots$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ \\\\ \\hline\n$t_{n-2}^{n}$ & $I_{n}^{*}$ & $t_{n-3}^{n *}$ & $I_{n}^{*}$ & $t_{n-4}^{n *}$ & $I_{n}^{*}$ & $\\cdots$ & $F_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ \\\\ \\hline\n$t_{n-1}^{n}$ & $I_{n}^{*}$ & $t_{n-2}^{n *}$ & $I_{n}^{*}$ & $t_{n-3}^{n *}$ & $I_{n}^{*}$ & $\\cdots$ & $t_{0}^{n *}$ & $T_{n}^{*}$ & $F_{n}$ & $F_{n}$ \\\\ \\hline\n$F_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ & $T_{n}$ & $F_{n}$ & $\\cdots$ &$T_{n}$ & $F_{n}$ & $T_{n}$ & $T_{n}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for the scenarios in $\\mathcal{RM}_{C_{n}}$}\n\\end{figure}\n\nOne sees that, if $\\nu(\\alpha)=t_{i}^{n}$, then the fact that $\\nu\\in\\mathcal{F}_{C_{n}}$ restricts the possible values of $2i+1$ formulas of the form $\\alpha^{j}$ or $\\alpha^{k}\\wedge\\neg(\\alpha^{k})$: $\\alpha\\wedge\\neg\\alpha$, $\\alpha^{1}$, $\\alpha^{1}\\wedge\\neg(\\alpha^{1}), \\dotsc  , \\alpha^{i}$ and $\\alpha^{i}\\wedge\\neg(\\alpha^{i})$.\n\nAlso worthy of note is the fact that, if we define $\\bot_{\\alpha}=(\\alpha\\wedge\\neg\\alpha)\\wedge\\alpha^{(n)}$ (the definable bottom of $C_{n}$), for all $\\nu\\in\\mathcal{F}_{C_{n}}$ one has $\\nu(\\bot_{\\alpha})=F_{n}$; and if we define ${\\sim}\\alpha=\\alpha\\rightarrow\\bot_{\\alpha}$, for $\\nu\\in\\mathcal{F}_{C_{n}}$ (the definable strong negation on $C_{n}$) we find that $\\nu(\\alpha)\\in D_{n}$ implies $\\nu({\\sim}\\alpha)=F_{n}$, while $\\nu(\\alpha)=F_{n}$ implies $\\nu({\\sim}\\alpha)=T_{n}$.\n\n\\begin{lemma}\\label{Scenarions for RMCn}\nDenote, for simplicity, $T_{n}$ by $t^{n}_{-1}$; for any formula $\\alpha$ on $C_{n}$, valuation $\\nu$ on $\\mathcal{F}_{C_{n}}$ and integer $1\\leq k\\leq n$, one has that:\n\\begin{enumerate}\n\\item if $\\nu(\\alpha)=t^{n}_{i}$, for $-1\\leq i\\leq k-2$, $\\nu(\\alpha^{k})=T_{n}$;\n\\item if $\\nu(\\alpha)=t^{n}_{k-1}$, $\\nu(\\alpha^{k})=F_{n}$;\n\\item if $\\nu(\\alpha)=t^{n}_{i}$, for $k\\leq i\\leq n-1$, $\\nu(\\alpha^{k})=t^{n}_{i-k}$;\n\\item if $\\nu(\\alpha)=F_{n}$, $\\nu(\\alpha^{k})=T_{n}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the lemma by induction on $k$, starting with the case $k=1$.\n\\begin{enumerate}\n\\item If $\\nu(\\alpha)=t^{n}_{-1}$, meaning $\\nu(\\alpha)=T_{n}$, $\\nu(\\neg\\alpha)=F_{n}$ and therefore $\\nu(\\alpha\\wedge\\neg\\alpha)=F_{n}$, implying $\\nu(\\alpha^{1})=T_{n}$.\n\\item If $\\nu(\\alpha)=t^{n}_{0}$, from the fact that $\\nu$ is in $\\mathcal{F}_{C_{n}}$ we derive that $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{n}$, and therefore $\\nu(\\alpha^{1})=F_{n}$.\n\\item If $\\nu(\\alpha)=t^{n}_{i}$, for $1\\leq i\\leq n-1$, again from the fact that $\\nu\\in\\mathcal{F}_{C_{n}}$ we find $\\nu(\\alpha^{1})=t^{n}_{i-1}$.\n\\item Finally, if $\\nu(\\alpha)=F_{n}$, $\\nu(\\alpha\\wedge\\neg\\alpha)=F_{n}$ and therefore $\\nu(\\alpha^{1})=T_{n}$.\n\\end{enumerate}\n\nNow, suppose the result holds for $k-1$.\n\n\\begin{enumerate}\n\\item If $\\nu(\\alpha)=t^{n}_{i}$, for $-1\\leq i\\leq k-2$, there are two cases to consider: if $i\\leq k-3$, by induction hypothesis $\\nu(\\alpha^{k-1})=T_{n}$, meaning $\\nu(\\neg(\\alpha^{k-1}))=F_{n}$, $\\nu(\\alpha^{k-1}\\wedge\\neg(\\alpha^{k-1}))=F_{n}$ and $\\nu(\\alpha^{k})=T_{n}$; if $i=k-2$, again by induction hypothesis $\\nu(\\alpha^{k-2})=F_{n}$, implying $\\nu(\\alpha^{k-1}\\wedge\\neg(\\alpha^{k-1}))=F_{n}$ and therefore $\\nu(\\alpha^{k})=T_{n}$.\n\n\\item If $\\nu(\\alpha)=t^{n}_{k-1}$, by induction hypothesis $\\nu(\\alpha^{k-1})=t^{n}_{0}$, and from the fact $\\nu$ is in $\\mathcal{F}_{C_{n}}$ we get $\\nu(\\alpha^{k-1}\\wedge\\neg(\\alpha^{k-1}))=T_{n}$, meaning $\\nu(\\alpha^{k})=F_{n}$.\n\n\\item If $\\nu(\\alpha)=t^{n}_{i}$, for $k\\leq i\\leq n-1$, by induction hypothesis we find that $\\nu(\\alpha^{k-1})=t^{n}_{i-k+1}$. Since $k\\leq i\\leq n-1$ and $1\\leq k\\leq n$, it follows that  $1\\leq i-k+1\\leq n-1$, and from the fact that $\\nu$ is in $\\mathcal{F}_{C_{n}}$ we obtain $\\nu(\\alpha^{k})=t^{n}_{i-k}$.\n\n\\item Finally, if $\\nu(\\alpha)=F_{n}$, again by induction hypothesis $\\nu(\\alpha^{k-1})=F_{n}$, meaning $\\nu(\\alpha^{k-1}\\wedge\\neg(\\alpha^{k-1}))=F_{n}$ and therefore $\\nu(\\alpha^{k})=T_{n}$.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lemma}\\label{restricted valuations and swap part 1}\nLet $\\nu$ be a valuation in $\\mathcal{F}_{C_{n}}$; then the mapping $\\mathsf{b}:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\textbf{2}$ given by $\\mathsf{b}(\\alpha):=\\nu(\\alpha)_{1}$ (that is, $\\mathsf{b}(\\alpha)=1$ if and only if $\\nu(\\alpha)\\in D_{n}$) is a $C_{n}$-bivaluation.\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item For any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\mathsf{b}(\\alpha\\#\\beta)=1$ if and only if $\\nu(\\alpha\\#\\beta)_{1}=1$; since $\\nu(\\alpha\\#\\beta)_{1}=\\nu(\\alpha)_{1}\\#\\nu(\\beta)_{1}$ by the definition of $\\tilde{\\#}$, we have that:\n\\begin{enumerate}\n\\item $\\mathsf{b}(\\alpha\\vee\\beta)=1$ if and only if either $\\mathsf{b}(\\alpha)=1$ or $\\mathsf{b}(\\beta)=1$ (clause $(B2)$ for being a $C_{n}$-bivaluation);\n\\item $\\mathsf{b}(\\alpha\\wedge\\beta)=1$ if and only if $\\mathsf{b}(\\alpha)=1$ and $\\mathsf{b}(\\beta)=1$ (clause $(B1)$);\n\\item $\\mathsf{b}(\\alpha\\rightarrow\\beta)=1$ if and only if $\\mathsf{b}(\\alpha)=0$ or $\\mathsf{b}(\\beta)=1$ (clause $(B3)$).\n\\end{enumerate}\n\n\\item If $\\mathsf{b}(\\alpha)=0$, this means $\\nu(\\alpha)_{1}=0$; since $\\nu(\\alpha)_{1}\\vee\\nu(\\alpha)_{2}=1$, by definition of $B_{n}$, and $\\nu(\\neg\\alpha)_{1}=\\nu(\\alpha)_{2}$, by definition of $\\tilde{\\neg}$, we find that $\\mathsf{b}(\\neg\\alpha)=\\nu(\\neg\\alpha)_{1}=1$, satisfying clause $(B4)$.\n\n\\item If $\\mathsf{b}(\\neg\\neg\\alpha)=1$, we have $\\nu(\\neg\\neg\\alpha)_{1}=1$; we have $\\nu(\\neg\\neg\\alpha)_{1}=\\nu(\\neg\\alpha)_{2}$ and $\\nu(\\neg\\alpha)_{2}\\leq \\nu(\\alpha)_{1}$ from the definition of $\\tilde{\\neg}$, and so $\\nu(\\alpha)_{1}\\geq 1$, what means that $\\mathsf{b}(\\alpha)=\\nu(\\alpha)_{1}=1$ and that clause $(B5)$ is satisfied.\n\n\\item If $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}(\\neg(\\alpha^{n-1}))$, then $\\nu(\\alpha^{n-1})_{1}=\\nu(\\neg(\\alpha^{n-1}))_{1}=\\nu(\\alpha^{n-1})_{2}$; from looking at the definition of $B_{n}$, we see this implies $\\nu(\\alpha^{n-1})\\in I_{n}$. From Lemma \\ref{Scenarions for RMCn}, we obtain $\\nu(\\alpha)=t^{n}_{n-1}$, and in this case $\\nu(\\alpha^{n})=F_{n}$, which implies $\\mathsf{b}(\\alpha^{n})=\\nu(\\alpha^{n})_{1}=0$.\n\nReciprocally, if $\\mathsf{b}(\\alpha^{n})=0$, $\\nu(\\alpha^{n})=F_{n}$, and from Lemma \\ref{Scenarions for RMCn} once again $\\nu(\\alpha)=t^{n}_{n-1}$: so $\\nu(\\alpha^{n-1})=t^{n}_{0}$, $\\nu(\\neg(\\alpha^{n-1}))\\in D_{n}$ and therefore \n\\[\\mathsf{b}(\\alpha^{n-1})=\\nu(\\alpha^{n-1})_{1}=1=\\nu(\\neg(\\alpha^{n-1}))_{1}=\\mathsf{b}(\\neg(\\alpha^{n-1})).\\]\nWe then find clause $(B6)_{n}$ is satisfied.\n\n\\item If $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)$, $\\nu(\\alpha)_{1}=\\nu(\\neg\\alpha)_{1}=\\nu(\\alpha)_{2}=1$, what means $\\nu(\\alpha)\\in I_{n}$; from Lemma \\ref{Scenarions for RMCn}, in this case we have that $\\nu(\\alpha^{1})\\in B_{n}\\setminus\\{T_{n}\\}$, and from the definition of $\\tilde{\\neg}$ one obtains that $\\nu(\\neg(\\alpha^{1}))\\in D_{n}$, and so $\\mathsf{b}(\\neg(\\alpha^{1}))=\\nu(\\neg(\\alpha^{1}))_{1}=1$.\n\nReciprocally, if $\\mathsf{b}(\\neg(\\alpha^{1}))=1$, $\\nu(\\neg(\\alpha^{1}))_{1}=1$ and therefore $\\nu(\\alpha^{1})_{2}=1$, meaning $\\nu(\\alpha^{1})\\in B_{n}\\setminus\\{T_{n}\\}$; this implies $\\nu(\\alpha)\\in I_{n}$, and by the definition of $\\tilde{\\neg}$, $\\nu(\\neg\\alpha)\\in D_{n}$. So $\\mathsf{b}(\\alpha)=\\nu(\\alpha)_{1}$ and $\\mathsf{b}(\\neg\\alpha)=\\nu(\\neg\\alpha)_{1}$ are both $1$, and we have handled clause $(B7)$.\n\n\\item If $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta)\\neq\\mathsf{b}(\\neg\\beta)$, $\\nu(\\alpha)_{1}\\neq\\nu(\\neg\\alpha)_{1}=\\nu(\\alpha)_{2}$ and $\\nu(\\beta)_{1}\\neq\\nu(\\neg\\beta)_{1}=\\nu(\\beta)_{2}$, meaning $\\nu(\\alpha), \\nu(\\beta)\\in \\{F_{n}, T_{n}\\}$. From the tables for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$ we see that $\\nu(\\alpha\\#\\beta)\\in\\{F_{n}, T_{n}\\}$, and again $\\mathsf{b}(\\alpha\\#\\beta)=\\nu(\\alpha\\#\\beta)_{1}$ differs from $\\mathsf{b}(\\neg(\\alpha\\#\\beta))=\\nu(\\neg(\\alpha\\#\\beta))_{1}$, what constitutes clause $(B8)$ and finishes the proof.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lemma}\\label{restricted valuations and swap part 2}\nLet $\\mathsf{b}$ be a $C_{n}$-bivaluation; then, the mapping $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}$, given by $\\nu(\\alpha)=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1}))$ is a valuation in $\\mathcal{F}_{C_{n}}$ such that $\\mathsf{b}(\\alpha)=1$ if and only if $\\nu(\\alpha)\\in D_{n}$ for every formula $\\alpha$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, we show $\\nu$ is a homomorphism.\n\\begin{enumerate}\n\\item We have that $\\nu(\\neg\\alpha)=(\\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\neg\\neg\\alpha), \\mathsf{b}((\\neg\\alpha)^{1}), \\dotsc  ,  \\mathsf{b}((\\neg\\alpha)^{n-1}))$, and so there are three cases to consider: if $\\nu(\\alpha)=T_{n}$, $\\mathsf{b}(\\neg\\alpha)=0$ and therefore $\\nu(\\neg\\alpha)=F_{n}$; if $\\nu(\\alpha)\\in I_{n}$, $\\mathsf{b}(\\neg\\alpha)=1$, implying $\\nu(\\neg\\alpha)\\in D_{n}$; finally, if $\\nu(\\alpha)=F_{n}$, $\\mathsf{b}(\\alpha)=0$ and $\\mathsf{b}(\\neg\\alpha)=1$, and from clause $(B5)$ for being a $C_{n}$-bivaluation we get $\\mathsf{b}(\\neg\\neg\\alpha)=0$, and so $\\nu(\\neg\\alpha)=T_{n}$. Regardless of the case, we find $\\nu(\\neg\\alpha)\\in \\tilde{\\neg}\\nu(\\alpha)$.\n\n\\item From the definition of $\\nu$, $\\nu(\\alpha\\vee\\beta)=(\\mathsf{b}(\\alpha\\vee\\beta), \\mathsf{b}(\\neg(\\alpha\\vee\\beta)), \\mathsf{b}((\\alpha\\vee\\beta)^{1}), \\dotsc  , \\mathsf{b}((\\alpha\\vee\\beta)^{n-1})$. \n\\begin{enumerate}\n\\item If $\\nu(\\alpha)$ or $\\nu(\\beta)$ equals $T_{n}$ and both are boolean valued, either $\\mathsf{b}(\\alpha)$ or $\\mathsf{b}(\\beta)$ equals $1$, meaning $\\mathsf{b}(\\alpha\\vee\\beta)=1$ from clause $(B2)$; and both $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta)\\neq\\mathsf{b}(\\neg\\beta)$, which implies from clause $(B8)$ that $\\mathsf{b}(\\alpha\\vee\\beta)\\neq\\mathsf{b}(\\neg(\\alpha\\vee\\beta))$ and therefore $\\nu(\\alpha\\vee\\beta)=T_{n}$.\n\\item If $\\nu(\\alpha)$ or $\\nu(\\beta)$ is in $I_{n}$, we have that either $\\mathsf{b}(\\alpha)=1$ or $\\mathsf{b}(\\beta)=1$ and from clause $(B2)$ one gets $\\mathsf{b}(\\alpha\\vee\\beta)=1$, meaning $\\nu(\\alpha\\vee\\beta)\\in D_{n}$.\n\\item If $\\nu(\\alpha)$ and $\\nu(\\beta)$ are both $F_{n}$, $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\beta)=0$ and so $\\mathsf{b}(\\alpha\\vee\\beta)=0$ from $(B2)$, meaning that $\\nu(\\alpha\\vee\\beta)=F_{n}$.\n\\end{enumerate}\nEither way, $\\nu(\\alpha\\vee\\beta)$ is in $\\nu(\\alpha)\\tilde{\\vee}\\nu(\\beta)$.\n\n\\item We have $\\nu(\\alpha\\wedge\\beta)=(\\mathsf{b}(\\alpha\\wedge\\beta), \\mathsf{b}(\\neg(\\alpha\\wedge\\beta)), \\mathsf{b}((\\alpha\\wedge\\beta)^{1}), \\dotsc  , \\mathsf{b}((\\alpha\\wedge\\beta)^{n-1})$. \n\\begin{enumerate}\n\\item If $\\nu(\\alpha)=\\nu(\\beta)=T_{n}$, $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\beta)=1$ and $\\mathsf{b}(\\neg\\alpha)=\\mathsf{b}(\\neg\\beta)=0$; from clause $(B1)$, $\\mathsf{b}(\\alpha\\wedge\\beta)=1$, and from $(B8)$ we get $\\mathsf{b}(\\neg(\\alpha\\wedge\\beta))=0$, meaning $\\nu(\\alpha\\wedge\\beta)=T_{n}$.\n\\item If $\\nu(\\alpha)$ or $\\nu(\\beta)$ equals $F_{n}$, either $\\mathsf{b}(\\alpha)$ or $\\mathsf{b}(\\beta)$ equals $0$, and therefore $\\mathsf{b}(\\alpha\\wedge\\beta)=0$ (from clause $(B1)$) and so $\\nu(\\alpha\\wedge\\beta)=F_{n}$.\n\\item In the remaining cases, when either $\\nu(\\alpha)$ or $\\nu(\\beta)$ is in $I_{n}$ but none of them equals $F_{n}$, one sees that $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\beta)=1$ and therefore $\\mathsf{b}(\\alpha\\wedge\\beta)=1$, meaning $\\nu(\\alpha\\wedge\\beta)\\in D_{n}$.\n\\end{enumerate}\nWe have just proved $\\nu(\\alpha\\wedge\\beta)\\in \\nu(\\alpha)\\tilde{\\wedge}\\nu(\\beta)$.\n\n\\item Clearly $\\nu(\\alpha\\rightarrow\\beta)=(\\mathsf{b}(\\alpha\\rightarrow\\beta), \\mathsf{b}(\\neg(\\alpha\\rightarrow\\beta)), \\mathsf{b}((\\alpha\\rightarrow\\beta)^{1}), \\dotsc  , \\mathsf{b}((\\alpha\\rightarrow\\beta)^{n-1})$. \n\\begin{enumerate}\n\\item If $\\nu(\\alpha)=F_{n}$ or $\\nu(\\beta)=T_{n}$, and both are boolean valued, $\\mathsf{b}(\\alpha)=0$ or $\\mathsf{b}(\\beta)=1$, and both $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta)\\neq\\mathsf{b}(\\neg\\beta)$. By clauses $(B3)$ and $(B8)$, this all means $\\mathsf{b}(\\alpha\\rightarrow\\beta)=1$ and $\\mathsf{b}(\\neg(\\alpha\\rightarrow\\beta))=0$, and so $\\nu(\\alpha\\rightarrow\\beta)=T_{n}$.\n\\item When $\\nu(\\beta)\\in I_{n}$, $\\mathsf{b}(\\beta)=1$ and so $\\mathsf{b}(\\alpha\\rightarrow\\beta)=1$ from clause $(B3)$, meaning $\\nu(\\alpha\\rightarrow\\beta)\\in D_{n}$.\n\\item If $\\nu(\\beta)=F_{n}$ and $\\nu(\\alpha)\\in D_{n}$, $\\mathsf{b}(\\beta)=0$ and $\\mathsf{b}(\\alpha)=1$, implying from clause $(B3)$ that $\\mathsf{b}(\\alpha\\rightarrow\\beta)=0$ and so $\\nu(\\alpha\\rightarrow\\beta)=F_{n}$.\n\\item Finally, if $\\nu(\\alpha)\\in I_{n}$ and $\\nu(\\beta)=T_{n}$, $\\mathsf{b}(\\alpha)=1$ and $\\mathsf{b}(\\beta)=1$; from clause $(B3)$ once again, $\\mathsf{b}(\\alpha\\rightarrow\\beta)=1$ and therefore $\\nu(\\alpha\\rightarrow\\beta)\\in D_{n}$.\n\\end{enumerate}\nIn all cases, $\\nu(\\alpha\\rightarrow\\beta)\\in\\nu(\\alpha)\\tilde{\\rightarrow}\\nu(\\beta)$.\n\\end{enumerate}\n\nNow, it remains to be shown that $\\nu$ is in $\\mathcal{F}_{C_{n}}$.\n\\begin{enumerate}\n\\item If $\\nu(\\alpha)=t^{n}_{0}$, we have $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=1$ (and so $\\mathsf{b}(\\alpha\\wedge\\neg\\alpha)=1$) and $\\mathsf{b}(\\neg(\\alpha\\wedge\\neg\\alpha))=\\mathsf{b}(\\alpha^{1})=0$, from what one derives $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{n}$.\n\\item If $\\nu(\\alpha)=t^{n}_{k-1}$, for $2\\leq k\\leq n$, we have $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=1$ (meaning $\\mathsf{b}(\\alpha\\wedge\\neg\\alpha)=1$) and $\\mathsf{b}(\\neg(\\alpha\\wedge\\neg\\alpha))=\\mathsf{b}(\\alpha^{1})=1$, and so $\\nu(\\alpha\\wedge\\neg\\alpha)\\in I_{n}$.\n\nFurthermore, $\\mathsf{b}((\\alpha^{1})^{k-1})=\\mathsf{b}(\\alpha^{k})=0$, from the fact that $\\nu(\\alpha)=t^{n}_{k-1}$, and since $\\mathsf{b}((\\alpha^{1})^{k-1})=0$ we find $\\nu(\\alpha^{1})=t^{n}_{k-2}$, what ends the proof.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $C_{n}$, $\\Gamma\\vdash_{C_{n}}\\varphi$ if, and only if, $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nFirst, suppose $\\Gamma\\vdash_{C_{n}}\\varphi$, and take a valuation $\\nu\\in\\mathcal{F}_{C_{n}}$ for which $\\nu(\\Gamma)\\subseteq D_{n}$: from Lemma \\ref{restricted valuations and swap part 1}, the function $\\mathsf{b}:F(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\textbf{2}$, defined by $\\mathsf{b}(\\alpha)=\\nu(\\alpha)_{1}$, is a bivaluation which, by hypothesis, satisfies $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$. Given the soundness of $C_{n}$ with respect to bivaluations and the fact that $\\Gamma\\vdash_{C_{n}}\\varphi$, it follows that $\\mathsf{b}(\\varphi)=1$, and therefore $\\nu(\\varphi)\\in D_{n}$. This, of course, shows that $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$.\n\nReciprocally, suppose $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$ and let $\\mathsf{b}$ be a $C_{n}$-bivaluation satisfying $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$: by Lemma \\ref{restricted valuations and swap part 2}, $\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow \\mathcal{A}_{C_{n}}$ defined by \n\\[\\nu(\\alpha)=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1})),\\]\nis a valuation in $\\mathcal{F}_{C_{n}}$ for which, in addition, $\\nu(\\Gamma)\\subseteq D_{n}$. Given $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$, we obtain that $\\nu(\\varphi)\\in D_{n}$, meaning $\\mathsf{b}(\\varphi)=1$ and, by completeness of $C_{n}$ with respect to bivaluations, $\\Gamma\\vdash_{C_{n}}\\varphi$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Decision methods}\\label{Decision methods}\n\nA finite Nmatrix $\\mathcal{M}=(\\mathcal{A}, D)$ which characterizes a logic $\\mathfrak{L}$ always leads to a decision method for when a formula $\\varphi$ is a tautology of $\\mathfrak{L}$, or when a finite set of premises $\\Gamma$ deduces $\\varphi$ in $\\mathfrak{L}$ as long as this logic satisfies the deduction meta-theorem (if $\\Gamma=\\{\\gamma_{1}, \\dotsc  , \\gamma_{n}\\}$, this is equivalent to testing whether $\\gamma_{1}\\rightarrow(\\gamma_{2}\\rightarrow\\cdots(\\gamma_{n}\\rightarrow\\varphi)\\cdots)$ is a tautology). The method is straightforward: one constructs a row-branching truth table for $\\varphi$, based on $\\mathcal{A}$, and $\\varphi$ is a tautology iff all rows contain a designated value on the column corresponding to $\\varphi$. Even more, if a logic is characterized by a finite class of finite Nmatrices, one still derives a decision method.\n\nThings are not as simple when dealing with RNmatrices: even the logic characterized by a single, finite RNmatrix may not have a decision method, given that the problem of finding out whether a certain homomorphism belongs to the set of restricted homomorphisms may not be decidable; an example of something like this happening is found by looking at the RNmatrix semantics obtained by Kearns for $S4$ (\\cite{Kearns}) in Example \\ref{Example Kearns}. However this is not always the case: some finite RNmatrices are indeed quite efficient in inducing decision methods, and this happens for all $\\mathcal{RM}_{C_{n}}$; this is due to the fact that finding those homomorphisms not relevant for a deduction in $\\mathcal{RM}_{C_{n}}$ is actually easy. We start by generalizing row-branching truth-tables to row-branching, row-eliminating truth-tables, and proceed to show how tableau semantics are defined from our RNmatrices characterizing $C_{n}$.\n\n\n\\subsection{Row-branching, row-eliminating truth tables}\\label{Row-branching,row-eliminating}\n\nTake a formula $\\varphi$ in the signature of $C_{n}$, and let $\\varphi_{1}, \\dotsc  , \\varphi_{k}=\\varphi$ be a sequence of all subformulas, proper or not, of $\\varphi$ ordered by complexity.\n\n\\begin{figure}[H]\n\\begin{center}\n\\scalebox{0.9}{\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|}\n\\hline\n$\\varphi_{1}$ & $\\varphi_{2}$ & $\\cdots$ & $\\varphi_{p}$\\\\\\hline\n\\multirow{13}{*}{$T_{n}$} & \\multirow{4}{*}{$T_{n}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\cline{2-4}\n& \\multirow{4}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\cline{2-4}\n& $\\vdots$ & $\\ddots$ & $\\vdots$ \\\\\\cline{2-4}\n& \\multirow{4}{*}{$F_{n}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\hline\n\\multirow{13}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$T_{n}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\cline{2-4}\n& \\multirow{4}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\cline{2-4}\n& $\\vdots$ & $\\ddots$ & $\\vdots$ \\\\\\cline{2-4}\n& \\multirow{4}{*}{$F_{n}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\hline\n$\\vdots$ & $\\vdots$ & $\\ddots$ & $\\vdots$\\\\\\hline\n\\multirow{13}{*}{$F_{n}$} & \\multirow{4}{*}{$T_{n}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\cline{2-4}\n& \\multirow{4}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\cline{2-4}\n& $\\vdots$ & $\\ddots$ & $\\vdots$ \\\\\\cline{2-4}\n& \\multirow{4}{*}{$F_{n}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$\\\\\\cline{4-4}\n& & & $t^{n}_{0}$\\\\\\cline{4-4}\n& & & $\\vdots$\\\\\\cline{4-4}\n& & & $F_{n}$\\\\\\hline\n\\end{tabular}}\n\\end{center}\n\\caption{Beginning a truth table: only propositional variables}\n\\label{Truth table propositional variables}\n\\end{figure}\n\\vspace{5mm}\n\nIn order to make things as clear as possible, we remember the complexity (or order) of a formula $\\varphi$ is $0$ iff the formula is a propositional variable or a nullary connective, and it is $r+1$ iff $\\varphi=\\sigma(\\alpha_{1}, \\dotsc  , \\alpha_{m})$, $\\sigma$ is a $m$-ary connective and the maximum of the orders of $\\alpha_{1}$ trough $\\alpha_{m}$ is $r$. We say ``a sequence'', rather than ``the sequence'', since different subformulas of $\\varphi$ may have the same order, and therefore may be interchanged without neglecting that the whole sequence remains ordered.\n\nOf course, given that the signature of $C_{n}$ has no nullary connectives, we can be certain that the first elements of the sequence are necessarily propositional variables, and so we start to build a truth table. For each $\\varphi_{i}$ which is a propositional variable we list all possible values it may assume in $\\mathcal{RM}_{C_{n}}$ given the values taken by $\\varphi_{1}$ trough $\\varphi_{i-1}$; since the values taken by propositional variables are independent of each other, we find that, for any combination of values for $\\varphi_{0}$ up to $\\varphi_{i-1}$, $\\varphi_{i}$ can assume $n+2$ values, $T_{n}, t^{n}_{0}, t^{n}_{1}, \\dotsc  , t^{n}_{n-1}$ and $F_{n}$. If $p\\geq 1$ propositional variables appear in $\\varphi$, this means we start by writing down $n+2$ rows for the column headed by $\\varphi_{1}$, each containing a value of $B_{n}$; for the column headed by $\\varphi_{2}$, assuming $p\\geq 2$, each of these $n+2$ rows is further subdivided into $n+2$ new rows, each filled with a value of $B_{n}$, to a total now of $(n+2)^{2}$ rows; and inductively, we have a table with $(n+2)^{p}$ rows, and $p$ columns (so far).\n\nWe summarize the situation so far with the table in Figure \\ref{Truth table propositional variables}, showing how the first part of the process, dealing only with propositional variables, goes. For simplicity, on all row-branching truth-tables we will always merge consecutive rows on a given column with repeated values; this is very helpful when truly trying to interpret those objects as ``row-branching''.\n\nSuppose we have written the table up to a column headed by $\\varphi_{q}$, for $q\\geq p$, and let us look at $\\varphi_{q+1}$: it must be of either the form ~(1)~ $\\varphi_{q+1}=\\varphi_{i}\\#\\varphi_{j}$, for some $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, or ~(2)~ $\\varphi_{q+1}=\\neg\\varphi_{i}$, for $1\\leq i, j\\leq q$, since the subformulas of $\\varphi_{q+1}$ are also subformulas of $\\varphi$ of order strictly smaller. In a normal truth-table, the new column, headed by $\\varphi_{q+1}$, would be filled on a given row by either $a\\tilde{\\#}b$, corresponding to case $(1)$, or $\\tilde{\\neg}a$, corresponding to case $(2)$, where: $a$ is the value taken by $\\varphi_{i}$ on the aforementioned row, $b$ is the value taken by $\\varphi_{j}$ and $\\tilde{\\#}$ is the (deterministic) operation corresponding to the connective $\\#$. In a truth-table arising from an Nmatrix, we must further subdivide the row in question in as many values are found in $a\\tilde{\\#}b$ and fill each new row with one of these values, whenever we find ourselves in case $(1)$, or branch the row in as many rows as there are elements in $\\tilde{\\neg}a$ and fill each sub-row with one of these, corresponding to case $(2)$, where now $\\tilde{\\#}$ is a multioperation.\n\nThe case with the RNmatrix $\\mathcal{RM}_{C_{n}}$ is similar, but has extra subtleties. Fix a given row for what follows.\n\\begin{enumerate}\n\\item If $\\varphi_{q+1}=\\varphi_{i}\\wedge\\neg\\varphi_{i}$ and $\\varphi_{i}$ takes the value $t^{n}_{0}$, then the row is not branched in the column headed by $\\varphi_{q+1}$ and assumes the value $T_{n}$.\n\\begin{center}\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{2cm}|p{1cm}|}\n\\hline\n$\\cdots$ & $\\varphi_{i}$ & $\\cdots$ & $\\neg\\varphi_{i}$ & $\\cdots$ & $\\varphi_{i}\\wedge\\neg\\varphi_{i}$ & $\\cdots$ \\\\\\hline\n$\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$\\\\\\hline\n\\multirow{4}{*}{$\\cdots$} & \\multirow{4}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$\\cdots$} & $T_{n}$ & $\\cdots$ & $T_{n}$ & $\\cdots$ \\\\\\cline{4-7}\n& & &$t^{n}_{0}$ & $\\cdots$ & $T_{n}$ & $\\cdots$ \\\\\\cline{4-7}\n& & &$\\vdots$ & $\\cdots$ & $\\vdots$ & $\\cdots$ \\\\\\cline{4-7}\n& & &$t^{n}_{n-1}$ & $\\cdots$ & $T_{n}$ & $\\cdots$ \\\\\\hline\n$\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$\\\\\\hline\n\\end{tabular}\n\\end{center}\n\n\\item If $\\varphi_{q+1}=\\varphi_{i}\\wedge\\neg\\varphi_{i}$ and $\\varphi_{i}$ takes the value $t^{n}_{l}$, for any $1\\leq l\\leq n-1$, the row is subdivided into $n$ new rows, each assigned a value of $I_{n}=\\{t_{0}^{n}, \\dotsc  , t_{n-1}^{n}\\}$.\n\n\\begin{center}\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1.5cm}|p{1cm}|}\n\\hline\n$\\cdots$ & $\\varphi_{i}$ & $\\cdots$ & $\\neg\\varphi_{i}$ & $\\cdots$ & $\\varphi_{i}\\wedge\\neg\\varphi_{i}$ & $\\cdots$ \\\\\\hline\n$\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ \\\\\\hline\n\\multirow{13}{*}{$\\cdots$} & \\multirow{13}{*}{$t^{n}_{l}$} & \\multirow{13}{*}{$\\cdots$} & \\multirow{4}{*}{$T_{n}$} & \\multirow{4}{*}{$\\cdots$} & $t^{n}_{0}$ & $\\cdots$ \\\\\\cline{6-7}\n&  &   &   &  & $\\vdots$ & $\\ddots$ \\\\\\cline{6-7}\n&  &   &   & & $t^{n}_{n-1}$ & $\\cdots$ \\\\\\cline{4-7}\n&  &   &  \\multirow{4}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$\\cdots$} & $t^{n}_{0}$ & $\\cdots$ \\\\\\cline{6-7}\n&  &   &   &  & $\\vdots$ & $\\ddots$ \\\\\\cline{6-7}\n&  &   &   &  & $t^{n}_{n-1}$ & $\\cdots$ \\\\\\cline{4-7}\n&  &   & $\\vdots$  & $\\ddots$ & $\\vdots$ & $\\ddots$ \\\\\\cline{4-7}\n&  &   &  \\multirow{4}{*}{$t^{n}_{n-1}$} & \\multirow{4}{*}{$\\cdots$} & $t^{n}_{0}$ & $\\cdots$ \\\\\\cline{6-7}\n&  &   &   &  & $\\vdots$ & $\\ddots$ \\\\\\cline{6-7}\n&  &   &   & & $t^{n}_{n-1}$ & $\\cdots$ \\\\\\hline\n$\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ \\\\\\hline\n\\end{tabular}\n\\end{center}\n\n\\item If $\\varphi_{q+1}=\\neg(\\varphi_{i}\\wedge\\neg\\varphi_{i})$ and $\\varphi_{i}$ takes the value $t^{n}_{l}$, for any $1\\leq l\\leq n-1$, the row is not branched any further in the column headed by $\\varphi_{q+1}$ and assumes the value $t^{n}_{l-1}$ on said column (look at Figure \\ref{Eliminate rows neg wedge neg}).\n\\end{enumerate}\n\nNotice all of these steps are easily algorithmically performed: after all, they boil down to identifying the form of a formula (namely $\\varphi_{q+1}$) and searching the value taken by another formula ($\\varphi_{i}$) on a given row; the intuition behind these procedures involves the fact that each row of a completed row-branching, row-eliminating truth-table should correspond to a possible homomorphism, and the previous three steps eliminate all homomorphisms which are not restricted.\n\nIf we are not in the cases described above, then one proceeds as if we were in an Nmatrix: assume that, on a fixed row, $\\varphi_{i}$ takes the value $a$ and $\\varphi_{j}$ takes the value $b$. Then, if $\\varphi_{q+1}=\\varphi_{i}\\#\\varphi_{j}$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, we divide our row into $|a\\tilde{\\#}b|$ new rows and assign each of the elements of the set $a\\tilde{\\#}b$ to one of these new rows, where $\\tilde{\\#}$ is the operation corresponding to the connective $\\#$ in the multialgebra $\\mathcal{A}_{C_{n}}$; notice that $|a\\tilde{\\#}b|$ may equal either $n+1$, when $a\\tilde{\\#}b=D_{n}=\\{T_{n}, t^{n}_{0}, \\dotsc  , t^{n}_{n-1}\\}$, or $1$, when $a\\tilde{\\#}b=\\{T_{n}\\}$ or $a\\tilde{\\#}b=\\{F_{n}\\}$. If $\\varphi_{q+1}=\\neg\\varphi_{i}$, the row is subdivided $|\\tilde{\\neg}a|$ times, and one element of $\\tilde{\\neg}a$ is placed on each of these; again we can have $|\\tilde{\\neg}a|=n+1$, when $\\tilde{\\neg}a=D_{n}$, or $|\\tilde{\\neg}a|=1$, when $\\tilde{\\neg}a=\\{T_{n}\\}$ or $\\tilde{\\neg}a=\\{F_{n}\\}$. For completeness sake, compare what the previous rules dictated against what would be expected if we were working with a mere Nmatrix.\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1.5cm}|p{1cm}|p{2cm}|p{1cm}|}\n\\hline\n$\\cdots$ & $\\varphi_{i}$ & $\\cdots$ & $\\neg\\varphi_{i}$ & $\\cdots$ & $\\varphi_{i}\\wedge\\neg\\varphi_{i}$ & $\\cdots$ & $\\neg(\\varphi_{i}\\wedge\\neg\\varphi_{i})$ & $\\cdots$\\\\\\hline\n$\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$\\\\\\hline\n\\multirow{13}{*}{$\\cdots$} & \\multirow{13}{*}{$t^{n}_{l}$} & \\multirow{13}{*}{$\\cdots$} & \\multirow{4}{*}{$T_{n}$} & \\multirow{4}{*}{$\\cdots$} & $t^{n}_{0}$ & $\\cdots$ & $t^{n}_{l-1}$ & $\\cdots$\\\\\\cline{6-9}\n&  &   &   &  & $\\vdots$ & $\\ddots$ &  $\\vdots$ & $\\ddots$\\\\\\cline{6-9}\n&  &   &   & & $t^{n}_{n-1}$ & $\\cdots$ &  $t^{n}_{l-1}$ & $\\cdots$\\\\\\cline{4-9}\n&  &   &  \\multirow{4}{*}{$t^{n}_{0}$} & \\multirow{4}{*}{$\\cdots$} & $t^{n}_{0}$ & $\\cdots$ &  $t^{n}_{l-1}$ & $\\cdots$\\\\\\cline{6-9}\n&  &   &   &  & $\\vdots$ & $\\ddots$ &  $\\vdots$ & $\\ddots$\\\\\\cline{6-9}\n&  &   &   &  & $t^{n}_{n-1}$ & $\\cdots$ &  $t^{n}_{l-1}$ & $\\cdots$\\\\\\cline{4-9}\n&  &   & $\\vdots$  & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$\\\\\\cline{4-9}\n&  &   &  \\multirow{4}{*}{$t^{n}_{n-1}$} & \\multirow{4}{*}{$\\cdots$} & $t^{n}_{0}$ & $\\cdots$ &  $t^{n}_{l-1}$ & $\\cdots$\\\\\\cline{6-9}\n&  &   &   &  & $\\vdots$ & $\\ddots$ &  $\\vdots$ & $\\ddots$\\\\\\cline{6-9}\n&  &   &   & & $t^{n}_{n-1}$ & $\\cdots$ &  $t^{n}_{l-1}$ & $\\cdots$\\\\\\hline\n$\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$ & $\\vdots$ & $\\ddots$\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Eliminating row in the case of a formula $\\neg(\\varphi_{i}\\wedge\\neg\\varphi_{i})$}\n\\label{Eliminate rows neg wedge neg}\n\\end{figure}\n\n\\begin{enumerate}\n\\item If $\\varphi_{q+1}=\\varphi_{i}\\wedge\\neg\\varphi_{i}$ and $\\varphi_{i}$ is $t^{n}_{0}$, $\\tilde{\\neg}t^{n}_{0}=D_{n}$ and so, for any value $b\\in \\tilde{\\neg}t^{n}_{0}$ that could be given to $\\neg\\varphi_{i}$, $t^{n}_{0}\\tilde{\\wedge}b=D_{n}$; so, only looking at $\\mathcal{A}_{C_{n}}$ would branch the row $n+1$ times, giving the values $T_{n}, t^{n}_{0}, \\dotsc  , t^{n}_{n-1}$ to $\\varphi_{q+1}$, instead of only $T_{n}$.\n\\item If $\\varphi_{q+1}=\\varphi_{i}\\wedge\\neg\\varphi_{i}$ and $\\varphi_{i}$ is $t^{n}_{l}$, for $1\\leq l\\leq n-1$, one has $\\tilde{\\neg}t^{n}_{l}=D_{n}$ and, for any value $b\\in\\tilde{\\neg}t^{n}_{l}$ the formula $\\neg\\varphi_{i}$ could assume, $t^{n}_{l}\\tilde{\\wedge}b=D_{n}$; this means that only considering $\\mathcal{A}_{C_{n}}$ would divide the row $n+1$ times, instead of $n$ times, and give $\\varphi_{q+1}$ the values $T_{n}, t^{n}_{0}, \\dotsc  , t^{n}_{n-1}$, instead of $t^{n}_{0}, \\dotsc  , t^{n}_{n-1}$.\n\\item If $\\varphi_{q+1}=\\neg(\\varphi_{i}\\wedge\\neg\\varphi_{i})$ and $\\varphi_{i}$ is $t^{n}_{l}$, for $1\\leq l\\leq n-1$, we obtain $\\tilde{\\neg}t^{n}_{l}=D_{n}$, for any value $b\\in\\tilde{\\neg}t^{n}_{l}$ that $\\neg\\varphi_{i}$ could assume $t^{n}_{l}\\tilde{\\wedge}b=D_{n}$ and, for any value $c\\in t^{n}_{l}\\tilde{\\wedge} b$ which $\\varphi_{i}\\wedge\\neg\\varphi_{i}$ could assume, $\\tilde{\\neg}c=\\{F_{n}\\}$ or $\\tilde{\\neg}c=D_{n}$; this means $\\varphi_{q+1}$ could, in principle, be assigned any value of $B_{n}$, and therefore subdivide its row into $n+2$ new ones. Instead, we demand that it does not branch its row any further and receives the value $t^{n}_{l-1}$. \n\\end{enumerate}\n\nSo, we are now in position to proceed writing our row-branching, row-eliminating truth-table as long as there are elements on the sequence $\\varphi_{1}, \\dotsc  , \\varphi_{k}=\\varphi$, giving us a table with $k$ columns and no more than $(n+2)^{k}$ rows. Of course, if at the end the column headed by $\\varphi_{k}$ is filled with nothing but designated values $D_{n}=\\{T_{n}, t^{n}_{0}, \\dotsc  , t^{n}_{n-1}\\}$, then $\\varphi$ is a tautology of $C_{n}$, being the converse also true: if $\\varphi$ is a tautology, by writing down its correspondent row-branching, row-eliminating truth-table, we obtain the column of $\\varphi$ has only designated values. We want, from here on forward, to prove that this is, in fact, the case, that is, that our truth-tables are sound and complete.\n\n\\begin{proposition}\\label{Rows of tables are homomorphisms}\nLet $\\Gamma$ be a finite set of formulas of $C_{n}$ closed by subformulas, that is, if $\\alpha\\in\\Gamma$ and $\\beta$ is a subformula of $\\alpha$, then $\\beta\\in\\Gamma$. Let $\\nu:\\Gamma\\rightarrow B_{n}$ be a function satisfying:\n\\begin{enumerate}\n\\item if $\\neg\\alpha\\in\\Gamma$, $\\nu(\\neg\\alpha)\\in\\tilde{\\neg}\\nu(\\alpha)$;\n\\item if $\\alpha\\#\\beta\\in\\Gamma$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, then $\\nu(\\alpha\\#\\beta)\\in\\nu(\\alpha)\\tilde{\\#}\\nu(\\beta)$;\n\\item if $\\alpha\\wedge\\neg\\alpha\\in\\Gamma$ and $\\nu(\\alpha)=t^{n}_{0}$, $\\nu(\\alpha\\wedge\\neg\\alpha)=T_{n}$;\n\\item if $\\alpha\\wedge\\neg\\alpha\\in\\Gamma$ and $\\nu(\\alpha)=t^{n}_{k}$, for some $1\\leq k\\leq n-1$, then $\\nu(\\alpha\\wedge\\neg\\alpha)\\in I_{n}=\\{t^{n}_{0}, \\dotsc  , t^{n}_{n-1}\\}$;\n\\item if $\\alpha^{1}=\\neg(\\alpha\\wedge\\neg\\alpha)\\in\\Gamma$ and $\\nu(\\alpha)=t^{n}_{k}$, for some $1\\leq k\\leq n-1$, then $\\nu(\\alpha^{1})=t^{n}_{k-1}$.\n\\end{enumerate}\nThen there exists a homomorphism $\\overline{\\nu}\\in\\mathcal{F}_{C_{n}}$ extending $\\nu$, \\textit{id est}, a homomorphism for which $\\overline{\\nu}(\\alpha)=\\nu(\\alpha)$, for every $\\alpha\\in\\Gamma$.\n\\end{proposition}\n\n\\begin{proof}\nWe define $\\overline{\\nu}$ by induction on the order of a formula, but whenever we define $\\overline{\\nu}(\\alpha)$ we will, simultaneously, define $\\overline{\\nu}(\\neg\\alpha)$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$ and $\\overline{\\nu}(\\alpha^{1})=\\overline{\\nu}(\\neg(\\alpha\\wedge\\neg\\alpha))$ as well.\n\nFirst, notice that, if $\\alpha\\not\\in\\Gamma$, then for no formula $\\beta\\in\\Gamma$ we have that $\\alpha$ is a subformula of $\\beta$, given $\\Gamma$ is closed by subformulas; of course this means that if $\\alpha\\not\\in \\Gamma$, then neither $\\neg\\alpha$, $\\alpha\\wedge\\neg\\alpha$ or $\\alpha^{1}$ are in $\\Gamma$. This is important since, whenever we define $\\overline{\\nu}(\\alpha)$ for a formula $\\alpha\\not\\in \\Gamma$, we need not worry if this definition may interfere with $\\nu(\\beta)$ for some $\\beta\\in\\Gamma$ of which $\\alpha$ is a subformula: such a $\\beta$ can not exist.\n\nFor a propositional variable $p$, $\\overline{\\nu}(p)$ is defined as $\\nu(p)$, whenever $p\\in\\Gamma$, and arbitrarily otherwise; we have then defined $\\overline{\\nu}(\\alpha)$, for every formula $\\alpha$ of order $0$, given $C_{n}$ has no nullary connectives, so it remains to define $\\overline{\\nu}(\\neg\\alpha)$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$ and $\\overline{\\nu}(\\alpha^{1})$. \n\\begin{enumerate}\n\\item If $\\neg p\\in \\Gamma$, $\\overline{\\nu}(\\neg p)$ is equal to $\\nu(\\neg p)$, and if $\\neg p\\not\\in \\Gamma$ we may take $\\overline{\\nu}(\\neg p)$ as equal to any value of the set $\\tilde{\\neg}\\overline{\\nu}(p)$.\n\\item If $p\\wedge\\neg p\\in\\Gamma$, we define $\\overline{\\nu}(p\\wedge\\neg p)$ as $\\nu(p\\wedge\\neg p)$; if $p\\wedge\\neg p\\not\\in\\Gamma$ and $\\overline{\\nu}(p)=t^{n}_{0}$, we must define $\\overline{\\nu}(p\\wedge\\neg p)$ as $T_{n}$, but if $p\\wedge\\neg p\\not\\in \\Gamma$ and $\\overline{\\nu}(p)=t^{n}_{k}$, for some $1\\leq k\\leq n-1$, we can make $\\overline{\\nu}(p\\wedge\\neg p)$ equal to an arbitrary value in $I_{n}=\\{t^{n}_{0}, \\dotsc   , t^{n}_{n-1}\\}$; if none of the previous cases apply, we define $\\overline{\\nu}(p\\wedge \\neg p)$ as an arbitrary element of $\\overline{\\nu}(p)\\tilde{\\wedge}\\overline{\\nu}(\\neg p)$.\n\\item If $p^{1}=\\neg(p\\wedge\\neg p)\\in \\Gamma$, we make $\\overline{\\nu}(p^{1})=\\nu(p^{1})$; if $p^{1}\\not\\in\\Gamma$ and $\\overline{\\nu}(p)=t^{n}_{k}$, for some $1\\leq k\\leq n-1$, we must make $\\overline{\\nu}(p^{1})$ equal to $t^{n}_{k-1}$; if none of the previous cases apply, we choose $\\overline{\\nu}(p^{1})$ as any element of $\\tilde{\\neg}\\overline{\\nu}(p\\wedge \\neg p)$.\n\\end{enumerate}\nThis finishes the base step of order $0$.\n\nNow we suppose that, for every formula $\\alpha$ of order at most $m$, for a $m\\geq 0$, all of $\\overline{\\nu}(\\alpha)$, $\\overline{\\nu}(\\neg\\alpha)$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$ and $\\overline{\\nu}(\\alpha^{1})$ are defined and satisfy the expected properties that would make of $\\overline{\\nu}$ a homomorphism, our induction hypothesis, and take a formula $\\alpha$ of order $m+1$; there are then two cases to consider.\n\\begin{enumerate}\n\\item If $\\alpha=\\neg\\beta$, $\\overline{\\nu}(\\alpha)=\\overline{\\nu}(\\neg\\beta)$ has been already defined and satisfies: \n\\begin{enumerate}\n\\item $\\overline{\\nu}(\\neg\\beta)=\\nu(\\neg\\beta)$, if $\\neg\\beta\\in\\Gamma$; \n\\item if $\\neg\\beta\\not\\in\\Gamma$ and $\\beta=\\gamma\\wedge\\neg\\gamma$, $\\overline{\\nu}(\\gamma^{1})$ equals $t^{n}_{k-1}$, in the case that $\\overline{\\nu}(\\gamma)=t^{n}_{k}$ (for a $1\\leq k\\leq n-1$), and any value in $\\tilde{\\neg}\\overline{\\nu}(\\gamma\\wedge\\neg\\gamma)$ otherwise; \n\\item and, if none of these is the case, one still has $\\overline{\\nu}(\\neg\\beta)\\in\\tilde{\\neg}\\overline{\\nu}(\\beta)$.\n\\end{enumerate}\nNow we define $\\overline{\\nu}(\\neg\\alpha)$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$ and $\\overline{\\nu}(\\alpha^{1})$.\n\\begin{enumerate}\n\\item If $\\neg\\alpha\\in \\Gamma$, we make $\\overline{\\nu}(\\neg\\alpha)=\\nu(\\neg\\alpha)$, otherwise is enough to require that $\\overline{\\nu}(\\neg\\alpha)\\in \\tilde{\\neg}\\overline{\\nu}(\\alpha)$. \n\n\\item If $\\alpha\\wedge\\neg\\alpha\\in\\Gamma$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)=\\nu(\\alpha\\wedge\\neg\\alpha)$; if $\\alpha\\wedge\\neg\\alpha\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)=t^{n}_{0}$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)=T_{n}$, and if $\\alpha\\wedge\\neg\\alpha\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)=t^{n}_{k}$ (for $1\\leq k\\leq n-1$), $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)\\in I_{n}$; finally, if none of these applies, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)\\in \\overline{\\nu}(\\alpha)\\tilde{\\wedge}\\overline{\\nu}(\\neg\\alpha)$.\n\n\\item If $\\alpha^{1}\\in\\Gamma$, obviously $\\overline{\\nu}(\\alpha^{1})=\\nu(\\alpha^{1})$; if $\\alpha^{1}\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)=t^{n}_{k}$ (for $1\\leq k\\leq n-1$), $\\overline{\\nu}(\\alpha^{1})=t^{n}_{k-1}$; and if $\\alpha^{1}\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)\\neq t^{n}_{k}$, we define arbitrarily $\\overline{\\nu}(\\alpha^{1})\\in\\tilde{\\neg}\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$.\n\\end{enumerate}\n\n\\item If $\\alpha=\\beta\\#\\gamma$, for some $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, we have that either: \n\\begin{enumerate}\n\\item $\\beta\\#\\gamma\\in\\Gamma$, when we simply define $\\overline{\\nu}(\\beta\\#\\gamma)$ as $\\nu(\\beta\\#\\gamma)$;\n\\item $\\beta\\#\\gamma\\not\\in\\Gamma$ but $\\#=\\wedge$ and $\\gamma=\\neg\\beta$, meaning $\\overline{\\nu}(\\beta\\wedge\\neg\\beta)$ has already been defined and is equal to $\\nu(\\beta\\wedge\\neg\\beta)$, if $\\beta\\wedge\\neg\\beta\\in\\Gamma$; or is equal to $T_{n}$, if $\\overline{\\nu}(\\beta)=t^{n}_{0}$; or lies in $I_{n}$, if $\\overline{\\nu}(\\beta)=t^{n}_{k}$ (for $1\\leq k\\leq n-1$); or lies in $\\overline{\\nu}(\\beta)\\tilde{\\wedge}\\overline{\\nu}(\\neg\\beta)$, if none of the previous cases applies;\n\\item $\\beta\\#\\gamma\\not\\in\\Gamma$, and $\\#\\neq\\wedge$ or $\\gamma\\neq\\neg\\beta$, when we can define $\\overline{\\nu}(\\beta\\#\\gamma)$ as any value of $\\overline{\\nu}(\\beta)\\tilde{\\#}\\overline{\\nu}(\\gamma)$.\n\\end{enumerate}\nWe are left now with the task of defining $\\overline{\\nu}(\\neg\\alpha)$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$ and $\\overline{\\nu}(\\alpha^{1})$.\n\\begin{enumerate}\n\\item If $\\neg(\\beta\\#\\gamma)\\in\\Gamma$, we make $\\overline{\\nu}(\\neg(\\beta\\#\\gamma))$ equal to $\\nu(\\neg(\\beta\\#\\gamma))$; if $\\neg(\\beta\\#\\gamma)\\not\\in\\Gamma$, $\\#=\\wedge$, $\\gamma=\\neg\\beta$, and $\\overline{\\nu}(\\beta)=t^{n}_{k}$ (for $1\\leq k\\leq n-1$), we have already defined $\\overline{\\nu}(\\neg(\\beta\\#\\gamma))=\\overline{\\nu}(\\beta^{1})$ to be equal to $t^{n}_{k-1}$; if we are in none of these cases, is sufficient to take $\\overline{\\nu}(\\neg(\\beta\\#\\gamma))$ as equal to any value of $\\tilde{\\neg}\\overline{\\nu}(\\beta\\#\\gamma)$.\n\n\\item If $\\alpha\\wedge\\neg\\alpha\\in\\Gamma$, we merely define $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)=\\nu(\\alpha\\wedge\\neg\\alpha)$; if $\\alpha\\wedge\\neg\\alpha\\notin\\Gamma$ and $\\overline{\\nu}(\\alpha)=t^{n}_{0}$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)=T_{n}$, and if $\\alpha\\wedge\\neg\\alpha\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)=t^{n}_{k}$, for $1\\leq k\\leq n-1$, $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$ may take any value on the set $I_{n}$; if none of these applies, we can assign any value of $\\overline{\\nu}(\\alpha)\\tilde{\\wedge}\\overline{\\nu}(\\neg\\alpha)$ to $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$.\n\n\\item Finally, if $\\alpha^{1}\\in\\Gamma$, $\\overline{\\nu}(\\alpha^{1})=\\nu(\\alpha^{1})$; if $\\alpha^{1}\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)=t^{n}_{k}$, for any $1\\leq k\\leq n-1$, $\\overline{\\nu}(\\alpha^{1})=t^{n}_{k-1}$; and if $\\alpha^{1}\\not\\in\\Gamma$ and $\\overline{\\nu}(\\alpha)\\neq t^{n}_{k}$ (for $1\\leq k\\leq n-1$), we simply make $\\overline{\\nu}(\\alpha^{1})$ equal to any value of $\\tilde{\\neg}\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{proof}\n\nSo, given a formula $\\varphi$ of $C_{n}$, construct its row-branching, row-eliminating truth-table as described before: the resulting table is finite and can be constructed in a finite number of steps; take $\\Gamma$ as the set formed by $\\varphi$ and all its subformulas and it is clear that it is closed by subformulas. By the procedure we used for constructing the table for $\\varphi$, its columns correspond to the elements of $\\Gamma$, while its rows correspond to functions $\\nu:\\Gamma\\rightarrow B_{n}$ satisfying the hypothesis of Proposition \\ref{Rows of tables are homomorphisms}. Of course, for every row and its $\\nu$, by Proposition \\ref{Rows of tables are homomorphisms} there exists a homomorphism $\\overline{\\nu}$ in $\\mathcal{F}_{C_{n}}$ extending $\\nu$; reciprocally, to every homomorphism $\\overline{\\nu}$ of $\\mathcal{F}_{C_{n}}$ one can find a row in correspondence with a $\\nu$ which is extended by $\\overline{\\nu}$ (it is sufficient to take $\\nu$ as the restriction of $\\overline{\\nu}$ to $\\Gamma$).\n\nSo, assume $\\vdash_{C_{n}}\\varphi$, and therefore $\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$; then, for any row of the table for $\\varphi$ and its $\\nu$, since $\\overline{\\nu}\\in\\mathcal{F}_{C_{n}}$ we have $\\overline{\\nu}(\\varphi)\\in D_{n}$, and therefore $\\nu(\\varphi)\\in D_{n}$, meaning all entries on the column headed by $\\varphi$ are designated and, therefore, $\\varphi$ is proved to be a tautology according to row-branching, row-eliminating truth-tables. Reciprocally, assume $\\varphi$ is proved according to row-branching, row-eliminating truth-tables: for any homomorphism $\\overline{\\nu}\\in\\mathcal{F}_{C_{n}}$, $\\nu=\\overline{\\nu}|_{\\Gamma}$ corresponds to a row of the table for $\\varphi$ and therefore satisfies $\\nu(\\varphi)\\in D_{n}$, meaning therefore that $\\overline{\\nu}(\\varphi)\\in D_{n}$; this implies $\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$, and so $\\vdash_{C_{n}}\\varphi$, what gives us soundness and completeness.\n\nWe can, in much the same way, define row-branching, row-eliminating truth-tables for $\\textbf{mbCcl}$ and $\\textbf{Cila}$ based on the RNmatrices $\\mathcal{M}_{\\textbf{mbCcl}}$ (see Section \\ref{MmbCcl}) and $\\mathcal{M}_{\\textbf{CILA}}$ (see Section \\ref{MCILA}); in both cases tables are proven to be decision methods for these logics, which are known to be not characterizable by finite Nmatrices yet are decidable by three-valued RNmatrices.\n\nNow, we present some examples; first of all, we stress a slight simplification: in the case that $\\Gamma=\\{\\gamma_{1}, \\dotsc  , \\gamma_{m}\\}$ is a finite set of premises, instead of writing down the table for $\\gamma_{1}\\rightarrow(\\cdots\\rightarrow(\\gamma_{m}\\rightarrow\\varphi)\\cdots)$ to verify the deduction $\\Gamma\\vdash_{C_{n}}\\varphi$, is enough to list the columns headed by the formulas $\\varphi$ and $\\gamma_{1}$ through $\\gamma_{m}$, and their subformulas and verify that, in every row where all premises assume designated values, so does $\\varphi$. This is due to the fact that all $C_{n}$ obey the deduction meta-theorem, meaning that $\\Gamma, \\psi\\vdash_{C_{n}}\\varphi$ if, and only if, $\\Gamma\\vdash_{C_{n}}\\psi\\rightarrow\\varphi$, and this may shorten significantly our truth-tables.\n\nTake a propositional variable $p$ and the deduction $p, \\neg p, \\neg(p\\wedge\\neg p)\\vdash_{C_{1}}\\neg\\neg p$. We obtain then the following series of tables that culminate in the completed row-branching, row-eliminating truth-table for that very argument.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{2cm}\n\\centering\n\\begin{tabular}{|p{1cm}|p{1cm}|}\n\\hline\n$p$ & Row\\\\ \\hline\n$T_{1}$ & \\nth{1}\\\\ \\hline\n$t^{1}_{0}$ & \\nth{2}\\\\ \\hline\n$F_{1}$ & \\nth{3}\\\\ \\hline\n\\end{tabular}\n\\caption*{First stage}\n\\end{minipage}\n\\hspace{3cm}\n\\begin{minipage}[t]{3cm}\n\\centering\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|}\n\\hline\n$p$ & $\\neg p$ & Row\\\\ \\hline\n$T_{1}$ & $F_{1}$ & \\nth{1}\\\\ \\hline\n\\multirow{2}{*}{$t^{1}_{0}$} & $T_{1}$ & \\nth{2}\\\\ \\cline{2-3}\n& $t^{1}_{0}$ & \\nth{3}\\\\ \\hline\n$F_{1}$ & $T_{1}$ & \\nth{4}\\\\ \\hline\n\\end{tabular}\n\\caption*{Second stage}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|}\n\\hline\n$p$ & $\\neg p$ & $\\neg\\neg p$ & Row\\\\ \\hline\n$T_{1}$ & $F_{1}$ & $T_{1}$ & \\nth{1}\\\\ \\hline\n\\multirow{3}{*}{$t^{1}_{0}$} & $T_{1}$ & $F_{1}$ & \\nth{2}\\\\ \\cline{2-4}\n& \\multirow{2}{*}{$t^{1}_{0}$} & $T_{1}$ & \\nth{3}\\\\ \\cline{3-4}\n& & $t^{1}_{0}$ & \\nth{4}\\\\ \\hline\n$F_{1}$ & $T_{1}$ & $F_{1}$ & \\nth{5}\\\\ \\hline\n\\end{tabular}\n\\caption*{Third stage}\n\\end{minipage}\n\\hspace{2.5cm}\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|}\n\\hline\n$p$ & $\\neg p$ & $\\neg\\neg p$ & $p\\wedge\\neg p$ & Row\\\\ \\hline\n$T_{1}$ & $F_{1}$ & $T_{1}$ & $F_{1}$ & \\nth{1}\\\\ \\hline\n\\multirow{3}{*}{$t^{1}_{0}$} & $T_{1}$ & $F_{1}$ & $T_{1}$ & \\nth{2}\\\\ \\cline{2-5}\n& \\multirow{2}{*}{$t^{1}_{0}$} & $T_{1}$ & $T_{1}$ & \\nth{3}\\\\ \\cline{3-5}\n& & $t^{1}_{0}$ & $T_{1}$ & \\nth{4}\\\\ \\hline\n$F_{1}$ & $T_{1}$ & $F_{1}$ & $F_{1}$ & \\nth{5}\\\\ \\hline\n\\end{tabular}\n\\caption*{Fourth stage}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{6.5cm}\n\\centering\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1.5cm}|p{1cm}|}\n\\hline\n$p$ & $\\neg p$ & $\\neg\\neg p$ & $p\\wedge\\neg p$ & $\\neg(p\\wedge\\neg p)$ & Row\\\\\\hline\n$T_{1}$ & $F_{1}$ & $T_{1}$ & $F_{1}$ & $T_{1}$ & \\nth{1}\\\\\\hline\n\\multirow{3}{*}{$t^{1}_{0}$} & $T_{1}$ & $F_{1}$ & $T_{1}$ & $F_{1}$ & \\nth{2}\\\\\\cline{2-6}\n& \\multirow{2}{*}{$t^{1}_{0}$} & $T_{1}$ & $T_{1}$ & $F_{1}$ & \\nth{3}\\\\\\cline{3-6}\n& & $t^{1}_{0}$ & $T_{1}$ & $F_{1}$ & \\nth{4}\\\\\\hline\n$F_{1}$ & $T_{1}$ & $F_{1}$ & $F_{1}$ & $T_{1}$ & \\nth{5}\\\\\\hline\n\\end{tabular}\n\\caption*{Fifth and final stage}\n\\end{minipage}\n\\end{figure}\n\nWe see that the deduction $p, \\neg p, \\neg(p\\wedge\\neg p)\\vdash_{C_{1}}\\neg\\neg p$ is valid: since in no rows one has $p$, $\\neg p$ and $\\neg(p\\wedge\\neg p)$ simultaneously assigned designated values, the deduction is vacuously true.\n\nNow, a final remark regarding our truth-tables: while we have presented row-bran\\-ching, row-eliminating truth-tables on which rows are discarded as soon as they violate some condition necessary for the corresponding homomorphism to lie in $\\mathcal{F}_{C_{n}}$, one could first write down the row-branching truth-table of the Nmatrix $(\\mathcal{A}_{C_{n}}, D_{n})$ to only then delete the rows corresponding to homomorphism outside $\\mathcal{F}_{C_{n}}$. The order in which this is done is not relevant: although eliminating the rows as soon as they are discovered not to be relevant is slightly more efficient, the alternative way is, arguably, more intuitive. To show one example of how this goes, consider again the deduction $p, \\neg p, \\neg(p\\wedge\\neg p)\\vdash_{C_{1}}\\neg\\neg p$; using only the Nmatrix $(\\mathcal{A}_{C_{1}}, D_{1})$ one obtains the following table.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{6.5cm}\n\\centering\n\\begin{center}\n\\begin{tabular}{|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1.5cm}|c|}\n\\hline\n$p$ & $\\neg p$ & $\\neg\\neg p$ & $p\\wedge\\neg p$ & $\\neg(p\\wedge\\neg p)$ & Row\\\\\\hline\n$T_{1}$ & $F_{1}$ & $T_{1}$ & $F_{1}$ & $T_{1}$ & \\nth{1}\\\\\\hline\n\\multirow{9}{*}{$t^{1}_{0}$} & \\multirow{3}{*}{$T_{1}$} & \\multirow{3}{*}{$F_{1}$} & $T_{1}$ & $F_{1}$ & \\nth{2}\\\\\\cline{4-6}\n& & & \\multirow{2}{*}{$t^{1}_{0}$}& $T_{1}$ & \\nth{3}\\\\\\cline{5-6}\n& & & & $t^{1}_{0}$ & \\nth{4}\\\\\\cline{2-6}\n& \\multirow{6}{*}{$t^{1}_{0}$} & \\multirow{3}{*}{$T_{1}$} & $T_{1}$ & $F_{1}$ & \\nth{5}\\\\\\cline{4-6}\n& & & \\multirow{2}{*}{$t^{1}_{0}$} & $T_{1}$ & \\nth{6}\\\\\\cline{5-6}\n& & & & $t^{1}_{0}$ & \\nth{7}\\\\\\cline{3-6}\n& & \\multirow{3}{*}{$t^{1}_{0}$} & $T_{1}$ & $F_{1}$ & \\nth{8}\\\\\\cline{4-6}\n& & & \\multirow{2}{*}{$t^{1}_{0}$} & $T_{1}$ & \\nth{9}\\\\\\cline{5-6}\n& & & & $t^{1}_{0}$ & \\nth{10}\\\\\\hline\n$F_{1}$ & $T_{1}$ & $F_{1}$ & $F_{1}$ & $T_{1}$ & \\nth{11}\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nIf one tries to evaluate the validity of the argument only from this table, it appears false: rows $3$ and $4$ have all premises designated but not the conclusion. But the rows, in the general case of $C_{n}$, to be eliminated are:\n\\begin{enumerate}\n\\item the ones where $\\alpha$ is assigned $t^{n}_{0}$ and $\\alpha\\wedge\\neg\\alpha$ is not assigned $T_{n}$;\n\\item the ones where $\\alpha$ is assigned the value $t^{n}_{k}$, for $1\\leq k\\leq n-1$, but $\\alpha\\wedge\\neg\\alpha$ is assigned the value $T_{n}$;\n\\item the ones where $\\alpha$ is $t^{n}_{k}$, for $1\\leq k\\leq n-1$, but $\\alpha^{1}$ is not $t^{n}_{k-1}$.\n\\end{enumerate}\nIn our specific case, rows $3$, $4$, $6$, $7$, $9$ and $10$ are eliminated, and we retrieve the previous table we had written for the deduction, which makes it valid.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Tableau semantics}\\label{Tableaux semantics for Cn}\n\nWe have presented, in Sections \\ref{RNmatrix for mbCcl}, \\ref{RNmatrix for CILA} and \\ref{The general case} shown RNmatrices that, through the row-branching, row-eliminating truth tables outlined in Section \\ref{Row-branching,row-eliminating}, become decision methods for the whole hierarchy of da Costa plus $\\textbf{Cila}$ and $\\textbf{mbCcl}$, although none of these logics is characterized by finite Nmatrices. However the corresponding truth tables can grow rather rapidly and implementing such a method may not be at all feasible.\n\nSo we offer a second decision procedure, based too on the RNmatrices $\\mathcal{RM}_{C_{n}}$, by means of tableau semantics\\index{Tableaux}. Although recent developments provided general approaches to obtaining tableau-like proof systems from finite Nmatrices (\\cite{Pawlowski}), the present study will make use more extensively of \\cite{con:far:per:21}, which appears to be more useful when dealing with RNmatrices defined by swap structures. Many of the ideas used to motivate tableaux from RNmatrices may be found in any classical account on the subject, e.g. \\cite{Smullyan}.\n\nOur tableaux will have as nodes labeled formulas, that is, pairs of labels $\\textsf{L}\\in \\textsf{B}_{n}=\\{\\textsf{T}_{n}, \\textsf{t}^{n}_{0}, \\dotsc  , \\textsf{t}^{n}_{n-1}, \\textsf{F}_{n}\\}$ and formulas $\\varphi$ of $C_{n}$. The labels correspond to the snapshots of $B_{n}$, but to the make the distinction between truth value and label clearer we use different fonts; furthermore, the pair formed by $\\textsf{L}$ and $\\varphi$ will be denoted by $\\textsf{L}(\\varphi)$. The labeled tableau system for $C_{n}$ will be denoted by $\\mathbb{T}_{n}$\\label{mathbbTn} and have the following $n+2$ rules of elimination for every connective (that is, $\\neg$, $\\wedge$, $\\vee$ and $\\rightarrow$, totaling $4n+8$ rules), where the pictorial rule\n\\[\\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{L}(\\varphi)}\\\\\\hline\\textsf{L}_{11}(\\varphi_{11}) & \\multirow{3}{*}{$\\cdots$} & \\textsf{L}_{m1}(\\varphi_{m1})\\\\ \\vdots& & \\vdots\\\\ \\textsf{L}_{1n_{1}}(\\varphi_{1n_{1}})& & \\textsf{L}_{mn_{m}}(\\varphi_{mn_{m}})\\end{array}\\]\nindicates that a branch of a tableau containing $\\textsf{L}(\\varphi)$ may be forked into $m$ new branches, each containing all formulas already present in the original branch and, additionally, $\\textsf{L}_{k1}(\\varphi_{k1})$ trough $\\textsf{L}_{kn_{k}}(\\varphi_{kn_{k}})$, for some $1\\leq k\\leq m$.\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{c}\n\n\\begin{array}{c|c|c|c|c|c|c|c}\\multicolumn{8}{c}{\\textsf{T}_{n}(\\varphi\\vee\\psi)} \\\\\\hline \\textsf{T}_{n}(\\varphi) & \\textsf{t}^{n}_{0}(\\varphi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{T}_{n}(\\psi) & \\textsf{t}^{n}_{n-1}(\\psi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\psi)\\end{array}\\quad (E_{\\vee}\\textsf{T}_{n}) \\\\\n\\\\\n\\begin{array}{c|c|c|c|c|c}\\multicolumn{6}{c}{\\textsf{t}^{n}_{0}(\\varphi\\vee\\psi)} \\\\\\hline \\textsf{t}^{n}_{0}(\\varphi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{t}^{n}_{0}(\\psi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\psi)\\end{array}\\quad (E_{\\vee}\\textsf{t}^{n}_{0})\\\\\n\\\\\n\\vdots\n\\\\\n\\begin{array}{c|c|c|c|c|c}\\multicolumn{6}{c}{\\textsf{t}^{n}_{n-1}(\\varphi\\vee\\psi)} \\\\\\hline \\textsf{t}^{n}_{0}(\\varphi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{t}^{n}_{0}(\\psi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\psi)\\end{array}\\quad (E_{\\vee}\\textsf{t}^{n}_{n-1})\\\\\n\\\\\n\n\\begin{array}{c} \\textsf{F}_{n}(\\varphi\\vee\\psi)\\\\\\hline \\textsf{F}_{n}(\\varphi)\\\\ \\textsf{F}_{n}(\\psi)\\end{array} \\quad (E_{\\vee}\\textsf{F}_{n})\n\\end{array}$\n\\vspace{0.5cm}\n\\caption*{The $n+2$ rules for eliminating disjunction}\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c|c|c}\n\\multicolumn{4}{c}{\\textsf{T}_{n}(\\neg\\varphi)}\\\\ \\hline \\textsf{t}^{n}_{0}(\\varphi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{F}_{n}(\\varphi)\n\\end{array} \\quad (E_{\\neg}\\textsf{T}_{n}) &\\quad\\quad\\quad &\n\\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{t}^{n}_{0}(\\neg\\varphi)}\\\\ \\hline \\textsf{t}^{n}_{0}(\\varphi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\varphi)\\\\\\end{array} \\quad (E_{\\neg}\\textsf{t}^{n}_{0}) & \\quad\\quad & \\cdots\\\\\n\\end{array}$\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{t}^{n}_{n-1}(\\neg\\varphi)}\\\\ \\hline \\textsf{t}^{n}_{0}(\\varphi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\varphi)\\\\\\end{array} \\quad (E_{\\neg}\\textsf{t}^{n}_{n-1}) &\\quad\\quad\\quad &\n\\begin{array}{c}\\textsf{F}_{n}(\\neg\\varphi)\\\\\\hline\\textsf{T}_{n}(\\varphi)\\\\\\end{array} \\quad (E_{\\neg}\\textsf{F}_{n}) &  & \\\\\n\\end{array}$\n\\vspace{0.5cm}\n\\caption*{The $n+2$ rules for eliminating negation}\n\\end{table}\n\n\\begin{landscape}\n\n\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{c}\n\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c}\\multicolumn{13}{c}{\\textsf{T}_{n}(\\varphi\\wedge\\psi)}\\\\ \\hline\\textsf{T}_{n}(\\varphi) & \\textsf{T}_{n}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{T}_{n}(\\varphi) & \\textsf{t}^{n}_{0}(\\varphi) & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{t}^{n}_{n-1}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi)\\\\ \\textsf{T}_{n}(\\psi) & \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) & \\textsf{T}_{n}(\\psi) & \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) &  & \\textsf{T}_{n}(\\psi) & \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) \\end{array} \\quad (E_{\\wedge}\\textsf{T}_{n}) \\\\\n\\\\\n\n\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c}\\multicolumn{13}{c}{\\textsf{t}^{n}_{0}(\\varphi\\wedge\\psi)}\\\\ \\hline \\textsf{T}_{n}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{T}_{n}(\\varphi)  & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$}  & \\textsf{t}^{n}_{n-1}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{t}^{n}_{0}(\\varphi)& \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi)\\\\ \n\\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) & \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) &  &  \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) & \\textsf{T}_{n}(\\psi) & & \\textsf{T}_{n}(\\psi) \\end{array} \\quad (E_{\\wedge}\\textsf{t}^{n}_{0})\\\\\n\\\\\n\\vdots\\\\\n\\\\\n\n\\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c}\\multicolumn{13}{c}{\\textsf{t}^{n}_{n-1}(\\varphi\\wedge\\psi)}\\\\ \\hline \\textsf{T}_{n}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{T}_{n}(\\varphi)  & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$}  & \\textsf{t}^{n}_{n-1}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi) & \\textsf{t}^{n}_{0}(\\varphi)& \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi)\\\\ \n\\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) & \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) &  &  \\textsf{t}^{n}_{0}(\\psi) &  & \\textsf{t}^{n}_{n-1}(\\psi) & \\textsf{T}_{n}(\\psi) & & \\textsf{T}_{n}(\\psi) \\end{array} \\quad (E_{\\wedge}\\textsf{t}^{n}_{n-1})\\\\\n\\\\\n\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}_{n}(\\varphi\\wedge\\psi)}\\\\\\hline \\textsf{F}_{n}(\\varphi) & \\textsf{F}_{n}(\\psi)\\\\\\end{array} \\quad (E_{\\wedge}\\textsf{F}_{n})\n\\end{array}$\n\\vspace{0.5cm}\n\\caption*{The $n+2$ rules for eliminating conjunction}\n\\end{table}\n\n\\vspace{1cm}\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{c}\n\\begin{array}{c|c|c|c|c}\\multicolumn{5}{c}{\\textsf{T}_{n}(\\varphi\\rightarrow\\psi)} \\\\\\hline \\textsf{F}_{n}(\\varphi) & \\textsf{T}_{n}(\\psi) & \\textsf{t}^{n}_{n-1}(\\psi) & \\cdots & \\textsf{t}^{n}_{n-1}(\\psi)\\end{array}\\quad (E_{\\rightarrow}\\textsf{T}_{n}) \\\\\n\\end{array}$\n\\end{table}\n\n\\end{landscape}\n\n\n\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{c}\n\\begin{array}{c|c|c|c|c|c}\\multicolumn{6}{c}{\\textsf{t}^{n}_{0}(\\varphi\\rightarrow\\psi)}\\\\\\hline \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi) & \\multirow{2}{*}{$\\textsf{t}^{n}_{0}(\\psi)$} & \\multirow{2}{*}{$\\cdots$} & \\multirow{2}{*}{$\\textsf{t}^{n}_{n-1}(\\psi)$}\\\\ \\textsf{T}_{n}(\\psi) & & \\textsf{T}_{n}(\\psi) & & &\\end{array} \\quad (E_{\\rightarrow}\\textsf{t}^{n}_{0})\\\\\n\\\\\n\\vdots\\\\\n\\\\\n\\begin{array}{c|c|c|c|c|c}\\multicolumn{6}{c}{\\textsf{t}^{n}_{n-1}(\\varphi\\rightarrow\\psi)}\\\\\\hline \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi) & \\multirow{2}{*}{$\\textsf{t}^{n}_{0}(\\psi)$} & \\multirow{2}{*}{$\\cdots$} & \\multirow{2}{*}{$\\textsf{t}^{n}_{n-1}(\\psi)$}\\\\ \\textsf{T}_{n}(\\psi) & & \\textsf{T}_{n}(\\psi) & & &\\end{array} \\quad (E_{\\rightarrow}\\textsf{t}^{n}_{n-1})\\\\\n\\\\\n\\begin{array}{c|c|c|c}\\multicolumn{4}{c}{\\textsf{F}_{n}(\\varphi\\rightarrow\\psi)}\\\\\\hline \\textsf{T}_{n}(\\varphi) & \\textsf{t}^{n}_{0}(\\varphi) & \\multirow{2}{*}{$\\cdots$} & \\textsf{t}^{n}_{n-1}(\\varphi)\\\\ \\textsf{F}_{n}(\\psi) & \\textsf{F}_{n}(\\psi) & & \\textsf{F}_{n}(\\psi)\\end{array}\\quad (E_{\\rightarrow}\\textsf{F}_{n})\n\\end{array}$\n\\vspace{0.5cm}\n\\caption*{The $n+2$ rules for eliminating implication}\n\\end{table}\n\n\n\n\n\nOne sees that the tableau rules for $C_{n}$ are rather difficult to write down given their sizes, so we give an alternative presentation that uses a more compact notation. Let $\\textsf{X}$ and $\\textsf{Y}$ be sets of labels.\n\\[\\begin{array}{ccccc}\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{L}(\\varphi)}\\\\\\hline \\textsf{L}^{*}(\\psi) & \\textsf{X}(\\psi)\\end{array}&\\quad\\quad\\quad\\quad&\n\\begin{array}{c}\\textsf{L}(\\varphi)\\\\\\hline \\textsf{L}^{*}(\\gamma)\\\\\\textsf{X}(\\psi)\\end{array}&\\quad\\quad\\quad\\quad&\n\\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{L}(\\varphi)}\\\\\\hline \\textsf{L}^{*}(\\gamma) & \\textsf{L}^{**}(\\psi) & \\textsf{X}(\\gamma)\\\\ \\textsf{X}(\\psi) & \\textsf{Y}(\\gamma) & \\textsf{Y}(\\psi)\\end{array}\n\\end{array}\\]\n\n\\begin{enumerate}\n\\item The leftmost rule states that a closed branch containing $\\textsf{L}(\\varphi)$ will also contain $\\textsf{L}^{*}(\\psi)$ or $\\textsf{x}(\\psi)$, for some label $\\textsf{x}\\in\\textsf{X}$; if $\\textsf{X}$ has $p$ elements, such a rule forks a branch into $p+1$ new ones.\n\n\\item For the rule in the middle, a closed branch containing $\\textsf{L}(\\varphi)$ must also contain both $\\textsf{L}^{*}(\\gamma)$ and $\\textsf{x}(\\psi)$, for some $\\textsf{x}\\in\\textsf{X}$. Notice that, since we do not have the vertical bar, it would appear the branch is not forked at all; but since $\\textsf{X}$ is a set, we still have multiple cases to consider, being the branch forked into $p$ new branches.\n\n\\item Finally, if $\\textsf{L}(\\varphi)$ is in a closed branch, the rightmost rule implies one of the following lies in the branch as well: $\\textsf{L}^{*}(\\gamma)$ and $\\textsf{x}(\\psi)$, for some $\\textsf{x}\\in\\textsf{X}$; or $\\textsf{L}^{**}(\\psi)$ and $\\textsf{y}(\\gamma)$, for some $\\textsf{y}\\in\\textsf{Y}$;\\footnote{Notice that, in this rule, we change the order of the formulas ($\\gamma$ above and $\\psi$ below) that was previously established in order to make the idea of a tableau as a tree growing downwards clearer, leaving the larger set of labels on the bottom.} or $\\textsf{x}(\\gamma)$ and $\\textsf{y}(\\psi)$, for $\\textsf{x}\\in\\textsf{X}$ and $\\textsf{y}\\in\\textsf{Y}$. If $\\textsf{Y}$ has $q$ elements, this rules forks a branch $pq+p+q$ times.\n\\end{enumerate}\n\nUsing these conventions and defining $\\textsf{I}_{n}=\\{\\textsf{t}^{n}_{0}, \\dotsc  , \\textsf{t}^{n}_{n-1}\\}$ and $\\textsf{D}_{n}=\\textsf{I}_{n}\\cup\\{\\textsf{T}_{n}\\}$, the rules of $\\mathbb{T}_{n}$ may be more compactly presented by the following, where $i$ takes any value in $\\{0, 1, \\dotsc  , n-1\\}$.\n\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}_{n}(\\neg\\varphi)}\\\\\\hline\\textsf{I}_{n}(\\varphi) & \\textsf{F}_{n}(\\varphi)\\end{array}\\quad (E_{\\neg}\\textsf{T}_{n}) & \\quad &\n\\begin{array}{c}\\textsf{t}^{n}_{i}(\\neg\\varphi)\\\\\\hline\\textsf{I}_{n}(\\varphi)\\end{array} \\quad (E_{\\neg}\\textsf{t}^{n}_{i})& \\quad &\n\\begin{array}{c}\\textsf{F}_{n}(\\neg\\varphi)\\\\\\hline\\textsf{T}_{n}(\\varphi)\\end{array}\\quad (E_{\\neg}\\textsf{F}_{n})\\\\\n&&&&\\\\\n\\begin{array}{c}\\textsf{T}_{n}(\\varphi\\wedge\\psi)\\\\\\hline\\textsf{D}_{n}(\\varphi)\\\\\\textsf{D}_{n}(\\psi)\\end{array}\\quad (E_{\\wedge}\\textsf{T}_{n}) & & \\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{t}^{n}_{i}(\\varphi\\wedge\\psi)}\\\\\\hline\\textsf{T}_{n}(\\varphi)&\\textsf{I}_{n}(\\varphi)&\\textsf{T}_{n}(\\psi)\\\\\\textsf{I}_{n}(\\psi)&\\textsf{I}_{n}(\\psi)&\\textsf{I}_{n}(\\varphi)\\end{array}\\quad (E_{\\wedge}\\textsf{t}^{n}_{i}) & & \n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}_{n}(\\varphi\\wedge\\psi)}\\\\\\hline\\textsf{F}_{n}(\\varphi)&\\textsf{F}_{n}(\\psi)\\end{array}\\quad(E_{\\wedge}\\textsf{F}_{n})\\\\\n\\end{array}$\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}_{n}(\\varphi\\vee\\psi)}\\\\\\hline\\textsf{D}_{n}(\\varphi)&\\textsf{D}_{n}(\\psi)\\end{array}\\quad(E_{\\vee}\\textsf{T}_{n}) & & \n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{t}^{n}_{i}(\\varphi\\vee\\psi)}\\\\\\hline\\textsf{I}_{n}(\\varphi)&\\textsf{I}_{n}(\\psi)\\end{array}\\quad (E_{\\vee}\\textsf{t}^{n}_{i})& &\n\\begin{array}{c}\\textsf{F}_{n}(\\varphi\\vee\\psi)\\\\\\hline\\textsf{F}_{n}(\\varphi)\\\\\\textsf{F}_{n}(\\psi)\\end{array}\\quad(E_{\\vee}\\textsf{F}_{n})\\\\\n&&&&\\\\\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}_{n}(\\varphi\\rightarrow\\psi)}\\\\\\hline\\textsf{F}_{n}(\\varphi)&\\textsf{D}_{n}(\\psi)\\end{array}\\quad(E_{\\rightarrow}\\textsf{T}_{n})& &\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{t}^{n}_{i}(\\varphi\\rightarrow\\psi)}\\\\\\hline\\textsf{T}_{n}(\\psi)&\\multirow{2}{*}{$\\textsf{I}_{n}(\\psi)$}\\\\\\textsf{I}_{n}(\\varphi)& \\end{array}\\quad(E_{\\rightarrow}\\textsf{t}^{n}_{i})& &\n\\begin{array}{c}\\textsf{F}_{n}(\\varphi\\rightarrow\\psi)\\\\\\hline\\textsf{F}_{n}(\\psi)\\\\\\textsf{D}_{n}(\\varphi)\\end{array}\\quad(E_{\\rightarrow}\\textsf{F}_{n})\n\\end{array}$\n\\end{table}\n\n\\begin{definition}\\label{Definitions about tableaux}\nA branch of a tableau in $\\mathbb{T}_{n}$ is closed\\index{Branch, Closed} whenever:\n\\begin{enumerate}\n\\item it contains $\\textsf{L}(\\varphi)$ and $\\textsf{L}^{*}(\\varphi)$ for labels $\\textsf{L}\\neq\\textsf{L}^{*}$;\n\\item it contains $\\textsf{t}^{n}_{0}(\\varphi)$ and $\\textsf{t}^{n}_{i}(\\varphi\\wedge\\neg\\varphi)$, for $0\\leq i\\leq n-1$;\n\\item it contains $\\textsf{t}^{n}_{k}(\\varphi)$, and either $\\textsf{T}_{n}(\\varphi\\wedge\\neg\\varphi)$ or $\\textsf{L}(\\varphi^{1})$, for $1\\leq k\\leq n-1$ and $\\textsf{L}\\neq \\textsf{t}^{n}_{k-1}$.\n\\end{enumerate}\n\nA branch $\\theta$ is said to be complete\\index{Branch, Complete} if, for every signed formula $\\textsf{L}(\\varphi)$ appearing in it with $\\varphi$ not a propositional variable, $\\theta$ also contains all the signed formulas of one of the branches of the only rule applicable to $\\textsf{L}(\\varphi)$. A complete branch is open if it is not closed.\n\nA tableau of $\\mathbb{T}_{n}$ is:\n\\begin{enumerate}\n\\item closed\\index{Tableau, Closed} if all of its branches are closed; \n\\item complete\\index{Tableau, Complete} if all of its branches are either closed or complete;\n\\item open\\index{Tableau, Open} if it is complete but not closed.\n\\end{enumerate}\n\\end{definition}\n\n\nThe rules of $\\mathbb{T}_{n}$ are analytic, meaning that if $\\textsf{L'}(\\psi)$ appears as a consequence of $\\textsf{L}(\\varphi)$ in some rule of $\\mathbb{T}_{n}$, the $\\psi$ is a proper subformula of $\\varphi$. In other words, the tableaux of $\\mathbb{T}_{n}$ can always be completed in a finite number of steps: in fact, if the complexity of $\\varphi$ is $m$, any completed tableau starting with $\\textsf{L}(\\varphi)$ is guaranteed to have at most $(n+1)^{2m}$ branches, of length at most $m$.\n\nWe set out now to prove that the method of tableaux here presented for $C_{n}$ is both sound and complete, using arguments very similar to those found in \\cite{Smullyan}; combined with the fact any such tableau may be completed in finite time, we will therefore have a different decision procedure for these systems.\n\n\\begin{definition}\nA formula $\\varphi$ of $C_{n}$ is provable according to tableaux of $\\mathbb{T}_{n}$, when we write $\\vdash_{\\mathbb{T}_{n}}\\varphi$, if there exists a closed tableau of $\\mathbb{T}_{n}$ started by $\\textsf{F}_{n}(\\varphi)$.\n\nGiven a finite set $\\Gamma=\\{\\gamma_{1}, \\dotsc  , \\gamma_{m}\\}$ of formulas of $C_{n}$, $\\varphi$ is said to be provable from $\\Gamma$ according to tableaux of $\\mathbb{T}_{n}$, when we write $\\Gamma\\vdash_{\\mathbb{T}_{n}}\\varphi$\\label{vdashmathbbTn}, if \n\\[\\vdash_{\\mathbb{T}_{n}}\\bigwedge_{i=1}^{m}\\gamma_{i}\\rightarrow\\varphi.\\]\n\\end{definition}\n\nGiven a valuation $\\nu$ in $\\mathcal{F}_{C_{n}}$, a signed formula $\\textsf{L}(\\varphi)$ is true under $\\nu$, also said to be satisfied by $\\nu$, if $\\nu(\\varphi)=L$, where $L$ is, of course, the element of $B_{n}$ corresponding to $\\textsf{L}$; if $\\textsf{L}(\\varphi)$ is not true under $\\nu$, we say it is false.\n\nThen a branch $\\theta$ of a tableau $\\mathcal{T}$ of $\\mathbb{T}_{n}$ is true under $\\nu$, or is satisfied by $\\nu$, if all of its signed formulas are true under $\\nu$; $\\mathcal{T}$ is true under $\\nu$, or satisfied by $\\nu$, if at least one of its branches is true under $\\nu$. We can see that a closed branch $\\theta$, and accordingly a closed tableau, is not satisfied under any $\\nu$ in $\\mathcal{F}_{C_{n}}$:\n\\begin{enumerate}\n\\item if $\\theta$ contains $\\textsf{L}(\\varphi)$ and $\\textsf{L}^{*}(\\varphi)$, for $\\textsf{L}\\neq \\textsf{L}^{*}$, $\\theta$ being true under $\\nu$ would mean $\\nu(\\varphi)=L$ and $\\nu(\\varphi)=L^{*}$, what is absurd;\n\\item if $\\theta$ contains $\\textsf{t}_{0}^{n}(\\varphi)$ and $\\textsf{t}_{i}^{n}(\\varphi\\wedge\\neg\\varphi)$, for $0\\leq i\\leq n-1$, and $\\theta$ is true under $\\nu$, from the definition of $\\mathcal{F}_{C_{n}}$ one has that $\\nu(\\varphi)=t_{0}^{n}$ would imply $\\nu(\\varphi\\wedge\\neg\\varphi)=T_{n}$, contradicting the fact that $\\nu(\\varphi\\wedge\\neg\\varphi)=t_{i}^{n}$;\n\\item and if $\\theta$ contains $\\textsf{t}^{n}_{k}(\\varphi)$, and either $\\textsf{T}_{n}(\\varphi\\wedge\\neg\\varphi)$ or $\\textsf{L}(\\varphi^{1})$, for $1\\leq k\\leq n-1$ and $\\textsf{L}\\neq \\textsf{t}^{n}_{k-1}$, then $\\theta$ being true under $\\nu$ would imply that $\\nu(\\varphi)=t_{k}^{n}$ and, from the definition of $\\mathcal{F}_{C_{n}}$, $\\nu(\\varphi\\wedge\\neg\\varphi)\\in\\{t_{0}^{n}, t_{1}^{n}, \\dotsc  , t_{n-1}^{n}\\}$ and $\\nu(\\varphi^{1})=t_{k-1}^{n}$, what in any of the two cases leads to a contradiction.\n\\end{enumerate}\n\n\\begin{lemma}\\label{validates consequence rule}\nFor a signed formula $\\textsf{L}(\\varphi)$, with $\\varphi$ not a propositional variable, and $\\nu$ a valuation in $\\mathcal{F}_{C_{n}}$ under which $\\textsf{L}(\\varphi)$ is true, all the formulas of at least one of the branches obtained from the application of a rule of $\\mathbb{T}_{n}$ to $\\textsf{L}(\\varphi)$ are also true under $\\nu$.\n\\end{lemma}\n\n\\begin{proof}\nThis is true from the definition of $\\mathcal{A}_{C_{n}}$ due to the form of the rules in $\\mathbb{T}_{n}$ and the very definition of being true under a valuation. For an example, look at the rule $E_{\\wedge}\\textsf{F}_{n}$, given by\n\\[\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}_{n}(\\varphi\\wedge\\psi)}\\\\\\hline\\textsf{F}_{n}(\\varphi)&\\textsf{F}_{n}(\\psi)\\end{array}.\\]\nIf $\\nu$ satisfies $\\textsf{F}_{n}(\\varphi\\wedge\\psi)$, this means $\\nu(\\varphi\\wedge\\psi)=F_{n}$; looking at the table for conjunction in $\\mathcal{A}_{C_{n}}$, presented below, we see that this implies either $\\nu(\\varphi)=F_{n}$ or $\\nu(\\psi)=F_{n}$, and so $\\nu$ satisfies either $\\textsf{F}_{n}(\\varphi)$ or $\\textsf{F}_{n}(\\psi)$, that is, precisely all formulas of one of the branches obtained from applying $E_{\\wedge}\\textsf{F}_{n}$ to $\\textsf{F}_{n}(\\varphi\\wedge\\psi)$.\n\\begin{center}\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\wedge$ & $F_{n}$ & $I_{n}$ & $T_{n}$ \\\\ \\hline\n$F_{n}$ & $\\{F_{n}\\}$ & $\\{F_{n}\\}$ & $\\{F_{n}\\}$ \\\\ \\hline\n$I_{n}$ & $\\{F_{n}\\}$  & $D_{n}$ & $D_{n}$ \\\\ \\hline\n$T_{n}$ & $\\{F_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\nOf course, a similar reasoning works for all the other rules of $\\mathbb{T}_{n}$.\n\\end{proof}\n\n\\begin{theorem}\\label{Correct tableaux}\nFor $\\Gamma\\cup\\{\\varphi\\}$ a finite set of formulas of $C_{n}$, if $\\Gamma\\vdash_{\\mathbb{T}_{n}}\\varphi$, then $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nGiven $\\Gamma=\\{\\gamma_{1}, \\dotsc  , \\gamma_{m}\\}$, it is clear that $\\bigwedge_{i=1}^{m}\\gamma_{i}\\vdash_{C_{n}}\\gamma_{j}$, for every $1\\leq j\\leq m$, and $\\Gamma\\vdash_{C_{n}}\\bigwedge_{i=1}^{m}\\gamma_{i}$; furthermore, $C_{n}$ satisfies the deduction meta-theorem, and by soundness and completeness of this system with respect to $\\mathcal{RM}_{C_{n}}$, it is enough to prove the theorem under the assumption that $\\Gamma=\\emptyset$.\n\nOne constructs a completed tableau for $\\textsf{F}_{n}(\\varphi)$ through a finite sequence of tableaux $\\mathcal{T}_{0}, \\dotsc  , \\mathcal{T}_{k}=\\mathcal{T}$, where $\\mathcal{T}_{0}$ contains only $\\textsf{F}_{n}(\\varphi)$ and each $\\mathcal{T}_{j+1}$, for $0\\leq j\\leq k-1$, is obtained from $\\mathcal{T}_{j}$ by application of a rule $R_{j}$ of $\\mathbb{T}_{n}$ to a, up to this moment, unused signed formula $\\textsf{L}_{j}(\\psi_{j})$ of a branch $\\theta_{j}$ of $\\mathcal{T}_{j}$.\n\nGiven a valuation $\\nu$ of $\\mathcal{F}_{C_{n}}$, we can then prove that if $\\mathcal{T}_{j}$ is true under $\\nu$, then so is $\\mathcal{T}_{j+1}$. To see this, suppose $\\nu$ indeed validates $\\mathcal{T}_{j}$, and therefore validates some branch $\\theta$ of $\\mathcal{T}_{j}$:\n\\begin{enumerate}\n\\item if $\\theta=\\theta_{j}$, that is, $\\mathcal{T}_{j+1}$ is obtained from $\\mathcal{T}_{j}$ by extension of precisely the branch satisfied by $\\nu$, by Lemma  \\ref{validates consequence rule} we find that all new formulas of one of the branches created by applying $R_{j}$ to $\\textsf{L}_{j}(\\psi_{j})$ are validated by $\\nu$; this branch, which extends $\\theta_{j}$, is then a branch of $\\mathcal{T}_{j+1}$ true under $\\nu$;\n\\item if $\\theta\\neq\\theta_{j}$, $\\theta$ is still a branch of $\\mathcal{T}_{j+1}$, and in particular it is true under $\\nu$.\n\\end{enumerate}\nIn any case, the conclusion must be that $\\nu$ validates $\\mathcal{T}_{j+1}$.\n\nSo suppose $\\varphi$ is not a tautology according to $\\mathcal{RM}_{C_{n}}$, and therefore there exists a valuation $\\nu\\in\\mathcal{F}_{C_{n}}$ with $\\nu(\\varphi)=F_{n}$. Since $\\nu$ satisfies $\\textsf{F}_{n}(\\varphi)$, and therefore $\\mathcal{T}_{0}$, by induction we prove that any completed tableau for $\\textsf{F}_{n}(\\varphi)$ is true under $\\nu$, and so it cannot be closed. Since every completed tableau for $\\textsf{F}_{n}(\\varphi)$ is open, $\\not\\vdash_{\\mathbb{T}_{n}}\\varphi$, and by the contrapositive we have the theorem.\n\\end{proof}\n\nTo prove completeness, we must define what are Hintikka sets for $\\mathbb{T}_{n}$.\n\n\\begin{definition}\\label{Hintikkaset}\nA non-empty set $\\Gamma$ of labeled formulas $\\textsf{L}(\\varphi)$, with $\\textsf{L}$ in $\\textsf{B}_{n}$ and $\\varphi$ a formula of $C_{n}$, is a Hintikka set for $\\mathbb{T}_{n}$ if all of the following properties are simultaneously satisfied, where $0\\leq i\\leq n-1$ and $0\\leq k\\leq n-2$.\n\\begin{enumerate}\n\\item If $\\textsf{L}(\\varphi), \\textsf{L}^{*}(\\varphi)\\in \\Gamma$, then $\\textsf{L}=\\textsf{L}^{*}$.\n\\item If $\\textsf{t}_{0}^{n}(\\varphi)$, no $\\textsf{t}_{j}^{n}(\\varphi\\wedge\\neg\\varphi)$, for $0\\leq j\\leq n-1$, is in $\\Gamma$.\n\\item If $\\textsf{t}_{k+1}^{n}(\\varphi)\\in\\Gamma$, then $\\textsf{T}_{n}(\\varphi\\wedge\\neg\\varphi)$ is not in $\\Gamma$.\n\\item If $\\textsf{t}_{k+1}^{n}(\\varphi), \\textsf{L}(\\varphi^{1})\\in\\Gamma$, then $\\textsf{L}=\\textsf{t}_{k}^{n}$.\n\\item If $\\textsf{T}_{n}(\\neg\\varphi)\\in\\Gamma$, at least one of $\\textsf{t}_{0}^{n}(\\varphi), \\dotsc  , \\textsf{t}_{n-1}^{n}(\\varphi), \\textsf{F}_{n}(\\varphi)$ is also in $\\Gamma$.\n\\item If $\\textsf{t}_{i}^{n}(\\neg\\varphi)\\in\\Gamma$, at least one of $\\textsf{t}_{0}^{n}(\\varphi), \\dotsc  , \\textsf{t}_{n-1}^{n}(\\varphi)$ is also in $\\Gamma$.\n\\item If $\\textsf{F}_{n}(\\neg\\varphi)\\in\\Gamma$, so is $\\textsf{T}_{n}(\\varphi)$.\n\\item If $\\textsf{T}_{n}(\\varphi\\wedge\\psi)\\in\\Gamma$, then:\n\\begin{enumerate}\n\\item either $\\textsf{T}_{n}(\\varphi), \\textsf{T}_{n}(\\psi)\\in\\Gamma$;\n\\item or $\\textsf{T}_{n}(\\varphi), \\textsf{t}_{j}^{n}(\\psi)\\in\\Gamma$, for some $0\\leq j\\leq n-1$;\n\\item or $\\textsf{t}_{j}^{n}(\\varphi), \\textsf{T}_{n}(\\psi)\\in \\Gamma$, for a $0\\leq j\\leq n-1$;\n\\item or $\\textsf{t}_{j}^{n}(\\varphi), \\textsf{t}_{l}^{n}(\\psi)\\in\\Gamma$, for $0\\leq j, l\\leq n-1$.\n\\end{enumerate}\n\\item If $\\textsf{t}_{i}^{n}(\\varphi\\wedge\\psi)\\in\\Gamma$, then either:\n\\begin{enumerate}\n\\item $\\textsf{T}_{n}(\\varphi), \\textsf{t}_{j}^{n}(\\psi)\\in\\Gamma$, for a $0\\leq j\\leq n-1$;\n\\item or $\\textsf{t}_{j}^{n}(\\varphi), \\textsf{T}_{n}(\\psi)\\in\\Gamma$, for some $0\\leq j\\leq n-1$;\n\\item or $\\textsf{t}_{j}^{n}(\\varphi), \\textsf{t}_{l}^{n}(\\psi)\\in\\Gamma$, for $0\\leq j, l\\leq n-1$.\n\\end{enumerate}\n\\item If $\\textsf{F}_{n}(\\varphi\\wedge\\psi)\\in\\Gamma$, either $\\textsf{F}_{n}(\\varphi)$ or $\\textsf{F}_{n}(\\psi)$ is in $\\Gamma$.\n\\item If $\\textsf{T}_{n}(\\varphi\\vee\\psi)$ is in $\\Gamma$, at least one of $\\textsf{T}_{n}(\\varphi)$, $\\textsf{T}_{n}(\\psi)$, $\\textsf{t}_{j}^{n}(\\varphi)$ or $\\textsf{t}_{l}^{n}(\\psi)$ is also in $\\Gamma$, for $0\\leq j, l\\leq n-1$.\n\\item If $\\textsf{t}_{i}^{n}(\\varphi\\vee\\psi)$ is in $\\Gamma$, either $\\textsf{t}_{j}^{n}(\\varphi)$ or $\\textsf{t}_{l}^{n}(\\psi)$, for $0\\leq j,l\\leq n-1$, is also in $\\Gamma$.\n\\item If $\\textsf{F}_{n}(\\varphi\\vee\\psi)\\in\\Gamma$, then both $\\textsf{F}_{n}(\\varphi)$ and $\\textsf{F}_{n}(\\psi)$ are in $\\Gamma$.\n\\item If $\\textsf{T}_{n}(\\varphi\\rightarrow\\psi)\\in\\Gamma$, at least one of $\\textsf{F}_{n}(\\varphi)$, $\\textsf{T}_{n}(\\psi)$ or $\\textsf{t}_{j}^{n}(\\psi)$, for $0\\leq j\\leq n-1$, is also in $\\Gamma$.\n\\item If $\\textsf{t}_{i}^{n}(\\varphi\\rightarrow\\psi)$ is in $\\Gamma$, either:\n\\begin{enumerate}\n\\item $\\textsf{t}_{j}^{n}(\\varphi)$ and $\\textsf{T}_{n}(\\psi)$, for a $0\\leq j\\leq n-1$, are both in $\\Gamma$;\n\\item or $\\textsf{t}_{j}^{n}(\\psi)$, for a $0\\leq j\\leq n-1$, is in $\\Gamma$.\n\\end{enumerate}\n\\item If $\\textsf{F}_{n}(\\varphi\\rightarrow\\psi)$ is in $\\Gamma$, then:\n\\begin{enumerate}\n\\item either $\\textsf{T}_{n}(\\varphi)$ and $\\textsf{F}_{n}(\\psi)$ are both in $\\Gamma$;\n\\item or $\\textsf{t}_{j}^{n}(\\varphi)$ and $\\textsf{F}_{n}(\\psi)$, for a $0\\leq j\\leq n-1$, are both in $\\Gamma$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{definition}\n\n\nWe must now prove that a Hintikka set for $\\mathbb{T}_{n}$ is satisfiable in $\\mathcal{F}_{C_{n}}$, that is, that there exists a $\\nu\\in\\mathcal{F}_{C_{n}}$ such that $\\textsf{L}(\\varphi)\\in\\Gamma$ implies $\\nu(\\varphi)=L$. We start by defining, for a Hintikka set $\\Gamma$ for $\\mathbb{T}_{n}$, $\\Gamma_{0}$ as the set of formulas of $C_{n}$ such that $\\textsf{L}(\\varphi)\\in\\Gamma$, for some label $\\textsf{L}$ of $\\textsf{B}_{n}$; then, we take $\\nu_{0}:\\Gamma_{0}\\rightarrow B_{n}$ defined by $\\nu_{0}(\\varphi)=L$ iff $\\textsf{L}(\\varphi)\\in \\Gamma$. That $\\nu_{0}$ is indeed a function one can prove by noticing clause $(1)$ of Definition \\ref{Hintikkaset}: if one had both $\\nu_{0}(\\varphi)=L$ and $\\nu_{0}(\\varphi)=L^{*}$, for $L\\neq L^{*}$, then it would follow that $\\textsf{L}(\\varphi), \\textsf{L}^{*}(\\varphi)\\in \\Gamma$, for $\\textsf{L}\\neq\\textsf{L}^{*}$, contradicting clause $(1)$.\n\nWe extend $\\nu_{0}$ to a function $\\nu:F(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow B_{n}$ by induction on the complexity of a formula in much the same way we proceeded in Proposition \\ref{Rows of tables are homomorphisms}. That is, for a propositional variable $p$, we define:\n\\begin{enumerate}\n\\item $\\nu(p)=\\nu_{0}(p)$, if $p\\in\\Gamma_{0}$, and arbitrarily otherwise;\n\\item $\\nu(\\neg p)=\\nu_{0}(\\neg p)$, if $\\neg p\\in\\Gamma_{0}$, and as any value in $\\tilde{\\neg}\\nu(p)$ otherwise;\n\\item $\\nu(p\\wedge\\neg p)=\\nu_{0}(p\\wedge\\neg p)$, in the case that $p\\wedge\\neg p\\in\\Gamma_{0}$; if $p\\wedge\\neg p\\not\\in \\Gamma_{0}$ and $\\nu(p)=t_{0}^{n}$, we make $\\nu(p\\wedge\\neg p)=T_{n}$, but if $p\\wedge\\neg p\\not\\in\\Gamma_{0}$ and $\\nu(p)=t_{k}^{n}$, for $1\\leq k\\leq n-1$, we define $\\nu(p\\wedge\\neg p)$ as an arbitrary value of $I_{n}$; if none of these is the case, $\\nu(p\\wedge\\neg p)$ may be given any value in $\\nu(p)\\tilde{\\wedge}\\nu(\\neg p)$;\n\\item $\\nu(p^{1})=\\nu_{0}(p^{1})$, whenever $p^{1}\\in \\Gamma_{0}$; in the case that $p^{1}\\not\\in\\Gamma_{0}$, if $\\nu(p)=t_{k}^{n}$, for a $1\\leq k\\leq n-1$, we make $\\nu(p^{1})=t_{k-1}^{n}$, and otherwise $\\nu(p^{1})$ can equal any value of $\\tilde{\\neg}\\nu(p\\wedge\\neg p)$.\n\\end{enumerate}\n\nIf $\\nu(\\alpha)$ and $\\nu(\\beta)$, as well as $\\nu(\\neg\\alpha)$, $\\nu(\\alpha\\wedge\\neg\\alpha)$, $\\nu(\\alpha^{1})$, $\\nu(\\neg\\beta)$, $\\nu(\\beta\\wedge\\neg\\beta)$, and $\\nu(\\beta^{1})$, have been already defined, we may also define $\\nu(\\varphi)$ and $\\nu(\\psi)$, as well as other correlated values of $\\nu$, for $\\varphi=\\neg\\alpha$ and $\\psi=\\alpha\\#\\beta$ (where $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$). For $\\varphi$ we have the following.\n\\begin{enumerate}\n\\item $\\nu(\\varphi)=\\nu(\\neg\\alpha)$ has already been defined.\n\\item If $\\neg\\varphi\\in\\Gamma_{0}$, make $\\nu(\\neg\\varphi)$ equal to $\\nu_{0}(\\neg\\varphi)$, otherwise it is enough to make $\\nu(\\varphi)$ equal to some value in $\\tilde{\\neg}\\nu(\\varphi)$.\n\\item If $\\varphi\\wedge\\neg\\varphi\\in\\Gamma_{0}$, one makes $\\nu(\\varphi\\wedge\\neg\\varphi)$ equal to $\\nu_{0}(\\varphi\\wedge\\neg\\varphi)$; if $\\varphi\\wedge\\neg\\varphi$ is not in $\\Gamma_{0}$ and $\\nu(\\varphi)\\in I_{n}$, we either make $\\nu(\\varphi\\wedge\\neg\\varphi)$ equal $T_{n}$, if $\\nu(\\varphi)=t^{n}_{0}$, or let it take any value in $I_{n}$, if $\\nu(\\varphi)=t_{k}^{n}$ for $1\\leq k\\leq n-1$; if $\\varphi\\wedge\\neg\\varphi\\not\\in\\Gamma_{0}$ and $\\nu(\\varphi)\\not\\in I_{n}$, it is enough for $\\nu(\\varphi\\wedge\\neg\\varphi)$ to lie in $\\nu(\\varphi)\\tilde{\\wedge}\\nu(\\neg\\varphi)$.\n\\item We make $\\nu(\\varphi^{1})$ equal to: $\\nu_{0}(\\varphi^{1})$ if $\\varphi^{1}\\in\\Gamma_{0}$; $t_{k-1}^{n}$ if $\\varphi^{1}\\not\\in\\Gamma_{0}$ and $\\nu(\\varphi)=t_{k}^{n}$, for $1\\leq k\\leq n-1$; and any value of $\\tilde{\\neg}\\nu(\\varphi\\wedge\\neg\\varphi)$ if $\\varphi^{1}\\not\\in\\Gamma_{0}$ and $\\nu(\\varphi)\\in\\{T_{n}, t_{0}^{n}, F_{n}\\}$.\n\\end{enumerate}\nFor $\\psi$ the conditions are as bellow.\n\\begin{enumerate}\n\\item If $\\psi\\in\\Gamma_{0}$, it is necessary to make $\\nu(\\psi)=\\nu_{0}(\\psi)$. In the case that $\\psi\\not\\in\\Gamma_{0}$, if $\\#=\\wedge$ and $\\beta=\\neg\\alpha$, $\\nu(\\psi)$ has already been defined; otherwise, $\\nu(\\psi)$ is given any value among those found in $\\nu(\\alpha)\\tilde{\\#}\\nu(\\beta)$.\n\\item To define $\\nu(\\neg\\psi)$, we again start by demanding that, in the case that $\\neg\\psi\\in\\Gamma_{0}$, it must equal $\\nu_{0}(\\neg\\psi)$. If $\\neg\\psi\\not\\in\\Gamma_{0}$, and $\\#=\\wedge$ and $\\beta=\\neg\\alpha$, $\\nu(\\neg\\psi)=\\nu(\\alpha^{1})$ has already been defined; otherwise $\\nu(\\neg\\psi)$ can be given any value found in $\\tilde{\\neg}\\nu(\\psi)$.\n\\item In the case that $\\psi\\wedge\\neg\\psi\\in\\Gamma_{0}$, $\\nu(\\psi\\wedge\\neg\\psi)=\\nu_{0}(\\psi\\wedge\\neg\\psi)$. When $\\psi\\wedge\\neg\\psi\\not\\in\\Gamma_{0}$: if $\\nu(\\psi)=t_{0}^{n}$, $\\nu(\\psi\\wedge\\neg\\psi)$ is defined as $T_{n}$; if $\\nu(\\psi)=t_{k}^{n}$ (where $1\\leq k\\leq n-1$), $\\nu(\\psi\\wedge\\neg\\psi)$ equals an arbitrary value of $I_{n}$; otherwise, $\\nu(\\psi\\wedge\\neg\\psi)$ can take any value on $\\nu(\\psi)\\tilde{\\wedge}\\nu(\\neg\\psi)$.\n\\item If $\\psi^{1}\\in\\Gamma_{0}$, $\\nu(\\psi^{1})$ is $\\nu_{0}(\\psi^{1})$. If $\\psi^{1}\\not\\in\\Gamma_{0}$ and $\\nu(\\psi)=t_{k}^{n}$, for a $1\\leq k\\leq n-1$, we make $\\nu(\\psi^{1})$ equal to $t_{k-1}^{n}$. And if $\\psi^{1}\\not\\in\\Gamma_{0}$ and $\\nu(\\psi)\\not\\in\\{t_{1}^{n}, \\dotsc  , t_{n-1}^{n}\\}$, we can give to $\\nu(\\psi^{1})$ any value from $\\tilde{\\neg}\\nu(\\psi\\wedge\\neg\\psi)$.\n\\end{enumerate}\n\nIt is easy to see both that $\\nu$ is a well defined function from $F(\\Sigma_{C}, \\mathcal{V})$ to $B_{n}$ and that, if $\\nu$ is a homomorphism, then it certainly lies in $\\mathcal{F}_{C_{n}}$. The difficulty here is convincing oneself that $\\nu$ is indeed a homomorphism: the problem here is that, while in Proposition \\ref{Rows of tables are homomorphisms} the relevant set of formulas was closed under subformulas, the set $\\Gamma_{0}$ is not; it would seem possible for a formula $\\beta\\in\\Gamma_{0}$ to have a subformula $\\alpha\\not\\in\\Gamma_{0}$ such that $\\nu(\\alpha)$, as defined above, is incompatible with $\\nu(\\beta)=\\nu_{0}(\\beta)$, in the sense that the function $\\nu$ cannot be a homomorphism. We will argue that this does not happen, although the complete picture will be left for the reader to fill out.\n\nOnly a few clauses of Definition \\ref{Hintikkaset} do not presuppose that the immediate subformulas of $\\beta$ are also in $\\Gamma_{0}$, specifically clauses $10$, $11$, $12$, $14$ and $15$ (item $(b)$); regarding the remaining clauses, the proof that $\\varphi$ behaves like a homomorphism should runs smoothly. So, take one of these difficult clauses to analyze more thoroughly, let us say $10$: so, there are formulas $\\varphi$ and $\\psi$ such that $\\textsf{F}_{n}(\\varphi\\wedge\\psi)\\in \\Gamma$, meaning then that $\\varphi\\wedge\\psi$ is in $\\Gamma_{0}$, and $\\textsf{F}_{n}(\\varphi)\\in \\Gamma$, and thus $\\varphi\\in\\Gamma_{0}$; the case in which $\\textsf{F}_{n}(\\varphi)$ is not in $\\Gamma$ but $\\textsf{F}_{n}(\\psi)$ is can be treated in an analogous way.\n\nBy the definition of $\\nu$ we have given, we have that $\\nu(\\varphi\\wedge\\psi)=\\nu(\\varphi)=F_{n}$, and $\\nu(\\psi)$ takes any value as long as $\\nu$ is still a well-defined function that satisfies all conditions to be in $\\mathcal{F}_{C_{n}}$, except perhaps being a homomorphism. We ask ourselves: is there any value we can give to $\\nu(\\psi)$ that would lead to $\\nu(\\varphi\\wedge\\psi)\\not\\in\\nu(\\varphi)\\tilde{\\wedge}\\nu(\\psi)$ and therefore force $\\nu$ not to be a homomorphism? The answer is no: since $\\nu(\\varphi)=F_{n}$, looking at the table for $\\tilde{\\wedge}$ just after Definition \\ref{Define operations in Cn}, we see that, for any value possibly taken by $\\nu(\\psi)$, $\\nu(\\varphi)\\tilde{\\wedge}\\nu(\\psi)=F_{n}\\tilde{\\wedge}\\nu(\\psi)=\\{F_{n}\\}$, which contains $\\nu(\\varphi\\wedge\\psi)$. Briefly put, the definition of Hintikka sets, despite not being strong enough to guarantee that those sets are closed under subformulas, carries enough information to make sure the $\\nu$, as defined above, is a homomorphism in $\\mathcal{F}_{C_{n}}$.\n\nFor another example, look now at clause $15(b)$: there are formulas $\\varphi$ and $\\psi$ and indices $0\\leq i, j\\leq n-1$ for which $\\textsf{t}_{i}^{n}(\\varphi\\rightarrow\\psi)$ and $\\textsf{t}_{j}^{n}(\\psi)$ are both in $\\Gamma$, and so $\\varphi\\rightarrow\\psi, \\psi\\in \\Gamma_{0}$: is it possible to find a value of $\\nu(\\varphi)$ that makes of $\\nu$ not a homomorphism? No: by looking at the table for $\\tilde{\\rightarrow}$ in $\\mathcal{A}_{C_{n}}$ below, we see that $\\nu(\\varphi\\rightarrow\\psi)=t_{i}^{n}$ is in $D_{n}=\\nu(\\varphi)\\tilde{\\rightarrow}t_{j}^{n}=\\nu(\\varphi)\\tilde{\\rightarrow}\\nu(\\psi)$, for any $\\nu(\\varphi)\\in B_{n}$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{|l|c|c|r|}\n\\hline\n$\\rightarrow$ & $F_{n}$ & $I_{n}$ & $T_{n}$ \\\\ \\hline\n$F_{n}$ & $\\{T_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n$I_{n}$ & $\\{F_{n}\\}$ & $D_{n}$ & $D_{n}$ \\\\ \\hline\n$T_{n}$ & $\\{F_{n}\\}$ & $D_{n}$ & $\\{T_{n}\\}$ \\\\ \\hline\n\\end{tabular}\n\\caption*{Table for Implication}\n\\end{figure}\n\nSummarizing what we have done just above, one gets the following result.\n\n\\begin{proposition}\\label{Hintikka's pre-lemma}\nFor any Hintikka set $\\Gamma$ for $\\mathbb{T}_{n}$, we define $\\Gamma_{0}=\\{\\varphi\\in\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V}): \\textsf{L}(\\varphi)\\in\\Gamma\\}$ and the function $\\nu_{0}:\\Gamma_{0}\\rightarrow B_{n}$ by $\\nu_{0}(\\varphi)=L$ iff $\\textsf{L}(\\varphi)\\in\\Gamma$. In this case, there is a homomorphism $\\nu\\in\\mathcal{F}_{C_{n}}$ extending $\\nu_{0}$.\n\\end{proposition}\n\nWe are now in position to trivially prove the equivalent to Hintikka's lemma in $\\mathbb{T}_{n}$.\n\n\\begin{theorem}\\label{Hintikka's lemma}\nFor any Hintikka set $\\Gamma$ for $\\mathbb{T}_{n}$, there exists a $\\nu\\in\\mathcal{F}_{C_{n}}$ satisfying that $\\textsf{L}(\\varphi)\\in\\Gamma$ implies $\\nu(\\varphi)=L$.\n\\end{theorem}\n\n\\begin{proof}\nTake $\\Gamma_{0}$ as the set of formulas $\\varphi$ for which there exists a label $\\textsf{L}$ with $\\textsf{L}(\\varphi)\\in\\Gamma$, and define the function $\\nu:\\Gamma_{0}\\rightarrow B_{n}$ by $\\nu_{0}(\\varphi)=L$ iff $\\textsf{L}(\\varphi)\\in\\Gamma$, as it would be expected. By Proposition \\ref{Hintikka's pre-lemma}, there exists $\\nu\\in\\mathcal{F}_{C_{n}}$ extending $\\nu_{0}$ and, of course, if $\\textsf{L}(\\varphi)\\in\\Gamma$, $\\nu(\\varphi)=\\nu_{0}(\\varphi)=L$.\n\\end{proof}\n\nHintikka sets for $\\mathbb{T}_{n}$, as it would be expected, correspond to open branches $\\theta$ of a complete tableau $\\mathcal{T}$ of $\\mathbb{T}_{n}$: in fact, define as $\\Gamma$ the set of labeled formulas appearing in $\\theta$. Since $\\theta$ is open, from Definition \\ref{Definitions about tableaux} it follows that if $\\textsf{L}(\\varphi)$ and $\\textsf{L}^{*}(\\varphi)$ are both in $\\Gamma$, meaning that both are in $\\theta$, then $\\textsf{L}=\\textsf{L}^{*}$, what amounts to clause $1$ in Definition \\ref{Hintikkaset}.\n\nIf $\\textsf{t}_{0}^{n}(\\varphi)$ and some $\\textsf{t}_{j}^{n}(\\varphi\\wedge\\neg\\varphi)$, for $0\\leq j\\leq n-1$, were both in $\\Gamma$, and so in $\\theta$, this would make of $\\theta$ a closed branch, contradicting our supposition that $\\theta$ is actually open; of course, we get from this that clause $2$ of Definition \\ref{Hintikkaset} is also satisfied. And if $\\textsf{t}^{n}_{k+1}(\\varphi)\\in\\Gamma$, for $0\\leq k\\leq n-2$, then neither $\\textsf{T}_{n}(\\varphi\\wedge\\neg\\varphi)$ nor $\\textsf{L}(\\varphi)$, for $\\textsf{L}\\neq \\textsf{t}_{k}^{n}$, can be in $\\Gamma$, corresponding to clauses $3$ and $4$. Finally, for $\\textsf{L}(\\neg\\varphi)$ or $\\textsf{L}(\\varphi\\#\\psi)$ (with $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$) in $\\Gamma$, one notices that $\\mathcal{T}$ being complete forces $\\theta$ to contain all labeled formulas of one branch of the rule in $\\mathbb{T}_{n}$ headed by $\\textsf{L}(\\neg\\varphi)$ or respectively $\\textsf{L}(\\varphi\\#\\psi)$, what leads to clauses $5$ through $16$ of Definition \\ref{Hintikkaset} being satisfied. \n\nThese observations, combined with Theorem \\ref{Hintikka's lemma}, allow one to prove the following result.\n\n\\begin{corollary}\\label{corollary to Hintika's lemma}\nFor $\\textsf{L}(\\varphi)$ a signed formula of $C_{n}$, let $\\theta$ be an open branch of a completed tableau $\\mathcal{T}$ in $\\mathbb{T}_{n}$ starting with $\\textsf{L}(\\varphi)$, and $\\Gamma$ be the set of labeled formulas appearing in $\\theta$. There exists a homomorphism $\\nu\\in\\mathcal{F}_{C_{n}}$ such that $\\nu(\\psi)=L$ iff $\\textsf{L}(\\psi)\\in\\Gamma$.\n\\end{corollary}\n\n\\begin{theorem}\\label{completeness tableaux}\nFor $\\Gamma\\cup\\{\\varphi\\}$ a finite set of formulas of $C_{n}$, if $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$, then $\\Gamma\\vdash_{\\mathbb{T}_{n}}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nAs it was done in Theorem \\ref{Correct tableaux}, we can assume that $\\Gamma=\\emptyset$: this is because, if $\\Gamma=\\{\\gamma_{1}, \\dotsc  , \\gamma_{m}\\}$, since $\\bigwedge_{i}^{m}\\gamma_{i}\\vdash_{C_{n}}\\gamma_{j}$, for every $1\\leq j\\leq m$, and $\\Gamma\\vdash_{C_{n}}\\bigwedge_{i=1}^{m}\\gamma_{i}$, we get that $\\Gamma\\vdash_{C_{n}}\\varphi$ if and only if $\\vdash_{C_{n}}\\bigwedge_{i=1}^{m}\\gamma_{i}\\rightarrow\\varphi$. Given that $\\Gamma\\vdash_{C_{n}}\\varphi$ iff $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$, and $\\Gamma\\vdash_{\\mathbb{T}_{n}}\\varphi$ iff $\\vdash_{\\mathbb{T}_{n}}\\bigwedge_{i=1}^{n}\\gamma_{i}\\rightarrow\\varphi$ by definition, it is indeed enough to take $\\Gamma=\\emptyset$.\n\nSo we suppose that $\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$: if $\\mathcal{T}$ is a completed tableau in $\\mathbb{T}_{n}$ starting with $\\textsf{F}_{n}(\\varphi)$ with an open branch $\\theta$, by Corollary \\ref{corollary to Hintika's lemma} there exists a homomorphism $\\nu\\in\\mathcal{F}_{C_{n}}$ such that $\\nu(\\psi)=L$ iff $\\textsf{L}(\\psi)$ appears in $\\theta$. Of course, since $\\textsf{F}_{n}(\\varphi)$ is in $\\theta$, $\\nu(\\varphi)=F_{n}$ and therefore $\\varphi$ is not a tautology, meaning that $\\not\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$ and leading to a contradiction.\n\nThis means that, if $\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$, every completed tableau in $\\mathbb{T}_{n}$ starting with $\\textsf{F}_{n}(\\varphi)$ must be closed, if there exist any; since it is always possible to construct a complete tableau in $\\mathbb{T}_{n}$ starting from any given labeled formula in a finite number of steps, it follows that $\\vdash_{\\mathbb{T}_{n}}\\varphi$.\n\\end{proof}\n\nOf course, Theorems \\ref{Correct tableaux} and \\ref{completeness tableaux} prove that the calculi $\\mathbb{T}_{n}$ are correct and complete in relation to the respective logics $C_{n}$. They are also decision methods, given that our tableaux can always be constructed in a finite number of steps, inspired by the RNmatrices $\\mathcal{RM}_{C_{n}}$ as our row-branching, row-eliminating truth-tables also were: it is easy to see that, if $\\varphi$ is valid in $C_{n}$, every complete tableau starting with $\\textsf{F}_{n}(\\varphi)$ in $\\mathbb{T}_{n}$ will be closed; and if $\\varphi$ is not valid in $C_{n}$, every complete tableau starting with $\\textsf{F}_{n}(\\varphi)$ in $\\mathbb{T}_{n}$ will be open, and each of its open branches produces a $\\nu\\in\\mathcal{F}_{C_{n}}$ with $\\nu(\\varphi)=F_{n}$.\n\n\\subsubsection{Examples and derived rules}\n\nIn this section we present some examples of actually using the calculi $\\mathbb{T}_{n}$ to test the validity of arguments.\n\nFor the argument $\\neg\\neg\\alpha\\vdash_{C_{1}}\\alpha$, which is indeed valid as shown below, keep in mind that we are listing, at the right side of the tableau, the row (or rows), as well as their label and lead connective, that lead to the inclusion of the present row; this way, one can check how the tableau rules of $\\mathbb{T}_{1}$ are being applied.\n\n\\begin{center}\n\\begin{prooftree}\n{\n}\n[\\textsf{F}_{1}(\\neg\\neg\\alpha\\rightarrow\\alpha)\n\t[\\textsf{T}_{1}(\\neg\\neg\\alpha), just=$\\textsf{F}_{1}\\rightarrow$:!u\n\t\t[\\textsf{F}_{1}(\\alpha), just=$\\textsf{F}_{1}\\rightarrow$:!uu\n\t\t\t[\\textsf{F}_{1}(\\neg\\alpha), just=$\\textsf{T}_{1}\\neg$:!uu\n\t\t\t\t[\\textsf{T}_{1}(\\alpha), just=$\\textsf{F}_{1}\\neg$:!u, close={:!uu, !c}]]\n\t\t\t[\\textsf{t}_{0}^{1}(\\neg\\alpha)\n\t\t\t\t[\\textsf{t}_{0}^{1}(\\alpha), close={:!uu, !c}]]]]\n\t[\\textsf{t}_{0}^{1}(\\neg\\neg\\alpha)\n\t\t[\\textsf{F}_{1}(\\alpha)\n\t\t\t[\\textsf{t}_{0}^{1}(\\neg\\alpha), just=$\\textsf{t}_{0}^{1}\\neg$:!uu\n\t\t\t\t[\\textsf{t}_{0}^{1}(\\alpha), just=$\\textsf{t}_{0}^{1}\\neg$:!u, close={:!uu, !c}]]]]]\n\\end{prooftree}\n\\end{center}\n\nTesting the same argument but in $C_{2}$, that is $\\neg\\neg\\alpha\\vdash_{C_{2}}\\alpha$, in the tableau of Figure \\ref{Test in C2} shows that it remains valid, but the complexity of the corresponding tableau grows substantially. \n\n\\begin{landscape}\n\nThis suggests, correctly, that the main problem found in dealing with the calculi $\\mathbb{T}_{n}$ is how fast their tableaux grow horizontally, meaning how ramified they become.\n\nTo simplify the task of writing tableaux in $\\mathbb{T}_{n}$, we can construct derived tableau rules: much like, when proving something axiomatically, we make use of intermediary results, we may obtain new tableaux rules derived from the ones already present in $\\mathbb{T}_{n}$. We begin by defining some sets of labels: for $0\\leq i\\leq n-1$, \n\\[\\textsf{D}_{n}^{\\leq i}=\\{\\textsf{T}_{n}, \\textsf{t}^{n}_{0}, \\dotsc  , \\textsf{t}^{n}_{i}\\},\\quad \\textsf{D}_{n}^{\\geq i}=\\{\\textsf{t}^{n}_{i}, \\dotsc  , \\textsf{t}^{n}_{n-1}\\}\\quad\\text{and}\\quad\\textsf{D}_{n}^{\\leq -1}=\\{\\textsf{T}_{n}\\}\\]\n(what somewhat reflects the heuristic that $T_{n}$ behaves like a $t^{n}_{-1}$ should).\n\n\\begin{figure}[H]\n\\centering\n\\begin{prooftree}\n{\n}\n[\\textsf{F}_{2}(\\neg\\neg\\alpha\\rightarrow\\alpha)\n\t[\\textsf{T}_{2}(\\neg\\neg\\alpha), just=$\\textsf{F}_{2}\\rightarrow$:!u\n\t\t[\\textsf{F}_{2}(\\alpha), just=$\\textsf{F}_{2}\\rightarrow$:!uu\n\t\t\t[\\textsf{t}_{0}^{2}(\\neg\\alpha), just=$\\textsf{T}_{2}\\neg$:!uu\n\t\t\t\t[\\textsf{t}_{0}^{2}(\\alpha), just=$\\textsf{t}_{0}^{2}\\neg$:!u, close={:!uu, !c}]\n\t\t\t\t[\\textsf{t}_{1}^{2}(\\alpha), close={:!uu, !c}]]\n\t\t\t[\\textsf{t}_{1}^{2}(\\neg\\alpha)\n\t\t\t\t[\\textsf{t}_{0}^{2}(\\alpha), just=$\\textsf{t}_{1}^{2}\\neg$:!u, close={:!uu, !c}]\n\t\t\t\t[\\textsf{t}_{1}^{2}(\\alpha), close={:!uu, !c}]]\n\t\t\t[\\textsf{F}_{2}(\\neg\\alpha)\n\t\t\t\t[\\textsf{T}_{2}(\\alpha), just=$\\textsf{F}_{2}\\neg$:!u, close={:!uu, !c}]]]]\n\t[\\textsf{t}_{0}^{2}(\\neg\\neg\\alpha)\n\t\t[\\textsf{F}_{2}(\\alpha)\n\t\t\t[\\textsf{t}_{0}^{2}(\\neg\\alpha), just=$\\textsf{t}_{0}^{2}\\neg$:!uu\n\t\t\t\t[\\textsf{t}_{0}^{2}(\\alpha), close={:!uu, !c}]\n\t\t\t\t[\\textsf{t}_{1}^{2}(\\alpha), close={:!uu, !c}]]\n\t\t\t[\\textsf{t}_{1}^{2}(\\neg\\alpha)\n\t\t\t\t[\\textsf{t}_{0}^{2}(\\alpha), close={:!uu, !c}]\n\t\t\t\t[\\textsf{t}_{1}^{2}(\\alpha), close={:!uu, !c}]]]]\n\t[\\textsf{t}_{1}^{2}(\\neg\\neg\\alpha)\n\t\t[\\textsf{F}_{2}(\\alpha)\n\t\t\t[\\textsf{t}_{0}^{2}(\\neg\\alpha), just=$\\textsf{t}_{1}^{2}\\neg$:!uu\n\t\t\t\t[\\textsf{t}_{0}^{2}(\\alpha), close={:!uu, !c}]\n\t\t\t\t[\\textsf{t}_{1}^{2}(\\alpha), close={:!uu, !c}]]\n\t\t\t[\\textsf{t}_{1}^{2}(\\neg\\alpha)\n\t\t\t\t[\\textsf{t}_{0}^{2}(\\alpha), close={:!uu, !c}]\n\t\t\t\t[\\textsf{t}_{1}^{2}(\\alpha), close={:!uu, !c}]]]]]\n\\end{prooftree}\n\\caption{Testing whether $\\neg\\neg\\alpha\\vdash_{C_{2}}\\alpha$}\n\\label{Test in C2}\n\\end{figure}\n\n\nTaking indices $0\\leq i\\leq n-1$, $0\\leq j\\leq n-2$, $k\\geq n$ and $l\\geq n-1$, we have the following derived rules for formulas of the form $\\varphi^{m}\\wedge\\neg \\varphi^{m}$.\n\n\\end{landscape}\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c} \\textsf{T}_{n}(\\varphi^{i}\\wedge\\neg\\varphi^{i})\\\\\\hline \\textsf{t}^{n}_{i}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c} \\textsf{t}^{n}_{i}(\\varphi^{j}\\wedge\\neg\\varphi^{j})\\\\\\hline \\textsf{D}_{n}^{\\geq j+1}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}_{n}(\\varphi^{i}\\wedge\\neg\\varphi^{i})}\\\\\\hline \\textsf{D}_{n}^{\\leq i-1}(\\varphi) & \\textsf{F}_{n}(\\varphi)\\end{array}\\\\\n& & & &\\\\\n\\begin{array}{c} \\textsf{T}_{n}(\\varphi^{k}\\wedge\\neg\\varphi^{k})\\\\\\hline \\otimes\\end{array}& \\quad\\quad\\quad & \\begin{array}{c} \\textsf{t}^{n}_{i}(\\varphi^{l}\\wedge\\neg\\varphi^{l})\\\\\\hline \\otimes\\end{array}& &\n\\end{array}$\n\\end{table}\n\nHere, the symbol $\\otimes$ at the bottom of a rule means that any branch where the labeled formula at the top of said rule appears is closed.\n\nProving these derived rules are valid is long and tedious, but rather trivial: to give one example, take the rule headed by $\\textsf{T}_{n}(\\varphi^{i}\\wedge\\neg\\varphi^{i})$, and $n=1$ and $i=0$; more complex cases should be treated with inductive arguments. Then we have the following tableau, which indeed only leaves us $\\textsf{t}^{1}_{0}(\\varphi)$ as the rule stated.\n\n\\begin{center}\n\\begin{prooftree}\n{\n}\n[\\textsf{T}_{1}(\\varphi\\wedge\\neg\\varphi)\n\t[\\textsf{T}_{1}(\\varphi), just=$\\textsf{T}_{1}\\wedge$:!u\n\t\t[\\textsf{T}_{1}(\\neg\\varphi),  just=$\\textsf{T}_{1}\\wedge$:!uu\n\t\t\t[\\textsf{t}^{1}_{0}(\\varphi), just=$\\textsf{T}_{1}\\neg$:!u, close={:!uu, !c}]\n\t\t\t[\\textsf{F}_{1}(\\varphi), close={:!uu, !c}]]]\n\t[\\textsf{T}_{1}(\\varphi)\n\t\t[\\textsf{t}^{1}_{0}(\\neg\\varphi)\n\t\t\t[\\textsf{t}^{1}_{0}(\\varphi), just=$\\textsf{t}^{1}_{0}\\neg$:!u, close={:!uu, !c}]]]\n\t[\\textsf{t}^{1}_{0}(\\varphi)\n\t\t[\\textsf{T}_{1}(\\neg\\varphi)\n\t\t\t[\\textsf{t}^{1}_{0}(\\varphi)]\n\t\t\t[\\textsf{F}_{1}(\\varphi), close={:!uu, !c}]]]\n\t[\\textsf{t}^{1}_{0}(\\varphi)\n\t\t[\\textsf{t}^{1}_{0}(\\neg\\varphi)\n\t\t\t[\\textsf{t}^{1}_{0}(\\varphi)]]]]\n\\end{prooftree}\n\\end{center}\n\nWe can also easily obtain derived rules for formulas of the form $\\varphi^{m}$, as we show below: take now indices $1\\leq r\\leq n$, $1\\leq s\\leq n-1$, $0\\leq t\\leq n-s-1$ and $n-s\\leq u\\leq n-1$.\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}_{n}(\\varphi^{r})}\\\\\\hline \\textsf{D}_{n}^{\\leq r-2}(\\varphi) & \\textsf{F}_{n}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{t}^{n}_{t}(\\varphi^{s})\\\\\\hline \\textsf{t}^{n}_{s+t}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{F}_{n}(\\varphi^{r})\\\\\\hline \\textsf{t}^{n}_{r-1}(\\varphi)\\end{array}\\\\\n & & & &\\\\\n& & \\begin{array}{c}\\textsf{t}^{n}_{u}(\\varphi^{s})\\\\\\hline \\otimes\\end{array} & & \n\\end{array}$\n\\end{table}\n\nAlso important are derived rules for labeled formulas of the form $\\neg\\varphi^{m}$, where the indices are again $0\\leq i\\leq n-1$, $1\\leq r\\leq n$ and $1\\leq s\\leq n-1$.\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c}\\textsf{T}_{n}(\\neg\\varphi^{r})\\\\\\hline \\textsf{D}_{n}^{\\geq r-1}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{t}^{n}_{i}(\\neg\\varphi^{s})\\\\\\hline \\textsf{D}_{n}^{\\geq s}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}_{n}(\\neg\\varphi^{r})}\\\\\\hline \\textsf{D}_{n}^{\\leq r-2}(\\varphi) & \\textsf{F}_{n}(\\varphi)\\end{array}\n\\end{array}$\n\\end{table}\n\nAll of these derived rules allow us to give more convoluted examples of tableaux, where the $DR$ to the right of a row will signify that a derived rule was used to obtain the said row. \n\n\n\\begin{landscape}\n\nWe start by showing that $\\alpha, \\neg\\alpha, \\alpha^{1}\\vdash_{C_{1}}\\neg\\neg\\alpha$, or what is equivalent, that $\\vdash_{C_{1}}(\\alpha\\wedge\\neg\\alpha)\\wedge\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\neg\\neg\\alpha$.\n\n\n\n\n\\begin{center}\n\\begin{prooftree}\n{\n}\n[\\textsf{F}_{1}(((\\alpha\\wedge\\neg \\alpha)\\wedge\\neg(\\alpha\\wedge\\neg \\alpha))\\rightarrow\\neg\\neg \\alpha)\n\t[\\textsf{T}_{1}((\\alpha\\wedge\\neg \\alpha)\\wedge\\neg(\\alpha\\wedge\\neg \\alpha)), just=$\\textsf{F}_{1}\\rightarrow$:!u\n\t\t[\\textsf{F}_{1}(\\neg\\neg \\alpha), just=$\\textsf{F}_{1}\\rightarrow$:!uu\n\t\t\t[\\textsf{t}^{1}_{0}(\\alpha\\wedge\\neg \\alpha), just=$DR$:!uu, close={:!c}]]]\n\t[\\textsf{t}^{1}_{0}((\\alpha\\wedge\\neg \\alpha)\\wedge\\neg(\\alpha\\wedge\\neg\\alpha))\n\t\t[\\textsf{F}_{1}(\\neg\\neg \\alpha), close={:!u}]]]\n\\end{prooftree}\n\\end{center}\n\n\\begin{figure}[H]\n\\centering\n\\begin{center}\n\\begin{prooftree}\n{\n}\n[\\textsf{T}_{2}((\\alpha\\wedge\\neg\\alpha)\\wedge\\alpha^{1})\n\t[\\textsf{T}_{2}(\\alpha\\wedge\\neg\\alpha), just=$\\textsf{T}_{2}\\wedge$:!u\n\t\t[\\textsf{T}_{2}(\\alpha^{1}), just=$\\textsf{T}_{2}\\wedge$:!uu\n\t\t\t[\\textsf{t}^{2}_{0}(\\alpha), just=$DR$:!uu\n\t\t\t\t[\\textsf{T}_{2}(\\alpha), close={:!u, !c}, just=$DR$:!uu]\n\t\t\t\t[\\textsf{F}_{2}(\\alpha), close={:!u, !c}]]]\n\t\t[\\textsf{t}^{2}_{0}(\\alpha^{1})\n\t\t\t[\\textsf{t}^{2}_{0}(\\alpha), just=$DR$:!u\n\t\t\t\t[\\textsf{t}^{2}_{1}(\\alpha), close={:!u, !c}]]]\n\t\t[\\textsf{t}^{2}_{1}(\\alpha^{1}), close={:!c}]]\n\t[\\textsf{t}^{2}_{0}(\\alpha\\wedge\\neg\\alpha)\n\t\t[\\textsf{T}_{2}(\\alpha^{1})\n\t\t\t[\\textsf{t}^{2}_{1}(\\alpha)\n\t\t\t\t[\\textsf{T}_{2}(\\alpha), close={:!u, !c}]\n\t\t\t\t[\\textsf{F}_{2}(\\alpha), close={:!u, !c}]]]\n\t\t[\\textsf{t}^{2}_{0}(\\alpha^{1})\n\t\t\t[\\textsf{t}^{2}_{1}(\\alpha)\n\t\t\t\t[\\textsf{t}^{2}_{1}(\\alpha)]]]\n\t\t[\\textsf{t}^{2}_{1}(\\alpha^{1}), close={:!c}]]\n\t[\\textsf{t}^{2}_{1}(\\alpha\\wedge\\neg\\alpha)\n\t\t[\\textsf{T}_{2}(\\alpha^{1})\n\t\t\t[\\textsf{t}^{2}_{1}(\\alpha)\n\t\t\t\t[\\textsf{T}_{2}(\\alpha), close={:!u, !c}]\n\t\t\t\t[\\textsf{F}_{2}(\\alpha), close={:!u, !c}]]]\n\t\t[\\textsf{t}^{2}_{0}(\\alpha^{1})\n\t\t\t[\\textsf{t}^{2}_{1}(\\alpha)\n\t\t\t\t[\\textsf{t}^{2}_{1}(\\alpha)]]]\n\t\t[\\textsf{t}^{2}_{1}(\\alpha^{1}), close={:!c}]]]\n\\end{prooftree}\n\\end{center}\n\\caption{Testing whether $\\alpha, \\neg\\alpha, \\alpha^{1}\\vdash_{C_{2}}\\beta$}\n\\label{Second test in C2}\n\\end{figure}\n\n\\end{landscape}\n\n\n\n\nNow, still in $C_{1}$, we show that $\\alpha^{1}, \\beta^{1}\\vdash_{C_{1}}(\\alpha\\vee\\beta)^{1}$ or, what is equivalent, $\\vdash_{C_{1}}(\\alpha^{1}\\wedge\\beta^{1})\\rightarrow(\\alpha\\vee\\beta)^{1}$.\n\n\\begin{center}\n\\begin{prooftree}\n{\n}\n[\\textsf{F}_{1}((\\alpha^{1}\\wedge\\beta^{1})\\rightarrow(\\alpha\\vee\\beta)^{1})\n\t[\\textsf{T}_{1}(\\alpha^{1}\\wedge\\beta^{1}), just=$\\textsf{F}_{1}\\rightarrow$:!u\n\t\t[\\textsf{F}_{1}((\\alpha\\vee\\beta)^{1}), just=$\\textsf{F}_{1}\\rightarrow$:!uu\n\t\t\t[\\textsf{t}^{1}_{0}(\\alpha\\vee\\beta), just=$DR$:!u\n\t\t\t\t[\\textsf{T}_{1}(\\alpha), just=$DR$:!uuu\n\t\t\t\t\t[\\textsf{T}_{1}(\\beta), just=$DR$:!uuuu\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\alpha), just=$\\textsf{t}^{1}_{0}\\vee$:!uuu, close={:!uu, !c}]\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\beta), close={:!u, !c}]]]\n\t\t\t\t[\\textsf{T}_{1}(\\alpha)\n\t\t\t\t\t[\\textsf{F}_{1}(\\beta)\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\alpha), close={:!uu, !c}]\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\beta), close={:!u, !c}]]]\n\t\t\t\t[\\textsf{F}_{1}(\\alpha)\n\t\t\t\t\t[\\textsf{T}_{1}(\\beta)\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\alpha), close={:!uu, !c}]\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\beta), close={:!u, !c}]]]\n\t\t\t\t[\\textsf{F}_{1}(\\alpha)\n\t\t\t\t\t[\\textsf{F}_{1}(\\beta)\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\alpha), just=$\\textsf{t}^{1}_{0}\\vee$:!uuu, close={:!uu, !c}]\n\t\t\t\t\t\t[\\textsf{t}^{1}_{0}(\\beta), close={:!u, !c}]]]]]]\n\t[\\textsf{t}^{1}_{0}(\\alpha^{1}\\wedge\\beta^{1})\n\t\t[\\textsf{F}_{1}((\\alpha\\vee\\beta)^{1}), close={:!u}]]]\n\\end{prooftree}\n\\end{center}\n\nFinally, in Figure \\ref{Second test in C2}, we show a negative result, now in $C_{2}$: it is true that $\\alpha, \\neg\\alpha, \\alpha^{1}$\\\\$\\vdash_{C_{2}}\\beta$ or, equivalently, $\\vdash_{C_{2}}((\\alpha\\wedge\\neg\\alpha)\\wedge\\alpha^{1})\\rightarrow\\beta$? The answer, obviously, is no, yet for simplicity we prove something slightly different, but equivalent: since $\\beta$ can assume any value, for this deduction to be true we must have that $(\\alpha\\wedge \\neg\\alpha)\\wedge\\alpha^{1}$ always equals $F_{2}$, what we show not to be the case; that is, there are homomorphisms on $\\mathcal{F}_{C_{2}}$, given by the open branches of the tableau in Figure \\ref{Second test in C2}, where $(\\alpha\\wedge\\neg\\alpha)\\wedge\\alpha^{1}$ indeed takes the value $T_{2}$.\n\n\n\\subsubsection{$\\mbCcl$ and $\\CILA$}\n\nBefore showing finite RNmatrices capable of characterizing the logics $C_{n}$, we had done the same for the logics $\\mbCcl$ and $\\CILA$ in Sections, respectively, \\ref{RNmatrix for mbCcl} and \\ref{RNmatrix for CILA} and, in exactly the same way we have motivated the introduction of the calculi $\\mathbb{T}_{n}$, we may motivate both $\\mathbb{T}_{\\mbCcl}$ and $\\mathbb{T}_{\\CILA}$. Starting with $\\mbCcl$, of course now our formulas are over the signature $\\Sigma_{\\textbf{LFI}}$ and our labels are $\\textsf{T}$, $\\textsf{t}$ or $\\textsf{F}$, but the procedure is the same: we look at the tables of the RNmatrix for $\\mbCcl$, obtaining the following rules.\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c|c|c}\\multicolumn{4}{c}{\\textsf{T}(\\varphi\\vee\\psi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{t}(\\varphi) & \\textsf{T}(\\psi) & \\textsf{t}(\\psi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c|c|c}\\multicolumn{4}{c}{\\textsf{t}(\\varphi\\vee\\psi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{t}(\\varphi) & \\textsf{T}(\\psi) & \\textsf{t}(\\psi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{F}(\\varphi\\vee\\psi)\\\\\\hline \\textsf{F}(\\varphi)\\\\ \\textsf{F}(\\psi)\\end{array}\\\\\n&&&&\\\\\n\\begin{array}{c|c|c|c}\\multicolumn{4}{c}{\\textsf{T}(\\varphi\\wedge\\psi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{T}(\\varphi) & \\textsf{t}(\\varphi) & \\textsf{t}(\\varphi)\\\\ \\textsf{T}(\\psi) & \\textsf{t}(\\psi) & \\textsf{T}(\\psi) & \\textsf{t}(\\psi) \\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c|c|c}\\multicolumn{4}{c}{\\textsf{t}(\\varphi\\wedge\\psi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{T}(\\varphi) & \\textsf{t}(\\varphi) & \\textsf{t}(\\varphi)\\\\ \\textsf{T}(\\psi) & \\textsf{t}(\\psi) & \\textsf{T}(\\psi) & \\textsf{t}(\\psi) \\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}(\\varphi\\wedge\\psi)}\\\\\\hline \\textsf{F}(\\varphi)& \\textsf{F}(\\psi)\\end{array}\n\\end{array}$\n\\end{table}\n\n\n\n\n\n\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccccc}\n\\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{T}(\\varphi\\rightarrow\\psi)}\\\\\\hline \\textsf{F}(\\varphi) & \\textsf{T}(\\psi) & \\textsf{t}(\\psi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c|c}\\multicolumn{3}{c}{\\textsf{t}(\\varphi\\rightarrow\\psi)}\\\\\\hline \\textsf{F}(\\varphi) & \\textsf{T}(\\psi) & \\textsf{t}(\\psi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{F}(\\varphi\\rightarrow\\psi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{t}(\\varphi)\\\\ \\textsf{F}(\\psi) & \\textsf{F}(\\psi)\\end{array}\\\\\n&&&&\\\\\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}(\\neg\\varphi)}\\\\\\hline \\textsf{t}(\\varphi) & \\textsf{F}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{t}(\\neg\\varphi)}\\\\\\hline \\textsf{t}(\\varphi) & \\textsf{F}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{F}(\\neg\\varphi)\\\\\\hline \\textsf{T}(\\varphi)\\end{array}\\\\\n&&&&\\\\\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}(\\circ\\varphi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{F}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{t}(\\circ\\varphi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{F}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{F}(\\circ\\varphi)\\\\\\hline \\textsf{t}(\\varphi)\\end{array}\n\\end{array}$\n\\end{table}\n\nThe closure conditions are the adequate translations of the ones in $\\mathbb{T}_{1}$: a branch is closed if it contains $\\textsf{L}(\\varphi)$ and $\\textsf{L}^{*}(\\varphi)$ with $\\textsf{L}\\neq\\textsf{L}^{*}$ or if it contains $\\textsf{t}(\\psi\\wedge\\neg\\psi)$.\n\nFor $\\CILA$, the situation is a bit simpler: one takes the calculus $\\mathbb{T}_{1}$ now with formulas over the signature $\\Sigma_{\\textbf{LFI}}$; exchange the labels $\\textsf{T}_{1}$, $\\textsf{t}^{1}_{0}$ and $\\textsf{F}_{1}$ by, respectively, $\\textsf{T}$, $\\textsf{t}$ and $\\textsf{F}$; and adds the following rules governing the connective $\\circ$.\n\n\\begin{table}[H]\n\\centering\n$\\begin{array}{ccc}\n\\begin{array}{c|c}\\multicolumn{2}{c}{\\textsf{T}(\\circ\\varphi)}\\\\\\hline \\textsf{T}(\\varphi) & \\textsf{F}(\\varphi)\\end{array} & \\quad\\quad\\quad & \\begin{array}{c}\\textsf{F}(\\circ\\varphi)\\\\\\hline \\textsf{t}(\\varphi)\\end{array}\n\\end{array}$\n\\end{table}\n\nThe closure conditions are the same as those of $\\mathbb{T}_{\\mbCcl}$, with one addition: a branch is also closed if it contains a labeled formula $\\textsf{t}(\\circ\\psi)$. We will not prove it, but $\\mathbb{T}_{\\mbCcl}$ and $\\mathbb{T}_{\\CILA}$ are decision methods for their respective logics.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{5}\n\\chapter{Restricted swap structures for da Costa's hierarchy}\\label{Chapter6}\\label{Chapter 6}\n\n\n\nThe RNmatrices we defined for $C_{n}$ are swap structures: roughly speaking, they are semantics built by assigning, simultaneously, truth values to a formula $\\alpha$ and several other, related, formulas; in our case, $\\neg\\alpha$, $\\alpha^{1}, \\dotsc , \\alpha^{n-1}$. Most frequently, these truth values are in the two-valued Boolean algebra $\\textbf{2}$, but it is possible, in many cases, to instead consider these values in an arbitrary Boolean algebra. One example would be of the $\\textbf{LFI}$ known as $\\textbf{mbC}$, whose swap structures may be recast over any non-trivial Boolean algebra (\\cite{ParLog}); this is made easier by the fact that, in $\\textbf{mbC}$'s case, we have only a single, finite Nmatrix. The end result of this process is a class of Nmatrices parameterized by Boolean algebras, which we call a swap structures semantics. This is done in order to produce a wider class of models, what allows one to study a given logic by adapting tools from algebraic logic and model theory to the context of non-deterministic algebras.\n\nHowever, swap structures have been presented, up to now, only as Nmatrices; given that the logics between $\\textbf{mbCcl}$ and $\\textbf{Cila}$ are not characterizable by finite Nmatrices, and that swap structures were coined to be decision methods (and henceforth finite), it is easy to see that there haven't been defined swap structures semantics for these logics so far. In Chapter \\ref{Chapter5} we changed this by finally defining, not classical swap structures, but their versions as RNmatrices over $\\textbf{2}$ for $C_{1}$ and the whole of da Costa's hierarchy; we continue by generalizing, in much the same way swap structures semantics generalize for arbitrary Boolean algebras the finite Nmatrix characterization of $\\textbf{mbC}$, the RNmatrices $\\mathcal{RM}_{C_{n}}$ to structures defined over any given, non-trivial, Boolean algebra $\\mathcal{B}$. As mentioned, this can have important applications in the model theory of $C_{n}$, being the class of RNmatrices obtained a very well-behaved category; but, more importantly, the methodology here outlined offers a new outlook of swap structures altogether.\n\nThe first step needed to make this generalization possible is to consider, instead of bivaluations, the concept of $\\mathcal{B}$-valuations; the rest of the process is actually rather straightforward, as we have already defined objects such as $B_{n}$, $D_{n}$, $\\mathcal{A}_{C_{n}}$ and $\\mathcal{F}_{C_{n}}$ in terms that facilitate exchanging $\\textbf{2}$ by an arbitrary $\\mathcal{B}$: that is, they were only defined in terms of the Boolean operators and the constants $0$ and $1$, present in any Boolean algebra. The only restriction that we make is to non-trivial, also known as non-degenerate, Boolean algebras, where $0\\neq 1$.\n\nThe research found in this chapter has been submitted as a preprint in \\cite{RestrictedSwap}.\n\n\\section{$\\mathcal{B}$-Valuations generalize bivaluations}\\label{B-valuations}\n\n\\begin{definition}\\label{B-valuation}\nGiven a Boolean algebra $\\mathcal{B}$ with universe $B$, a $\\mathcal{B}$-valuation\\index{Valuation, $\\mathcal{B}$-} for $C_{n}$ is a function $\\mathsf{b}:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow B$ satisfying:\n\\begin{enumerate}\n\\item[$(V1)$] $\\mathsf{b}(\\alpha\\#\\beta)=\\mathsf{b}(\\alpha)\\#\\mathsf{b}(\\beta)$, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$;\n\\item[$(V2)$] ${\\sim}\\mathsf{b}(\\alpha)\\leq\\mathsf{b}(\\neg\\alpha)$;\n\\item[$(V3)$] $\\mathsf{b}(\\neg\\neg\\alpha)\\leq\\mathsf{b}(\\alpha)$;\n\\item[$(V4)_{n}$] $\\mathsf{b}(\\alpha^{n})={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1})))$;\n\\item[$(V5)$] $\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)$;\n\\item[$(V6)_{n}$] $\\mathsf{b}(\\alpha^{(n)})\\wedge\\mathsf{b}(\\beta^{(n)})\\leq\\mathsf{b}((\\alpha\\#\\beta)^{(n)})$, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\\label{Some properties of B-valuations}\nLet $\\mathcal{B}$ be a Boolean algebra and $\\mathsf{b}$ a $\\mathcal{B}$-valuation for $C_{n}$: let us denote, for a formula $\\alpha$, the $n+1$-tuple $(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1}))$ in $B^{n+1}$ by $z=(z_{1}, \\dotsc  , z_{n+1})$.\n\\begin{enumerate}\n\\item For any $1\\leq k\\leq n-1$, \n\\[\\mathsf{b}(\\neg(\\alpha^{k}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i}),\\]\nmeaning at most one among $z_{1}, \\dotsc  , z_{n}$ and $z_{n+1}$ equals $0$; furthermore, if $z_{k}=0$, then $z_{i}=1$ for every $k+1\\leq i\\leq n+1$.\n\n\\item For $1\\leq k\\leq n-1$, $\\mathsf{b}(\\alpha^{(k)})=\\bigwedge_{i=1}^{k}\\mathsf{b}(\\alpha^{i})$ and\n\\[\\mathsf{b}(\\alpha^{n})={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})).\\]\n\n\\item One has $\\mathsf{b}(\\alpha^{(n)})={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})$.\n\n\\item We have $\\mathsf{b}(\\alpha)\\vee\\mathsf{b}(\\neg\\alpha)=1$, $(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\vee\\mathsf{b}(\\alpha^{1})=1$ and, for every $1<k\\leq n-1$,\n\\[(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i}))\\vee\\mathsf{b}(\\alpha^{k})=1.\\]\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\begin{proof}\n\n\n\\begin{enumerate}\n\\item A simple argument by induction is sufficient: the case $k=1$ is done, from clause $(V5)$ of Definition \\ref{B-valuation}; so, suppose for induction hypothesis that $\\mathsf{b}(\\alpha^{k-1})=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-2}\\mathsf{b}(\\alpha^{i})$, and we have that, again from clause $(V5)$,\n\\[\\mathsf{b}(\\neg(\\alpha^{k}))=\\mathsf{b}(\\neg((\\alpha^{k-1})^{1}))=\\mathsf{b}(\\alpha^{k-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{k-1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i}).\\]\nIf $\\mathsf{b}(\\alpha)=0$, from $(V2)$ we must have $\\mathsf{b}(\\neg\\alpha)=1$; and from the previous formula for $\\mathsf{b}(\\neg(\\alpha^{k}))$, we find $\\mathsf{b}(\\neg(\\alpha^{1}))=\\cdots=\\mathsf{b}(\\neg(\\alpha^{n-1}))=0$, meaning, again from $(V2)$, that $\\mathsf{b}(\\alpha^{1})=\\cdots=\\mathsf{b}(\\alpha^{n-1})=1$. If $\\mathsf{b}(\\neg\\alpha)=0$, by the same argument one obtains $\\mathsf{b}(\\neg(\\alpha^{1}))=\\cdots=\\mathsf{b}(\\neg(\\alpha^{n-1}))=0$ and the desired result.\n\nFinally, if $\\mathsf{b}(\\alpha^{k})=0$, for a $k+1\\leq j\\leq n-1$, one obtains $\\mathsf{b}(\\neg(\\alpha^{j}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{j-1}\\mathsf{b}(\\alpha^{i})=0$, and therefore $\\mathsf{b}(\\alpha^{j})=1$.\n\n\\item For the first part, we proceed again by induction: for $k=1$, the result is trivial given the definition $\\alpha^{(1)}=\\alpha^{1}$; assuming $\\mathsf{b}(\\alpha^{(k-1)})=\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i})$, we get \n\\[\\mathsf{b}(\\alpha^{(k)})=\\mathsf{b}(\\alpha^{k}\\wedge\\alpha^{(k-1)})=\\mathsf{b}(\\alpha^{k})\\wedge\\mathsf{b}(\\alpha^{(k-1}))=\\bigwedge_{i=1}^{k}\\mathsf{b}(\\alpha^{i}).\\]\n\nNow, remembering $(V4)_{n}$ and the previous item of the proposition,, \n\\[\\mathsf{b}(\\alpha^{n})={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1})))={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})).\\]\n\n\\item From the previous item,\n\\[\\mathsf{b}(\\alpha^{(n)})=\\mathsf{b}(\\alpha^{n}\\wedge\\alpha^{(n-1)})=\\mathsf{b}(\\alpha^{n})\\wedge\\mathsf{b}(\\alpha^{(n-1)})={\\sim}\\big[\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})\\big]\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})=\\]\n\\[\\big[{\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\vee{\\sim}\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})\\big]\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})=\\]\n\\[\\big[{\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})\\big]\\vee\\big[{\\sim}\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})\\big]=\\]\n\\[({\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i}))\\vee 0={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i}).\\]\n\n\n\n\n\\item From $(V2)$, ${\\sim}\\mathsf{b}(\\alpha)\\leq\\mathsf{b}(\\neg\\alpha)$, and since $\\mathsf{b}(\\alpha)\\vee{\\sim}\\mathsf{b}(\\alpha)=1$ we find $\\mathsf{b}(\\alpha)\\vee\\mathsf{b}(\\neg\\alpha)=1$; \nfrom the first item of this proposition, $\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)$, and since $\\mathsf{b}(\\alpha)\\vee\\mathsf{b}(\\neg\\alpha)=1$ works for any formula $\\alpha$, by replacing in the latter equation $\\alpha$ with $\\alpha^{1}$ one finds \n\\[\\mathsf{b}(\\alpha^{1})\\vee(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))=\\mathsf{b}(\\alpha^{1})\\vee\\mathsf{b}(\\neg(\\alpha^{1}))=1;\\]\nfinally, again from the first item of the proposition, for every $1<k\\leq n-1$, $\\mathsf{b}(\\neg(\\alpha^{k}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i})$, and therefore\n\\[\\mathsf{b}(\\alpha^{k})\\vee(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i}))=\\mathsf{b}(\\alpha^{k})\\vee\\mathsf{b}(\\neg(\\alpha^{k}))=1.\\]\n\\end{enumerate}\n\\end{proof}\n\n\\begin{proposition}\\label{Negations of B-valuations}\nGiven a Boolean algebra $\\mathcal{B}$, let $\\mathsf{b}$ be a $\\mathcal{B}$-valuation for $C_{n}$.\n\\begin{enumerate}\n\\item For any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, if $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$ and $\\mathsf{b}(\\neg\\beta)={\\sim}\\mathsf{b}(\\beta)$, then $\\mathsf{b}(\\neg(\\alpha\\#\\beta))={\\sim}\\mathsf{b}(\\alpha\\#\\beta)$;\n\\item if $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$, then $\\mathsf{b}(\\neg\\neg\\alpha)={\\sim}\\mathsf{b}(\\neg\\alpha)$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Assume $\\mathsf{b}$ is a $\\mathcal{B}$-valuation for $C_{n}$ for which $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$ and $\\mathsf{b}(\\neg\\beta)={\\sim}\\mathsf{b}(\\beta)$. Then $\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)=\\mathsf{b}(\\beta)\\wedge\\mathsf{b}(\\neg\\beta)=0$ and therefore ${\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))={\\sim}(\\mathsf{b}(\\beta)\\wedge\\mathsf{b}(\\neg\\beta))=1$.\n\nSince, from Proposition \\ref{Some properties of B-valuations}, $\\mathsf{b}(\\neg(\\alpha^{k}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i})$ for every $1\\leq k\\leq n-1$, we find $\\mathsf{b}(\\neg(\\alpha^{k}))=\\mathsf{b}(\\neg(\\beta^{k}))=0$ for every $1\\leq k\\leq n-1$, implying from $(V2)$ (in Definition \\ref{B-valuation}) that $\\mathsf{b}(\\alpha^{k})=\\mathsf{b}(\\beta^{k})=1$ for all $1\\leq k\\leq n-1$. Again from Proposition \\ref{Some properties of B-valuations}, $\\mathsf{b}(\\alpha^{(n)})={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})$, and so $\\mathsf{b}(\\alpha^{(n)})=\\mathsf{b}(\\beta^{(n)})=1$, meaning from $(V6)_{n}$ that $\\mathsf{b}((\\alpha\\#\\beta)^{(n)})=1$.\n\nBut, again from the fact that, for any formula $\\alpha$, $\\mathsf{b}(\\alpha^{(n)})={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})$, we obtain that $\\mathsf{b}((\\alpha\\#\\beta)^{(n)})\\leq{\\sim}[\\mathsf{b}(\\alpha\\#\\beta)\\wedge\\mathsf{b}(\\neg(\\alpha\\#\\beta))]$, and thus $\\mathsf{b}(\\alpha\\#\\beta)\\wedge\\mathsf{b}(\\neg(\\alpha\\#\\beta))=0$, which means $\\mathsf{b}(\\neg(\\alpha\\#\\beta))={\\sim}\\mathsf{b}(\\alpha\\#\\beta)$ from $(V2)$.\n\n\\item Now suppose $\\mathsf{b}$ is a $\\mathcal{B}$-valuation for $C_{n}$ for which $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$: by $(V3)$, \n\\[\\mathsf{b}(\\neg\\neg\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\leq\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)=0,\\]\nand since $\\mathsf{b}(\\neg\\neg\\alpha)\\vee\\mathsf{b}(\\neg\\alpha)=1$ from $(V2)$, we find $\\mathsf{b}(\\neg\\neg\\alpha)={\\sim}\\mathsf{b}(\\neg\\alpha)$.\n\n\\end{enumerate}\n\\end{proof}\n\nOne notices that, as a corollary to this last proposition, whenever $\\mathcal{V}_{0}$ is a non-empty subset of $\\mathcal{V}$ and $\\mathsf{b}$ is a $\\mathcal{B}$-valuation for $C_{n}$ satisfying that $\\mathsf{b}(\\neg p)={\\sim}\\mathsf{b}(p)$, for every $p\\in\\mathcal{V}_{0}$, one has $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$ for every formula $\\alpha$ in $F(\\Sigma_{\\textbf{C}}, \\mathcal{V}_{0})$.\n\nA simple argument by structural induction is sufficient, being the base case one of the hypothesis: if the result holds for $\\alpha$ and $\\beta$, from Proposition \\ref{Negations of B-valuations} $\\mathsf{b}(\\neg(\\alpha\\#\\beta))={\\sim}\\mathsf{b}(\\alpha\\#\\beta)$, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$; and, again by the proposition, $\\mathsf{b}(\\neg\\neg\\alpha)={\\sim}\\mathsf{b}(\\neg\\alpha)$. Since $\\{\\alpha\\in F(\\Sigma_{\\textbf{C}}, \\mathcal{V}_{0})\\ :\\  \\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)\\}$ contains $\\mathcal{V}_{0}$ and is closed under the connectives in $\\{\\neg, \\vee, \\wedge, \\rightarrow\\}$ (all of $\\Sigma_{\\textbf{C}}$), we obtain this set is the whole of $F(\\Sigma_{\\textbf{C}}, \\mathcal{V}_{0})$.\n\n Additionally, under those hypothesis, we have that for every $\\alpha\\in F(\\Sigma_{\\textbf{C}}, \\mathcal{V}_{0})$ we have $\\mathsf{b}(\\alpha^{(n)})=1$: we can see that from the equality $\\mathsf{b}(\\alpha^{(n)})={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))\\wedge\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i})$. Since $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$, $\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)=0$ and therefore ${\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha))=1$; furthermore, for any $1\\leq k\\leq n-1$, $\\mathsf{b}(\\neg(\\alpha^{k}))={\\sim}\\mathsf{b}(\\alpha^{k})$ and $\\mathsf{b}(\\neg(\\alpha^{k}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i})$, thus we find\n \\[\\mathsf{b}(\\alpha^{k})={\\sim}(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge\\bigwedge_{i=1}^{k-1}\\mathsf{b}(\\alpha^{i})),\\]\n and in our current position this means $\\mathsf{b}(\\alpha^{k})=1$ (for all $1\\leq k\\leq n-1$), what gives the desired result $\\mathsf{b}(\\alpha^{(n)})=1$.\n \n \n \\begin{lemma}\\label{B-valuations on n-consistency}\nGiven a bivaluation $\\mathsf{b}$ for $C_{n}$, $\\mathsf{b}(\\alpha^{(n)})=1$ if and only if $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$.\n \\end{lemma}\n \n \\begin{proof}\n We first prove that, if $\\mathsf{b}(\\alpha^{(n)})=1$, then $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$. Start by noticing that if $\\mathsf{b}$ is a bivaluation for $C_{n}$, then $\\mathsf{b}(\\alpha^{(k)})=\\bigwedge_{i=1}^{k}\\mathsf{b}(\\alpha^{i})$ for $k\\geq 1$: this is obviously true for $k=1$, given $\\alpha^{(1)}=\\alpha^{1}$; supposing this is true for $k-1$, we obtain that\n \\[\\mathsf{b}(\\alpha^{(k)})=\\mathsf{b}(\\alpha^{(k-1)}\\wedge\\alpha^{k})=\\mathsf{b}(\\alpha^{(k-1)})\\wedge\\mathsf{b}(\\alpha^{k})=\\bigwedge_{i=1}^{k}\\mathsf{b}(\\alpha^{i}).\\]\nSo, since $\\mathsf{b}(\\alpha^{(n)})=\\bigwedge_{i=1}^{n}\\mathsf{b}(\\alpha^{i})=1$, $\\mathsf{b}(\\alpha^{1})=\\cdots=\\mathsf{b}(\\alpha^{n})=1$; from $(B6)_{n}$ of Definition \\ref{B-valuation}, $\\mathsf{b}(\\alpha^{n})=1$ implies $\\mathsf{b}(\\neg(\\alpha^{n-1}))=0$.\n\nNow, from $(B7)$, since $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}((\\alpha^{n-2})^{1})=0$ one finds out that $\\mathsf{b}(\\neg(\\alpha^{n-2}))=0$; inductively, we obtain $\\mathsf{b}(\\neg(\\alpha^{1}))=\\cdots=\\mathsf{b}(\\neg(\\alpha^{n-1}))=0$, and with one final application of $(B7)$, $\\mathsf{b}(\\neg(\\alpha^{1}))=0$ implies $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$.\n\nReciprocally, $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ implies $\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)=0$ and so $\\mathsf{b}(\\alpha^{1})=1$, while by $(B7)$ we obtain that $\\mathsf{b}(\\neg(\\alpha^{1}))=0$; this, of course, implies that $\\mathsf{b}(\\alpha^{2})=\\cdots=\\mathsf{b}(\\alpha^{n})=1$, and therefore $\\mathsf{b}(\\alpha^{(n)})=\\bigwedge_{i=1}^{n}\\mathsf{b}(\\alpha^{i})=1$.\n\\end{proof}\n\n\\begin{theorem}\nGiven the Boolean algebra of two elements $\\textbf{2}$, a function $\\mathsf{b}:F(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\{0,1\\}$ is a bivaluation for $C_{n}$ if, and only if, it is a $\\textbf{2}$-valuation for $C_{n}$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose, first of all, that $\\mathsf{b}$ is a bivaluation. Condition $(V1)$ follows from $(B1)$, $(B2)$ and $(B3)$. If $\\mathsf{b}(\\alpha)=0$, from $(B4)$ we find $\\mathsf{b}(\\neg\\alpha)=1$ and therefore ${\\sim}\\mathsf{b}(\\alpha)=1\\leq 1=\\mathsf{b}(\\neg\\alpha)$; if $\\mathsf{b}(\\alpha)=1$, regardless of the value of $\\mathsf{b}(\\neg\\alpha)$ we obtain ${\\sim}\\mathsf{b}(\\alpha)=0\\leq \\mathsf{b}(\\neg\\alpha)$, which means that, in all cases, ${\\sim}\\mathsf{b}(\\alpha)\\leq\\mathsf{b}(\\neg\\alpha)$, corresponding to clause $(V2)$.\n\nIf $\\mathsf{b}(\\neg\\neg\\alpha)=1$, condition $(B5)$ implies that $\\mathsf{b}(\\alpha)=1$, and therefore $\\mathsf{b}(\\neg\\neg\\alpha)\\leq\\mathsf{b}(\\alpha)$; if $\\mathsf{b}(\\neg\\neg\\alpha)=0$, $\\mathsf{b}(\\neg\\neg\\alpha)\\leq\\mathsf{b}(\\alpha)$ is true regardless of the value of $\\mathsf{b}(\\alpha)$, meaning condition $(V3)$ always holds. Now,  from condition $(B4)$ we can not have $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}(\\neg(\\alpha^{n-1}))=0$, so there remain three cases to consider: if $\\mathsf{b}(\\alpha^{n-1})=0$ and $\\mathsf{b}(\\neg(\\alpha^{n-1}))=1$, or vice-versa, from $(B6)_{n}$ we obtain $\\mathsf{b}(\\alpha^{n})=1$ and therefore \n\\[\\mathsf{b}(\\alpha^{n})=1={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1})));\\]\nif $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}(\\neg(\\alpha^{n-1}))=1$, again from $(B6)_{n}$ we obtain $\\mathsf{b}(\\alpha^{n})=0$, and so\n\\[\\mathsf{b}(\\alpha^{n})=0={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1}))),\\]\nproving that $(V4)_{n}$ is valid. As mentioned earlier, we can not have $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=0$, according to $(B4)$, and so if $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=1$, from $(B7)$ we get $\\mathsf{b}(\\neg(\\alpha^{1}))=1$ and therefore $\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)$; if $\\mathsf{b}(\\alpha)=0$ or $\\mathsf{b}(\\neg\\alpha)=0$, again from $(B7)$ we find $\\mathsf{b}(\\neg(\\alpha^{1}))=0$, meaning once more that $\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)$ and that $(V5)$ holds. \n\nFinally, notice that if $\\mathsf{b}(\\alpha^{(n)})=0$ or $\\mathsf{b}(\\beta^{(n)})=0$, $(V6)_{n}$ already holds true, so let us assume that $\\mathsf{b}(\\alpha^{(n)})=1$ and $\\mathsf{b}(\\beta^{(n)})=1$: this means, by Lemma \\ref{B-valuations on n-consistency}, that $\\mathsf{b}(\\alpha)\\neq \\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta)\\neq\\mathsf{b}(\\neg\\beta)$; from $(B8)$, this translates to $\\mathsf{b}(\\alpha\\#\\beta)\\neq\\mathsf{b}(\\neg(\\alpha\\#\\beta))$ for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, and again by Lemma \\ref{B-valuations on n-consistency} this becomes $\\mathsf{b}((\\alpha\\#\\beta)^{(n)})=1$, which verifies $(V6)_{n}$ and finishes proving $\\mathsf{b}$ is a $\\textbf{2}$-valuation.\n\nReciprocally, suppose $\\mathsf{b}$ is a $\\textbf{2}$-valuation. $(B1)$, $(B2)$ and $(B3)$ clearly follow from $(V1)$. If $\\mathsf{b}(\\alpha)=0$, from $(V2)$ we obtain $\\mathsf{b}(\\neg\\alpha)\\geq{\\sim}\\mathsf{b}(\\alpha)=1$, and therefore $\\mathsf{b}(\\neg\\alpha)=1$, thus satisfying $(B4)$. If $\\mathsf{b}(\\neg\\neg\\alpha)=1$, $\\mathsf{b}(\\alpha)\\geq\\mathsf{b}(\\neg\\neg\\alpha)=1$ from $(V3)$, and so $\\mathsf{b}(\\alpha)=1$, meaning $(B5)$ is also validated.\n\nNow, if $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}(\\neg(\\alpha^{n-1}))$, since both can not be $0$ given $(V2)$, both equal $1$ and from $(V4)_{n}$ we get $\\mathsf{b}(\\alpha^{n})={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1})))=0$, and one direction of $(B6)_{n}$ holds. Reciprocally, if $\\mathsf{b}(\\alpha^{n})=0$, given $\\mathsf{b}(\\alpha^{n})={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1})))$ from $(V4)_{n}$ we get $\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1}))=1$, and therefore $\\mathsf{b}(\\alpha^{n-1})=\\mathsf{b}(\\neg(\\alpha^{n-1}))=1$, meaning that both directions of $(B6)_{n}$ hold.\n\nTo analyze $(B7)$, there are again two directions to consider: if $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)$, this means that both are equal to $1$ (given $(V2)$), and so $\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)=1$, in accordance to $(V5)$; if $\\mathsf{b}(\\neg(\\alpha^{1}))=1$, again from $(V5)$ we obtain $1=\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)$, meaning $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\neg\\alpha)=1$ and that $(B7)$ holds true.\n\nFinally, suppose $\\mathsf{b}(\\alpha)\\neq\\mathsf{b}(\\neg\\alpha)$ and $\\mathsf{b}(\\beta)\\neq\\neg\\mathsf{b}(\\beta)$, meaning that $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$ and $\\mathsf{b}(\\neg\\beta)={\\sim}\\mathsf{b}(\\beta)$: from Proposition \\ref{Negations of B-valuations}, $\\mathsf{b}(\\neg(\\alpha\\#\\beta))={\\sim}\\mathsf{b}(\\alpha\\#\\beta)$, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, and so $\\mathsf{b}(\\neg(\\alpha\\#\\beta))\\neq \\mathsf{b}(\\alpha\\#\\beta)$. This implies $(B8)$ is true, and that $\\mathsf{b}$ is then a bivaluation.\n\\end{proof}\n\nFix a Boolean algebra $\\mathcal{B}$ and take formulas $\\Gamma\\cup\\{\\varphi\\}\\subseteq F(\\Sigma_{\\textbf{C}}, \\mathcal{V})$: we say $\\Gamma$ proves $\\varphi$ according to $\\mathcal{B}$-valuations for $C_{n}$, and write $\\Gamma\\vDash_{C_{n}}^{\\mathcal{B}}\\varphi$\\label{vDashBCn}, if every $\\mathcal{B}$-valuation $\\mathsf{b}$ satisfying $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$ also has the property that $\\mathsf{b}(\\varphi)=1$.\n\n\n\\begin{theorem}\nLet $n\\in\\mathbb{N}\\setminus\\{0\\}$ and $\\Gamma\\cup\\{\\varphi\\}$ be a set of formulas in the signature $\\Sigma_{\\textbf{C}}$: then $\\Gamma\\vdash_{C_{n}}\\varphi$ if and only if $\\Gamma\\vDash_{C_{n}}^{\\mathcal{B}}\\varphi$ for every Boolean algebra $\\mathcal{B}$.\n\\end{theorem}\n\n\\begin{proof}\nFirst assume that $\\Gamma\\vdash_{C_{n}}\\varphi$, and let us fix a Boolean algebra $\\mathcal{B}$. It is a straightforward exercise to prove that, if $\\varphi$ is an instance of an axiom of $C_{n}$, then $\\mathsf{b}(\\varphi)=1$ for any $\\mathcal{B}$-valuation $\\mathsf{b}$ for $C_{n}$; now, again for a $\\mathcal{B}$-valuation $\\mathsf{b}$ for $C_{n}$, if $\\mathsf{b}(\\alpha)=\\mathsf{b}(\\alpha\\rightarrow\\beta)=1$, since $\\mathsf{b}(\\alpha\\rightarrow\\beta)=\\mathsf{b}(\\alpha)\\rightarrow\\mathsf{b}(\\beta)$ from $(V1)$, one derives that $\\mathsf{b}(\\beta)=1$, meaning ``$\\vDash_{C_{n}}^{\\mathcal{B}}$'' models all axiom schemata and rules of inference of $C_{n}$.\n\nSo, if $\\varphi_{1}, \\dotsc  , \\varphi_{n}$, with $\\varphi_{n}=\\varphi$, is a derivation of $\\varphi$ from $\\Gamma$, we prove by induction that, if $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$, then $\\mathsf{b}(\\varphi_{1})=\\cdots=\\mathsf{b}(\\varphi_{n})=1$, and therefore if $\\mathsf{b}$ is a $\\mathcal{B}$-valuation for $C_{n}$, $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$ implies $\\mathsf{b}(\\varphi)=1$, meaning $\\Gamma\\vDash_{C_{n}}^{\\mathcal{B}}\\varphi$. This is rather easy: $\\varphi_{1}$ is either in $\\Gamma$, where the fact that $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$ implies $\\mathsf{b}(\\varphi_{1})=1$, or is an instance of an axiom schema, meaning from the previous observations that $\\mathsf{b}(\\varphi_{1})=1$. So, assume that $\\mathsf{b}(\\varphi_{1})=\\cdots=\\mathsf{b}(\\varphi_{i-1})=1$, and there are three cases to consider: if $\\varphi_{i}$ is in $\\Gamma$ or is an instance of an axiom schema, proceed as before; if there exist $\\varphi_{j}$ and $\\varphi_{k}$, with $1\\leq j, k<i$, such that $\\varphi_{j}=\\varphi_{k}\\rightarrow\\varphi_{i}$ or $\\varphi_{k}=\\varphi_{j}\\rightarrow\\varphi_{i}$, since $\\mathsf{b}(\\varphi_{j})=\\mathsf{b}(\\varphi_{k})=1$ we find $\\mathsf{b}(\\varphi_{i})=1$, and we are done.\n\nReciprocally, assume $\\Gamma\\vDash_{C_{n}}^{\\mathcal{B}}\\varphi$ for every Boolean algebra $\\mathcal{B}$; in particular, $\\Gamma\\vDash_{C_{n}}^{\\textbf{2}}\\varphi$, and since bivaluations characterize $C_{n}$, we derive that $\\Gamma\\vdash_{C_{n}}\\varphi$.\n\\end{proof}\n\n\n\n\n\n\\section{The RNmatrices $\\mathcal{RM}^{\\mathcal{B}}_{C_{n}}$}\\label{RNmatrices RMBCn}\n\nNow that we have the appropriate generalization of a bivaluation to arbitrary Boolean algebras, our goal is to generalize the RNmatrix $\\mathcal{RM}_{C_{n}}=(\\mathcal{A}_{C_{n}}, D_{n}. \\mathcal{F}_{C_{n}})$ to a new restricted Nmatrix built around $\\mathcal{B}$ by using the notion of snapshots in a broader context, in much the same way it would be done while dealing with swap structures. This will be achieved by generalizing both $\\mathcal{A}_{C_{n}}$ and $D_{n}$ to, respectively, the $\\Sigma_{\\textbf{C}}$-multialgebra $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ and the subset of its universe $D_{n}^{\\mathcal{B}}$; this, of course, takes the Nmatrix $\\mathcal{M}_{C_{n}}=(\\mathcal{A}_{C_{n}}, D_{n})$ to the, still, Nmatrix $\\mathcal{M}_{C_{n}}^{\\mathcal{B}}=(\\mathcal{A}_{C_{n}}^{\\mathcal{B}}, D_{n}^{\\mathcal{B}})$. The decisive step, however, is generalizing the set of restricted valuations from $\\mathcal{F}_{C_{n}}$ to $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$, what will allow to trade the RNmatrix $\\mathcal{RM}_{C_{n}}=(\\mathcal{A}_{C_{n}}, D_{n}, \\mathcal{F}_{C_{n}})$ for $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}=(\\mathcal{A}_{C_{n}}^{\\mathcal{B}}, D_{n}^{\\mathcal{B}}, \\mathcal{F}_{C_{n}}^{\\mathcal{B}})$.\n\nFor a Boolean algebra $\\mathcal{B}$, consider a $\\mathcal{B}$-valuation $\\mathsf{b}$ for $C_{n}$ and the $(n+1)$-tuple $z=(z_{1}, \\dotsc  , z_{n+1})$ in $\\mathcal{B}^{n+1}$ such that $z_{1}=\\mathsf{b}(\\alpha)$, $z_{2}=\\mathsf{b}(\\neg\\alpha)$, $z_{3}=\\mathsf{b}(\\alpha^{1}), \\dotsc  , z_{n+1}=\\mathsf{b}(\\alpha^{n-1})$, for a formula $\\alpha$ of $C_{n}$; these will be the aforementioned generalized snapshots over which we shall work. We can be sure that the following definition works by referring back to Proposition \\ref{Some properties of B-valuations}.\n\n\\begin{definition}\nGiven a Boolean algebra $\\mathcal{B}$, the set of $\\mathcal{B}$-snapshots for $C_{n}$ is\\label{BnB}\n\\[B_{n}^{\\mathcal{B}}=\\{z\\in B^{n+1}\\ :\\  (\\bigwedge_{i=1}^{k}z_{i})\\vee z_{k+1}=1,\\quad \\text{for every} \\quad 1\\leq k\\leq n\\}.\\]\n\\end{definition}\n\nOther sets that will be useful to us are:\n\\begin{enumerate}\n\\item $D_{n}^{\\mathcal{B}}=\\{z\\in B_{n}^{\\mathcal{B}}\\ :\\  z_{1}=1\\}$, the set of designated values;\\label{DnB}\n\\item $Boo_{n}^{\\mathcal{B}}=\\{z\\in B_{n}^{\\mathcal{B}}\\ :\\  z_{1}\\wedge z_{2}=0\\}$, the set of Boolean values;\\label{BoonB}\n\\item $U_{n}^{\\mathcal{B}}=B_{n}^{\\mathcal{B}}\\setminus D_{n}^{\\mathcal{B}}$, the set of undesignated values.\n\\end{enumerate}\n\nAlthough somewhat obvious, it is still relevant to point out the set of designated elements correspond to formulas which are true under a $\\mathcal{B}$-valuation: that is, if $z=(z_{1}, \\dotsc  , z_{n+1})$ satisfies $z_{1}=1$, and $z=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1}))$, then $\\mathsf{b}(\\alpha)=1$, that is, $\\alpha$ is true according to $\\mathsf{b}$. For simplicity, and to maintain the notation we had already adopted, we will denote the $i$-th coordinate of $z\\in B^{n+1}$ by $z_{i}$.\n\nNotice that, from the definition of $B_{n}^{\\mathcal{B}}$, for any snapshot $z$ one has $z_{1}\\vee z_{2}=1$; so, if $z\\in Boo_{n}^{\\mathcal{B}}$, $z_{1}\\wedge z_{2}=0$ and therefore $z_{2}={\\sim}z_{1}$; then, from the fact that\n\\[1=(\\bigwedge_{i=1}^{k}z_{i})\\vee z_{k+1}=(z_{1}\\wedge z_{2}\\wedge\\bigwedge_{i=3}^{k}z_{i})\\vee z_{k+1}=(0\\wedge\\bigwedge_{i=3}^{k}z_{i})\\vee z_{k+1}=0\\vee z_{k+1}=z_{k+1},\\]\nfor every $3\\leq k\\leq n$, we discover that $z\\in Boo_{n}^{\\mathcal{B}}$ if, and only if, there exists $a$ in $\\mathcal{B}$ such that $z=(a, {\\sim}a, 1, \\dotsc  , 1)$, what establishes a bijection between $\\mathcal{B}$ and $Boo_{n}^{\\mathcal{B}}$.\n\n\\begin{definition}\nGiven a Boolean algebra $\\mathcal{B}$, the full swap structure for $C_{n}$ over $\\mathcal{B}$ is the $\\Sigma_{\\textbf{C}}$-multialgebra $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$\\label{ACnB}, with universe $B_{n}^{\\mathcal{B}}$, with operations defined as, for any two $z, w\\in B_{n}^{\\mathcal{B}}$,\n\\[\\tilde{\\neg}z=\\{w\\in B_{n}^{\\mathcal{B}}\\ :\\  w_{1}=z_{2}\\quad\\text{and}\\quad w_{2}\\leq z_{1}\\}\\]\nand\n\\[\\text{for}\\quad \\#\\in\\{\\vee, \\wedge, \\rightarrow\\},\\quad z\\tilde{\\#}w=\\begin{cases*}\n\\{u\\in B_{n}^{\\mathcal{B}}\\ :\\  u_{1}=z_{1}\\# w_{1}\\}\\cap Boo_{n}^{\\mathcal{B}} & if $z, w\\in Boo_{n}^{\\mathcal{B}}$\\\\\n\\{u\\in B_{n}^{\\mathcal{B}}\\ :\\  u_{1}=z_{1}\\# w_{1}\\} & otherwise\n\\end{cases*}.\\]\n\\end{definition}\n\nNotice that this endows $Boo_{n}^{\\mathcal{B}}$ with a structure of a Boolean algebra, top element $(1, 0, 1, \\dotsc  , 1)$ and bottom $(0, 1, 1, \\dotsc  , 1)$, isomorphic to that of $\\mathcal{B}$, giving us a copy of $\\mathcal{B}$ inside of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$: in all of this, ``$\\tilde{\\neg}$'' plays the role of the negation of $Boo_{n}^{\\mathcal{B}}$ as a Boolean algebra, meaning $\\tilde{\\neg}z=\\{({\\sim}a, a, 1, \\dotsc  , 1)\\}$, if $z=(a, {\\sim}a, 1, \\dotsc  , 1)$. This is true since, appealing to its definition, \n\\[\\tilde{\\neg}z=\\{w\\in B_{n}^{\\mathcal{B}}\\ :\\  w_{1}={\\sim}a\\quad\\text{and}\\quad w_{2}\\leq a\\},\\]\nwhat means that $w_{1}\\wedge w_{2}\\leq {\\sim}a\\wedge a=0$, and therefore $w\\in Boo_{n}^{\\mathcal{B}}$ is such that $w_{1}={\\sim}a$, and there is only one element satisfying this property, namely $w=({\\sim}a, a, 1, \\dotsc  , 1)$.\n\nFurthermore, one sees that $B_{n}\\subseteq B_{n}^{\\mathcal{B}}$, given that $\\textbf{2}$ is necessarily a subalgebra of $\\mathcal{B}$, and $D_{n}^{\\mathcal{B}}\\cap B_{n}=D_{n}$ and $Boo_{n}^{\\mathcal{B}}\\cap B_{n}=Boo_{n}$; the multioperations of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$, when restricted to $B_{n}$, become those of $\\mathcal{A}_{C_{n}}$, and it is clear then that $\\mathcal{A}_{C_{n}}$ is a submultialgebra of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$.\n\n\\begin{definition}\\label{restricted valuations for RMBCn}\nGiven a Boolean algebra $\\mathcal{B}$, the restricted Nmatrix for $C_{n}$ over $\\mathcal{B}$ is the RNmatrix\\label{RMCnB} \n\\[\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}=(\\mathcal{A}_{C_{n}}^{\\mathcal{B}}. D_{n}^{\\mathcal{B}}, \\mathcal{F}_{C_{n}}^{\\mathcal{B}}),\\]\nwhere $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$\\label{FCnB} is the set of valuations $\\nu$ for $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$, meaning homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow \\mathcal{A}_{C_{n}}^{\\mathcal{B}}$, satisfying, for any two formulas $\\alpha$ and $\\beta$ of $C_{n}$:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\wedge\\neg\\alpha)\\in\\{z\\in \\nu(\\alpha)\\tilde{\\wedge}\\nu(\\neg\\alpha)\\ :\\  z_{2}=\\nu(\\alpha)_{3}\\}$;\n\\item $\\nu(\\alpha^{1})=(\\nu(\\alpha)_{3}, \\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}, \\nu(\\alpha)_{4}, \\dotsc  , \\nu(\\alpha)_{n+1}, {\\sim}(\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}))$;\n\\item for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})\\in D_{n}^{\\mathcal{B}}$.\n\\end{enumerate}\n\\end{definition}\n\nThe following is a somewhat obvious result, but we prove it for completeness sake.\n\n\\begin{lemma}\\label{index formula}\nFor every formula $\\alpha$ of $C_{n}$, $j, k\\in\\mathbb{N}$ and endomorphism $\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$,\n\\[\\sigma(\\alpha^{j})=\\sigma(\\alpha)^{j}\\quad\\text{and}\\quad\\sigma(\\alpha^{(k)})=\\sigma(\\alpha)^{(k)}.\\]\n\\end{lemma}\n\n\\begin{proof}\nOne has $\\sigma(\\alpha^{0})=\\sigma(\\alpha)=\\sigma(\\alpha)^{0}$, so assume that $\\sigma(\\alpha^{j})=\\sigma(\\alpha)^{j}$:\n\\[\\sigma(\\alpha^{j+1})=\\sigma(\\neg(\\alpha^{j}\\wedge\\neg\\alpha^{j}))=\\neg(\\sigma(\\alpha^{j})\\wedge\\neg\\sigma(\\alpha^{j}))=\\neg(\\sigma(\\alpha)^{j}\\wedge\\neg\\sigma(\\alpha)^{j})=\\sigma(\\alpha)^{j+1}.\\]\nNow $\\sigma(\\alpha^{(0)})=\\sigma(\\alpha)=\\sigma(\\alpha)^{(0)}$ and $\\sigma(\\alpha^{(1)})=\\sigma(\\alpha^{1})=\\sigma(\\alpha)^{1}=\\sigma(\\alpha)^{(0)}$, so assume $\\sigma(\\alpha^{(k)})=\\sigma(\\alpha)^{(k)}$:\n\\[\\sigma(\\alpha^{(k+1)})=\\sigma(\\alpha^{(k)}\\wedge\\alpha^{k+1})=\\sigma(\\alpha^{(k)})\\wedge\\sigma(\\alpha^{k+1})=\\sigma(\\alpha)^{(k)}\\wedge\\sigma(\\alpha)^{k+1}=\\sigma(\\alpha)^{(k+1)}.\\]\n\\end{proof}\n\n\\begin{proposition}\nFor any homomorphism $\\nu$ in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ and any endomorphism $\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$, $\\nu\\circ\\sigma$ lies in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$.\n\\end{proposition}\n\n\\begin{proof}\nOf course $\\nu\\circ\\sigma:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ is still a homomorphism of $\\Sigma_{\\textbf{C}}$-multialgebras.\n\n\\begin{enumerate}\n\\item That $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)$ is in $\\nu\\circ\\sigma(\\alpha)\\tilde{\\wedge}\\nu\\circ\\sigma(\\neg\\alpha)$ is a given, since $\\nu\\circ\\sigma$ is a homomorphism. Now, given $\\sigma$ is a homomorphism of $\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})$, $\\nu\\circ\\sigma(\\alpha\\wedge\\neg\\alpha)=\\nu(\\sigma(\\alpha\\wedge\\neg\\alpha))$ equals $\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))$; since $\\nu$ lies in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$, \n\\[\\nu(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))_{2}=\\nu(\\sigma(\\alpha))_{3}=\\nu\\circ\\sigma(\\alpha)_{3},\\]\nas we needed to prove.\n\n\\item We use, again, that $\\sigma$ is a change of variables: from Lemma \\ref{index formula}, $\\sigma(\\alpha^{1})=\\sigma(\\alpha)^{1}$; keeping in mind that $\\nu\\circ\\sigma(\\alpha)_{i}=\\nu(\\sigma(\\alpha))_{i}$,\n\\[\\nu\\circ\\sigma(\\alpha^{1})=\\nu(\\sigma(\\alpha)^{1})=\\]\n\\[\\Big(\\nu(\\sigma(\\alpha))_{3}, \\nu(\\sigma(\\alpha))_{1}\\wedge\\nu(\\sigma(\\alpha))_{2}, \\nu(\\sigma(\\alpha))_{4}, \\dotsc  , \\nu(\\sigma(\\alpha))_{n+1}, {\\sim}(\\bigwedge_{i=1}^{n+1}\\nu(\\sigma(\\alpha))_{i})\\Big)=\\]\n\\[\\Big(\\nu\\circ\\sigma(\\alpha)_{3}, \\nu\\circ\\sigma(\\alpha)_{1}\\wedge\\nu\\circ\\sigma(\\alpha)_{2}, \\nu\\circ\\sigma(\\alpha)_{4}, \\dotsc  , \\nu\\circ\\sigma(\\alpha)_{n+1}, {\\sim}(\\bigwedge_{i=1}^{n+1}\\nu\\circ\\sigma(\\alpha)_{i})\\Big).\\]\n\n\\item Finally, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, \n\\[\\nu\\circ\\sigma\\Big((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)}\\Big)=\\nu\\Big((\\sigma(\\alpha^{(n)})\\wedge\\nu(\\beta^{(n)}))\\rightarrow\\sigma((\\alpha\\#\\beta)^{(n)})\\Big);\\]\nfrom Lemma \\ref{index formula}, this equals \n\\[\\nu\\Big((\\sigma(\\alpha)^{(n)}\\wedge\\sigma(\\beta)^{(n)})\\rightarrow\\sigma(\\alpha\\#\\beta)^{(n)}\\Big)=\\nu\\Big((\\sigma(\\alpha)^{(n)}\\wedge\\sigma(\\beta)^{(n)})\\rightarrow(\\sigma(\\alpha)\\#\\sigma(\\beta))^{(n)}\\Big),\\]\nwhich is in $D_{n}^{\\mathcal{B}}$ since $\\nu$ is in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ (and $\\sigma(\\alpha)$ and $\\sigma(\\beta)$ are still formulas of $C_{n}$).\n\\end{enumerate}\n\\end{proof}\n\n\nIt follows, quite easily, that $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ is structural.\n\n\\begin{definition}\nTo signify that $\\varphi$ follows from $\\Gamma$ according to the semantical consequence relation with respect to $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ we write $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}}^{RN}\\varphi$\\label{vDashRNRM}.\n\nThe semantical consequence relation with respect to the restricted swap structures semantics for $C_{n}$, i.e. the class $\\mathcal{RS}_{C_{n}}$ of $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ for all Boolean algebras $\\mathcal{B}$, will be similarly represented by $\\Gamma\\vDash_{\\mathcal{RS}_{C_{n}}}^{RN}\\varphi$: this means that\\label{vDashRNRS} \n\\[\\Gamma\\vDash_{\\mathcal{RS}_{C_{n}}}^{RN}\\varphi\\quad\\text{if, and only if,}\\quad \\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}}^{RN}\\varphi\\quad\\text{for all Boolean algebras $\\mathcal{B}$}.\\]\n\\end{definition}\n\n\\begin{lemma}\\label{Finding the coordinates}\nFor $2\\leq k\\leq n-2$, we have that, for a homomorphism $\\nu\\in\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$,\n\\[\\nu(\\alpha^{k})=(\\nu(\\alpha)_{k+2}, \\bigwedge_{i=1}^{k+1}\\nu(\\alpha)_{i}, \\nu(\\alpha)_{k+3}, \\dotsc  ,  \\nu(\\alpha)_{n+1}, {\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}, 1, \\dotsc  , 1);\\]\nof course, we then have \n\\[\\nu(\\alpha^{n-1})=(\\nu(\\alpha)_{n+1}, \\bigwedge_{i=1}^{n}\\nu(\\alpha)_{i}, {\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}, 1, \\dotsc  , 1)\\]\nand \n\\[\\nu(\\alpha^{n})=({\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}, \\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}, 1, \\dotsc  , 1).\\]\n\\end{lemma}\n\n\\begin{proof}\nFor $k=2$, the result follows from the definition of $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ and the fact that $\\alpha^{2}=(\\alpha^{1})^{1}$:\n\\begin{enumerate}\n\\item $\\nu(\\alpha^{2})_{1}=\\nu(\\alpha^{1})_{3}=\\nu(\\alpha)_{4}$; \n\\item $\\nu(\\alpha^{2})_{2}=\\nu(\\alpha^{1})_{1}\\wedge\\nu(\\alpha^{1})_{2}=\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}\\wedge\\nu(\\alpha)_{3}$;\n\\item for $3\\leq j\\leq n-1$, $\\nu(\\alpha^{2})_{j}=\\nu(\\alpha^{1})_{j+1}=\\nu(\\alpha)_{j+2}$;\n\\item $\\nu(\\alpha^{2})_{n}=\\nu(\\alpha^{1})_{n+1}={\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}$;\n\\item $\\nu(\\alpha^{2})_{n+1}={\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha^{1})_{i}={\\sim}(\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}\\wedge{\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i})=1$.\n\\end{enumerate}\n\nFor the general case, suppose the result holds for $k-1$, and remember $\\alpha^{k}=(\\alpha^{k-1})^{1}$.\n\\begin{enumerate}\n\\item $\\nu(\\alpha^{k})_{1}=\\nu(\\alpha^{k-1})_{3}=\\nu(\\alpha)_{k+3}$;\n\\item $\\nu(\\alpha^{k})_{2}=\\nu(\\alpha^{k-1})_{1}\\wedge\\nu(\\alpha^{k-1})_{2}=\\nu(\\alpha)_{k+1}\\wedge\\bigwedge_{i=1}^{k}\\nu(\\alpha)_{i}=\\bigwedge_{i=1}^{k+1}\\nu(\\alpha)_{i}$;\n\\item for $3\\leq j\\leq n-k+1$, $\\nu(\\alpha^{k})_{j}=\\nu(\\alpha^{k-1})_{j+1}=\\nu(\\alpha)_{j+k}$;\n\\item $\\nu(\\alpha^{k})_{n-k+2}=\\nu(\\alpha^{k-1})_{n-k+3}={\\sim}\\bigwedge_{i=1}^{k+1}\\nu(\\alpha)_{i}$;\n\\item since $\\bigwedge_{i=1}^{n-k+2}=\\bigwedge_{i=1}^{k+1}\\nu(\\alpha)_{i}\\wedge{\\sim}\\bigwedge_{i=1}^{k+1}\\nu(\\alpha)_{i}=0$, all following coordinates must equal $1$.\n\\end{enumerate}\n\nThe cases $\\nu(\\alpha^{n-1})$ and $\\nu(\\alpha^{n})$ follow analogously.\n\\end{proof}\n\nUsing the previous lemma, specifically the fact that $\\nu(\\alpha^{k})_{1}=\\nu(\\alpha)_{k+2}$, is not difficult to prove that, for $1\\leq k\\leq n$, $\\nu(\\alpha^{(k)})_{1}=\\bigwedge_{i=3}^{k+2}\\nu(\\alpha)_{i}$: the base case is rather obvious, since $\\nu(\\alpha^{(1)})_{1}=\\nu(\\alpha^{1})_{1}=\\nu(\\alpha)_{3}$; assuming the result holds for an arbitrary $1\\leq k\\leq n-1$, \n\\[\\nu(\\alpha^{(k+1)})_{1}=\\nu(\\alpha^{(k)})_{1}\\wedge\\nu(\\alpha^{k+1})_{1}=\\bigwedge_{i=3}^{k+2}\\nu(\\alpha)_{i}\\wedge\\nu(\\alpha)_{k+3}=\\bigwedge_{i=3}^{k+3}\\nu(\\alpha)_{i}.\\]\n\nThen, one has, again from Lemma \\ref{Finding the coordinates},\n\\[\\nu(\\alpha^{(n)})_{1}=\\nu(\\alpha^{(n-1)})_{1}\\wedge\\nu(\\alpha^{n})_{1}=\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\wedge{\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}=\\]\n\\[\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\wedge{\\sim}\\Big(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}\\wedge\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\Big)=\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\wedge\\Big({\\sim}(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2})\\vee{\\sim}\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\Big)=\\]\n\\[\\Big(\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\wedge{\\sim}(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2})\\Big)\\vee \\Big(\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\wedge{\\sim}\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}\\Big)=\\]\n\\[{\\sim}(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2})\\wedge\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}={\\sim}(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2})\\wedge\\nu(\\alpha^{(n-1)})_{1}\\]\n\n\\begin{proposition}\\label{Value of strong negation}\nLet ${\\sim}\\alpha=\\neg\\alpha\\wedge\\alpha^{(n)}$ be the definable strong negation in $C_{n}$; for every $\\nu\\in\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$, one has:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\wedge{\\sim}\\alpha)=F_{n}$;\n\\item $\\nu(\\alpha\\vee{\\sim\\alpha})\\in D_{n}^{\\mathcal{B}}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\n\\begin{enumerate}\n\\item One has $\\nu(\\neg\\alpha)_{1}=\\nu(\\alpha)_{2}$ and $\\nu(\\alpha^{(n)})_{1}={\\sim}(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2})\\wedge\\nu(\\alpha^{(n-1)})_{1}$, and therefore\n\\[\\nu(\\neg\\alpha\\wedge\\alpha^{(n)})_{1}=\\nu(\\alpha)_{2}\\wedge{\\sim}(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2})\\wedge\\nu(\\alpha^{(n-1)})_{1}=\\nu(\\alpha)_{2}\\wedge({\\sim}\\nu(\\alpha)_{1}\\vee{\\sim}\\nu(\\alpha)_{2})\\wedge\\nu(\\alpha^{(n-1)})_{1}=\\]\n\\[\\Big(\\nu(\\alpha)_{2}\\wedge{\\sim}\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha^{(n-1)})_{1}\\Big)\\vee\\Big(\\nu(\\alpha)_{2}\\wedge{\\sim}\\nu(\\alpha)_{2}\\wedge\\nu(\\alpha^{(n-1)})_{1}\\Big)=\\nu(\\alpha)_{2}\\wedge{\\sim}\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha^{(n-1)})_{1}.\\]\nSo, \n\\[\\nu(\\alpha\\wedge{\\sim}\\alpha)_{1}=\\nu(\\alpha)_{1}\\wedge\\Big(\\nu(\\alpha)_{2}\\wedge{\\sim}\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha^{(n-1)})_{1}\\Big)=0,\\]\nwhat means that $\\nu(\\alpha\\wedge{\\sim}\\alpha)=F_{n}$.\n\n\n\\item Since $\\nu(\\alpha)$ is in $B_{n}^{\\mathcal{B}}$, $\\nu(\\alpha)_{1}\\vee\\nu(\\alpha)_{2}=1$ and, for any $1\\leq k\\leq n$, $(\\bigwedge_{i=1}^{k}\\nu(\\alpha)_{i})\\vee\\nu(\\alpha)_{k+1}=1$, meaning \n\\[{\\sim}\\nu(\\alpha)_{k+1}\\leq \\bigwedge_{i=1}^{k}\\nu(\\alpha)_{i}\\leq \\nu(\\alpha)_{1}.\\]\nThis implies that, for all $3\\leq k\\leq n+1$, ${\\sim}\\nu(\\alpha)_{k}\\leq \\nu(\\alpha)_{1}$, and so ${\\sim}\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i}=\\bigvee_{i=3}^{n+1}{\\sim}\\nu(\\alpha)_{i}\\leq \\nu(\\alpha)_{1}$, what implies in turn that\n\\[\\nu(\\alpha\\vee{\\sim}\\alpha)_{1}=\\nu(\\alpha)_{1}\\vee\\Big(\\nu(\\alpha)_{2}\\wedge{\\sim}\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha^{(n-1)})_{1}\\Big)=\\]\n\\[(\\nu(\\alpha)_{1}\\vee\\nu(\\alpha)_{2})\\wedge(\\nu(\\alpha)_{1}\\vee{\\sim}\\nu(\\alpha)_{1})\\wedge(\\nu(\\alpha)_{1}\\vee\\bigwedge_{i=3}^{n+1}\\nu(\\alpha)_{i})=1.\\]\nOf course, this finishes proving that $\\nu(\\alpha\\vee{\\sim}\\alpha)\\in D_{n}^{\\mathcal{B}}$.\n\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{sound arb}\nLet $\\nu$ be a valuation in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$; then, the mapping $\\mathsf{b}:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow |\\mathcal{B}|$\\label{|B|}, given by $\\mathsf{b}(\\alpha)=\\nu(\\alpha)_{1}$, is a $\\mathcal{B}$-valuation for $C_{n}$ such that $\\mathsf{b}(\\alpha)=1$ if, and only if, $\\nu(\\alpha)\\in D_{n}^{\\mathcal{B}}$ for every formula $\\alpha$.\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Clause $(V1)$ is quite obvious: since $\\nu$ is a homomorphism, $\\nu(\\alpha\\#\\beta)\\in \\nu(\\alpha)\\tilde{\\#}\\nu(\\beta)$, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, and given that $u\\in z\\tilde{\\#}w$ if, and only if, $u_{1}=z_{1}\\# w_{1}$, we obtain that $\\nu(\\alpha\\#\\beta)_{1}=\\nu(\\alpha)_{1}\\#\\nu(\\beta)_{1}$ and therefore $\\mathsf{b}(\\alpha\\#\\beta)=\\mathsf{b}(\\alpha)\\#\\mathsf{b}(\\beta)$.\n\n\\item We have $\\mathsf{b}(\\neg\\alpha)=\\nu(\\neg\\alpha)_{1}$: since $\\nu(\\neg\\alpha)\\in \\tilde{\\neg}\\nu(\\alpha)$, given $\\nu$ is a homomorphism, and $w\\in \\tilde{\\neg}z$ if, and only if $w_{1}=z_{2}$ and $w_{2}\\leq z_{1}$, we obtain $\\nu(\\neg\\alpha)_{1}=\\nu(\\alpha)_{2}$; from the definition of $B_{n}^{\\mathcal{B}}$, $\\nu(\\alpha)_{1}\\vee\\nu(\\alpha)_{2}=1$, implying $\\nu(\\alpha)_{2}\\geq {\\sim}\\nu(\\alpha)_{1}$. Of course, this means $\\mathsf{b}(\\neg\\alpha)\\geq{\\sim}\\mathsf{b}(\\alpha)$, which corresponds to clause $(V2)$.\n\n\\item Again from the definition of $\\tilde{\\neg}$, $\\nu(\\neg\\neg\\alpha)_{1}=\\nu(\\neg\\alpha)_{2}$, and $\\nu(\\neg\\alpha)_{2}\\leq \\nu(\\alpha)_{1}$, meaning $\\mathsf{b}(\\neg\\neg\\alpha)=\\nu(\\neg\\neg\\alpha)_{1}\\leq \\nu(\\alpha)_{1}=\\mathsf{b}(\\alpha)$. This corresponds to clause $(V3)$.\n\n\\item From Lemma \\ref{Finding the coordinates}, we have $\\nu(\\alpha^{n})_{1}={\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}$, $\\nu(\\alpha^{n-1})_{1}=\\nu(\\alpha)_{n+1}$ and $\\nu(\\neg(\\alpha^{n-1}))_{1}=\\nu(\\alpha^{n-1})_{2}=\\bigwedge_{i=1}^{n}\\nu(\\alpha)_{i}$, meaning therefore that \n\\[\\mathsf{b}(\\alpha^{n})={\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}={\\sim}(\\nu(\\alpha)_{n+1}\\wedge \\bigwedge_{i=1}^{n}\\nu(\\alpha)_{i})={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1}))),\\]\nthat is, clause $(V4)_{n}$.\n\n\\item We have $\\mathsf{b}(\\neg(\\alpha^{1}))=\\nu(\\neg(\\alpha^{1}))_{1}=\\nu(\\alpha^{1})_{2}=\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}=\\nu(\\alpha)_{1}\\wedge\\nu(\\neg\\alpha)_{1}=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)$. That validates clause $(V5)$.\n\n\\item From the third condition of Definition \\ref{restricted valuations for RMBCn}, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$ one has $\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})\\in D_{n}^{\\mathcal{B}}$, meaning $(\\nu(\\alpha^{(n)})_{1}\\wedge\\nu(\\beta^{(n)})_{1})\\rightarrow\\nu((\\alpha\\#\\beta)^{(n)})_{1}=\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})_{1}=1$. Of course, this implies $\\mathsf{b}(\\alpha^{(n)})\\wedge\\mathsf{b}(\\beta^{(n)})=\\nu(\\alpha^{(n)})_{1}\\wedge\\nu(\\beta^{(n)})_{1}\\leq \\nu((\\alpha\\#\\beta)^{(n)})_{1}=\\mathsf{b}((\\alpha\\#\\beta)^{(n)})$. This shows condition $(V6)_{n}$ is also validated.\n\\end{enumerate}\n\nOf course, $\\mathsf{b}(\\alpha)=1$ if, and only if, $\\nu(\\alpha)_{1}=1$, which is equivalent to $\\nu(\\alpha)\\in D_{n}^{\\mathcal{B}}$.\n\\end{proof}\n\n\\begin{lemma}\\label{comp arb}\nLet $\\mathsf{b}$ be a $\\mathcal{B}$-valuation for $C_{n}$; then, the mapping $\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow B_{n}^{\\mathcal{B}}$ given by $\\nu(\\alpha)=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1}))$ is a valuation in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ such that $\\mathsf{b}(\\alpha)=1$ if, and only if, $\\nu(\\alpha)\\in D_{n}^{\\mathcal{B}}$ for every formula $\\alpha$.\n\\end{lemma}\n\n\\begin{proof}\nFirst of all, we prove $\\nu$ is a homomorphism.\n\\begin{enumerate}\n\\item One sees that, by definition of $\\nu$, $\\nu(\\neg\\alpha)_{1}=\\mathsf{b}(\\neg\\alpha)=\\nu(\\alpha)_{2}$ and, by $(V3)$, $\\nu(\\neg\\alpha)_{2}=\\mathsf{b}(\\neg\\neg\\alpha)\\leq \\mathsf{b}(\\alpha)=\\nu(\\alpha)_{1}$, proving that $\\nu(\\neg\\alpha)\\in \\tilde{\\#}\\nu(\\alpha)$.\n\n\\item For $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, from condition $(V1)$ one gets $\\nu(\\alpha\\#\\beta)_{1}=\\mathsf{b}(\\alpha\\#\\beta)=\\mathsf{b}(\\alpha)\\#\\mathsf{b}(\\beta)=\\nu(\\alpha)_{1}\\#\\nu(\\beta)_{1}$. Furthermore, $\\nu(\\alpha), \\nu(\\beta)\\in Boo_{n}^{\\mathcal{B}}$ if and only if $\\nu(\\alpha)_{1}={\\sim}\\nu(\\alpha)_{2}$ and $\\nu(\\beta)_{1}={\\sim}\\nu(\\beta)_{2}$, or equivalently, $\\mathsf{b}(\\neg\\alpha)={\\sim}\\mathsf{b}(\\alpha)$ and $\\mathsf{b}(\\neg\\beta)={\\sim}\\mathsf{b}(\\beta)$; from Proposition \\ref{Negations of B-valuations}, this implies $\\mathsf{b}(\\neg(\\alpha\\#\\beta))={\\sim}\\mathsf{b}(\\alpha\\#\\beta)$, that is, $\\nu(\\alpha\\#\\beta)\\in Boo_{n}^{\\mathcal{B}}$. With all of this, we find that, regardless of the values of $\\nu(\\alpha)$ and $\\nu(\\beta)$, $\\nu(\\alpha\\#\\beta)\\in \\nu(\\alpha)\\tilde{\\#}\\nu(\\beta)$.\n\\end{enumerate}\n\nNow, we need only to prove that $\\nu$ is in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$.\n\n\\begin{enumerate}\n\\item From the fact $\\nu$ is a homomorphism, $\\nu(\\alpha\\wedge\\neg\\alpha)\\in \\nu(\\alpha)\\tilde{\\wedge}\\nu(\\neg\\alpha)$; moreover, $\\nu(\\alpha\\wedge\\neg\\alpha)_{2}=\\mathsf{b}(\\neg(\\alpha\\wedge\\neg\\alpha))=\\mathsf{b}(\\alpha^{1})=\\nu(\\alpha)_{3}$, what proves the first condition for being in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ of Definition \\ref{restricted valuations for RMBCn} is satisfied.\n\n\\item $\\nu(\\alpha^{1})_{1}=\\mathsf{b}(\\alpha^{1})=\\nu(\\alpha)_{3}$, from the definition of $\\nu$; from property $(V5)$, $\\nu(\\alpha^{1})_{2}=\\mathsf{b}(\\neg(\\alpha^{1}))=\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)=\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}$; for $3\\leq k\\leq n$, $\\nu(\\alpha^{1})_{k}=\\mathsf{b}((\\alpha^{1})^{k-2})=\\mathsf{b}(\\alpha^{k-1})=\\nu(\\alpha)_{k+1}$.\n\nFinally, we have from $(V4)_{n}$ that $\\nu(\\alpha^{1})_{n+1}=\\mathsf{b}(\\alpha^{n})={\\sim}(\\mathsf{b}(\\alpha^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-1})))$; from $(V5)$, $\\mathsf{b}(\\neg(\\alpha^{n-1}))=\\mathsf{b}(\\alpha^{n-2})\\wedge\\mathsf{b}(\\neg(\\alpha^{n-2}))$, and proceeding recursively, one obtains $\\nu(\\alpha^{1})_{n+1}=(\\mathsf{b}(\\alpha)\\wedge\\mathsf{b}(\\neg\\alpha)\\wedge{\\sim}\\bigwedge_{i=1}^{n-1}\\mathsf{b}(\\alpha^{i}))={\\sim}\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}$.\n\n\\item For any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, from $(V6)_{n}$ we find that $\\mathsf{b}(\\alpha^{(n)})\\wedge\\mathsf{b}(\\beta^{(n)})\\leq\\mathsf{b}((\\alpha\\#\\beta)^{(n)})$, meaning $\\nu(\\alpha^{(n)}\\wedge\\beta^{(n)})_{1}=\\nu(\\alpha^{(n)})_{1}\\wedge\\nu(\\beta^{(n)})_{1}\\leq \\nu((\\alpha\\#\\beta)^{(n)})_{1}$, and therefore $\\nu(\\alpha^{(n)}\\wedge\\beta^{(n)})_{1}\\rightarrow\\nu((\\alpha\\#\\beta)^{(n)})_{1}=\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})_{1}=1$, which is equivalent to $\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})\\in D_{n}^{\\mathcal{B}}$.\n\\end{enumerate}\n\nClearly, $\\mathsf{b}(\\alpha)=1$ if, and only if, $\\nu(\\alpha)_{1}=1$, which is in turn equivalent to $\\nu(\\alpha)\\in D_{n}^{\\mathcal{B}}$.\n\\end{proof}\n\n\n\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $C_{n}$, $\\Gamma\\vdash_{C_{n}}\\varphi$ if, and only if, $\\Gamma\\vDash_{\\mathcal{RS}_{C_{n}}}^{RN}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nWe start by assuming that $\\Gamma\\vdash_{C_{n}}\\varphi$, and taking a valuation $\\nu$ lying in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ and satisfying $\\nu(\\Gamma)\\subseteq D_{n}^{\\mathcal{B}}$: from Lemma \\ref{sound arb}, the function \n\\[\\mathsf{b}:F(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\mathcal{B},\\]\nsuch that $\\mathsf{b}(\\alpha)=\\nu(\\alpha)_{1}$ for any formula $\\alpha$ of $C_{n}$, is a $\\mathcal{B}$-valuation for which, by hypothesis, $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$. We know the method of $\\mathcal{B}$-valuations to be sound for $C_{n}$, so $\\Gamma\\vdash_{C_{n}}\\varphi$ implies that $\\mathsf{b}(\\varphi)=1$, and therefore $\\nu(\\varphi)\\in D_{n}^{\\mathcal{B}}$. This proves $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}}\\varphi$, and hence the semantics of restricted swap structures is sound.\n\nReciprocally, assume $\\Gamma\\vDash_{\\mathcal{RS}_{C_{n}}}^{RN}\\varphi$ and suppose $\\mathsf{b}$ is a $\\mathcal{B}$-valuation with $\\mathsf{b}(\\Gamma)\\subseteq\\{1\\}$. From Lemma \\ref{comp arb}, \n\\[\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow \\mathcal{A}_{C_{n}}^{\\mathcal{B}}\\] \ndefined as\n\\[\\nu(\\alpha)=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1})),\\]\nis a valuation in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ with the additional property that $\\nu(\\Gamma)\\subseteq D_{n}^{\\mathcal{B}}$. Since $\\Gamma\\vDash_{\\mathcal{RS}_{C_{n}}}^{RN}\\varphi$, it follows that $\\nu(\\varphi)\\in D_{n}^{\\mathcal{B}}$, what means $\\mathsf{b}(\\varphi)=1$ and, by completeness of $C_{n}$ with respect to $\\mathcal{B}$-valuations, $\\Gamma\\vdash_{C_{n}}\\varphi$.\n\\end{proof}\n\n\n\\subsection{$\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ is not trivial}\\label{RM is not trivial}\n\nA natural question is then if $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ adds anything to the already defined $\\mathcal{RM}_{C_{n}}$: that is, is there a valuation for $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ which is not a valuation for $\\mathcal{RM}_{C_{n}}$? This is mostly equivalent to the question of whether there exists a (non-trivial) Boolean algebra $\\mathcal{B}$ and a $\\mathcal{B}$-valuation which is not a bivaluation.\n\nSo, take a non-trivial Boolean algebra $\\mathcal{B}$, and remember Definition \\ref{B-valuation} of a $\\mathcal{B}$-valuation . We will construct by structural induction a $\\mathcal{B}$-valuation $\\mathsf{b}$ for $C_{n}$, extending any given function $b$ from $\\mathcal{V}$ into the universe of $\\mathcal{B}$, and by taking a $b$ for which there exists a $p\\in\\mathcal{V}$ with $b(\\neg p)\\neq {\\sim}b(p)$ we will obtain that $\\mathsf{b}$ satisfies:\n\\begin{enumerate}\n\\item that there exists a formula $\\alpha$ for $C_{n}$ with $\\mathsf{b}(\\neg\\alpha)\\neq{\\sim}\\mathsf{b}(\\alpha)$;\n\\item that the image $\\{\\mathsf{b}(\\alpha) : \\alpha\\in F(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\}$ of $\\mathsf{b}$ is not contained in $\\{0,1\\}$.\n\\end{enumerate}\n\nFurthermore, by taking the homomorphism $\\nu(\\alpha)=(\\mathsf{b}(\\alpha), \\mathsf{b}(\\neg\\alpha), \\mathsf{b}(\\alpha^{1}), \\dotsc  , \\mathsf{b}(\\alpha^{n-1}))$, we find a valuation in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ whose image is not contained in $B_{n}$ or $Boo_{n}^{\\mathcal{B}}$.\n\nNow, concerning the task of defining $\\mathsf{b}$, satisfying clauses $(V1)$ through $(V5)$ of Definition \\ref{B-valuation} is easy, but in order to guarantee that clause $(V6)_{n}$ is respected we must be very careful. We have that $\\mathsf{b}$ must be equal to $b$ over formulas of complexity $0$, namely variables; we must also define, for our recursion to work, $\\mathsf{b}(\\neg p)$, $\\mathsf{b}(p^{i})$ and $\\mathsf{b}(\\neg p^{i})$, for all $i\\in\\mathbb{N}$ and $p\\in\\mathcal{V}$, but we will show how this is done in the more general setting where $p$ is replaced by an arbitrary formula $\\alpha$ of bounded complexity. So, assume for a moment that $\\mathsf{b}(\\alpha)$, $\\mathsf{b}(\\neg\\alpha)$, $\\mathsf{b}(\\alpha^{i})$ and $\\mathsf{b}(\\neg\\alpha^{i})$, for any $i\\in\\mathbb{N}$, have been defined for all formulas $\\alpha$ of complexity at most $m$: then a formula of complexity $m+1$ is either: $\\alpha\\#\\beta$, for $\\alpha$ and $\\beta$ of complexity less or equal to $m$ and $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, when we define, in order to satisfy $(V1)$,\n\\[\\mathsf{b}(\\alpha\\#\\beta)=\\mathsf{b}(\\alpha)\\#\\mathsf{b}(\\beta);\\]\nor $\\neg\\alpha$, for an $\\alpha$ of complexity equal to $m$, when $\\mathsf{b}(\\neg\\alpha)$ has already been defined. So we must now specify, for $\\alpha$ and $\\beta$ of complexity at most $m$, $\\mathsf{b}(\\neg\\neg \\alpha)$, $\\mathsf{b}((\\neg\\alpha)^{i})$, $\\mathsf{b}(\\neg(\\neg\\alpha)^{i})$, $\\mathsf{b}(\\neg(\\alpha\\#\\beta))$, $\\mathsf{b}((\\alpha\\#\\beta)^{j})$ and $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{j}))$, for $1\\leq i,j\\leq n$.\\footnote{We do not need to worry about defining $\\mathsf{b}(\\alpha^{j})$ or $\\mathsf{b}(\\neg\\alpha^{j})$ for $j> n$, for any formula $\\alpha$, since they must always take the values, respectively, $1$ and $0$.} We start by looking at $\\mathsf{b}(\\neg\\neg\\alpha)$, which we make equal to\n\\begin{enumerate}\n\\item $\\mathsf{b}(\\gamma)\\wedge\\mathsf{b}(\\neg\\gamma)$, if $\\neg\\alpha=\\gamma^{1}$ (thus validating $(V5)$);\n\\item any value between $\\mathsf{b}(\\alpha)$ and ${\\sim}\\mathsf{b}(\\neg\\alpha)$, both included, if the previous case does not apply (validating $(V2)$ and $(V3)$).\n\\end{enumerate}\nRegarding $\\mathsf{b}((\\neg\\alpha)^{i})$ and $\\mathsf{b}(\\neg(\\neg\\alpha)^{i})$, for $1\\leq i\\leq n-1$, there isn't a lot we must do: supposing we have defined $\\mathsf{b}((\\neg\\alpha)^{i-1})$ and $\\mathsf{b}(\\neg(\\neg\\alpha)^{i-1})$, it is sufficient to demand that $\\mathsf{b}((\\neg\\alpha)^{i})$ is greater or equal to \n\\[{\\sim}[\\mathsf{b}((\\neg\\alpha)^{i-1}\\wedge\\neg(\\neg\\alpha)^{i-1})],\\]\nin order to satisfy clause $(V2)$, and that $\\mathsf{b}(\\neg(\\neg\\alpha)^{i})$ equals $\\mathsf{b}((\\neg\\alpha)^{i-1})\\wedge\\mathsf{b}(\\neg(\\neg\\alpha)^{i-1})$, in order to validate $(V5)$ (unless $\\neg\\alpha=\\gamma^{k}$ for some $1\\leq k\\leq n-1$ such that $k+i=n$, in which case $\\mathsf{b}((\\neg\\alpha)^{i})=\\mathsf{b}(\\gamma^{n})$ is defined according to $(V4)_{n}$). Of course, $\\mathsf{b}((\\neg\\alpha)^{n})$ and $\\mathsf{b}(\\neg(\\neg\\alpha)^{n})$ are given values according to clauses, respectively, $(V4)_{n}$ and $(V5)$, being expressed in terms of the already defined $\\mathsf{b}((\\neg\\alpha)^{n-1})$ and $\\mathsf{b}(\\neg(\\neg\\alpha)^{n-1})$.\n\n\nDifficulties finally arise when defining $\\mathsf{b}(\\neg(\\alpha\\#\\beta))$, $\\mathsf{b}((\\alpha\\#\\beta)^{i})$ and $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{i})$, because we must be mindful of clause $(V6)_{n}$: denote by $a_{0}$ the value $\\mathsf{b}(\\alpha_{0})=\\mathsf{b}(\\alpha)\\#\\mathsf{b}(\\beta)$, and by $a_{\\#}$ the value $\\mathsf{b}(\\alpha^{(n)})\\wedge\\mathsf{b}(\\beta^{(n)})=\\bigwedge_{j=1}^{n}\\mathsf{b}(\\alpha^{j})\\wedge\\mathsf{b}(\\beta^{j})$ (all of $\\mathsf{b}(\\alpha^{j})$ and $\\mathsf{b}(\\beta^{j})$, by hypothesis, already defined).\n\n\\begin{enumerate}[align=left]\n\\item[\\underline{$\\mathsf{b}(\\neg(\\alpha\\#\\beta))$} - ] The value for $\\mathsf{b}(\\neg(\\alpha\\#\\beta))$ will be $a_{\\neg}$: on the off chance that $\\#=\\wedge$, $\\alpha=\\gamma^{n-1}$ and $\\beta=\\neg\\gamma^{n-1}$, we make $a_{\\neg}$ equal to ${\\sim}a_{0}={\\sim}[\\mathsf{b}(\\gamma^{n-1})\\wedge\\mathsf{b}(\\neg(\\gamma^{n-1}))]$ (as required by $(V4)_{n}$); otherwise, $a_{\\neg}$ can be any value such that \n\\[a_{\\neg}\\geq {\\sim}a_{0}\\quad\\text{and}\\quad a_{\\neg}\\wedge a_{0}\\leq{\\sim}a_{\\#}\\]\n(this is possible, one example of such an element being $a_{\\neg}={\\sim}a_{0}$); this last case of course implies $\\mathsf{b}(\\neg(\\alpha\\#\\beta))=a_{\\neg}\\geq{\\sim}a_{0}={\\sim}\\mathsf{b}(\\alpha\\#\\beta)$, and thus also validates $(V2)$. \n\n\\item[\\underline{$\\mathsf{b}((\\alpha\\#\\beta)^{1})$} - ] Having defined $a_{\\neg}$, we define $\\mathsf{b}((\\alpha\\#\\beta)^{1})$ to be any value $a_{1}$ such that $a_{1}\\geq {\\sim}(a_{0}\\wedge a_{\\neg})$ (and therefore we have\n\\[\\mathsf{b}\\big(\\neg((\\alpha\\#\\beta)\\wedge\\neg(\\alpha\\#\\beta))\\big)=\\mathsf{b}((\\alpha\\#\\beta)^{1})=a_{1}\\geq {\\sim}(a_{0}\\wedge a_{\\neg})={\\sim}\\mathsf{b}\\big((\\alpha\\#\\beta)\\wedge\\neg(\\alpha\\#\\beta)\\big),\\]\nand thus $(V2)$ remains valid) and ${\\sim}(a_{0}\\wedge a_{\\neg})\\wedge a_{1}\\geq a_{\\#}$ (what would still be possible, being enough to choose, for one, $a_{1}={\\sim}(a_{0}\\wedge a_{\\neg})$).\n\nAnd we make $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{1})=\\mathsf{b}(\\alpha\\#\\beta)\\wedge\\mathsf{b}(\\neg(\\alpha\\#\\beta))=a_{0}\\wedge a_{\\neg}$, thus respecting clause $(V5)$ (and incidentally also $(V2)$ and $(V3)$).\n\n\\item[\\underline{Inductive step} - ] Supposing we have defined $a_{1}=\\mathsf{b}((\\alpha\\#\\beta)^{1})$ trough $a_{k}=\\mathsf{b}((\\alpha\\#\\beta)^{k})$, for $1\\leq k\\leq n-2$, we make $\\mathsf{b}((\\alpha\\#\\beta)^{k+1})$ equal to any value $a_{k+1}$ satisfying \n\\[a_{k+1}\\geq{\\sim}(a_{0}\\wedge a_{\\neg}\\wedge \\bigwedge_{j=1}^{k}a_{j})\\quad\\text{and}\\quad{\\sim}(a_{0}\\wedge a_{\\neg})\\wedge\\bigwedge_{j=2}^{k}a_{j}\\geq a_{\\#};\\]\n\nInductively, we get $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{k})=a_{0}\\wedge a_{\\neg}\\wedge \\bigwedge_{j=1}^{k-1}a_{j}$, and so: first of all, this means that $\\mathsf{b}((\\alpha\\#\\beta)^{k})\\wedge\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{k})=a_{0}\\wedge a_{\\neg}\\wedge \\bigwedge_{j=1}^{k}a_{j}$ and therefore \n\\[\\mathsf{b}((\\alpha\\#\\beta)^{k+1})=\\mathsf{b}(\\neg((\\alpha\\#\\beta)^{k}\\wedge\\neg(\\alpha\\#\\beta)^{k})\\geq {\\sim}\\big(\\mathsf{b}((\\alpha\\#\\beta)^{k})\\wedge\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{k})\\big),\\]\nmeaning $(V2)$ is respected; second, by making $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{k+1})=\\mathsf{b}((\\alpha\\#\\beta)^{k})\\wedge \\mathsf{b}(\\neg(\\alpha\\#\\beta)^{k})$, so as to respect $(V5)$, we obtain that $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{k+1})=a_{0}\\wedge a_{\\neg}\\wedge \\bigwedge_{j=1}^{k}a_{j}$ and the pattern remains.\n\n\\item[\\underline{$\\mathsf{b}((\\alpha\\#\\beta)^{n})$} - ] Completing the process outlined in the previous items, we make $\\mathsf{b}((\\alpha\\#\\beta)^{n})$ equal to \n\\[a_{n}={\\sim}(\\mathsf{b}((\\alpha\\#\\beta)^{n-1})\\wedge \\mathsf{b}(\\neg(\\alpha\\#\\beta)^{n-1}))={\\sim}(a_{0}\\wedge a_{\\neg}\\wedge \\bigwedge_{j=1}^{n-1}a_{j}),\\]\ngiven clause $(V4)_{n}$; and define $\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{n})$ to be simply $\\mathsf{b}((\\alpha\\#\\beta)^{n-1})\\wedge\\mathsf{b}(\\neg(\\alpha\\#\\beta)^{n-1})$, that is, $a_{0}\\wedge a_{\\neg}\\wedge\\bigwedge_{j=1}^{n-1}a_{j}$.\n\\end{enumerate}\n\nFinally, now that $a_{1}$ trough $a_{n}$ are defined, given that $(\\alpha\\#\\beta)^{(n)}=\\bigwedge_{j=1}^{n}(\\alpha\\#\\beta)^{j}$, one obtains \n\\[\\mathsf{b}((\\alpha\\#\\beta)^{(n)})=\\bigwedge_{j=1}^{n}\\mathsf{b}((\\alpha\\#\\beta)^{j})=a_{n}\\wedge\\bigwedge_{j=1}^{n-1}a_{j};\\]\nso\n\\[\\mathsf{b}((\\alpha\\#\\beta)^{(n)})=[{\\sim}(a_{0}\\wedge a_{\\neg})\\vee{\\sim}\\bigwedge_{j=1}^{n-1}a_{j})]\\wedge\\bigwedge_{j=1}^{n-1}a_{j}={\\sim}(a_{0}\\wedge a_{\\neg})\\wedge\\bigwedge_{j=1}^{n-1}a_{j}\\geq a_{\\#}=\\mathsf{b}(\\alpha^{(n)})\\wedge\\mathsf{b}(\\beta^{(n)}),\\]\nsatisfying $(V6)_{n}$ and proving, with some difficulty, $\\mathsf{b}$ is a $\\mathcal{B}$-valuation, as we wanted to show.\n\n\n\n\n\n\n\n\n\\section{Counting snapshots}\\label{Counting snapshots}\n\nWe would like to prove, for completeness sake, that if $\\mathcal{B}$ is a finite Boolean algebra with $2^{m}$ elements, then $B_{n}^{\\mathcal{B}}$ has $(n+2)^{m}$ elements. To this end, we begin with a few observations intended to make our job easier. We notice that the very useful equality\n\\[B_{n+1}^{\\mathcal{B}}=\\{(a_{1}, \\dotsc  , a_{n+2})\\in |\\mathcal{B}|^{n+2}\\ :\\  (a_{1}, \\dotsc  , a_{n+1})\\in B_{n}^{\\mathcal{B}}\\quad\\text{and}\\quad a_{n+2}\\vee\\bigwedge_{i=1}^{n+1}a_{i}=1\\}\\]\nholds for any $n\\in\\mathbb{N}\\setminus\\{0\\}$.\n\nNow, given all finite Boolean algebras are isomorphic to the field of subsets of some set, we will simply assume all finite Boolean algebras here involved are, in fact, fields of subsets: in that case, the Boolean algebra with $2^{m}$ elements is $\\mathcal{P}(X_{m})$, for $X_{m}=\\{x_{1}, \\dotsc  , x_{m}\\}$\\label{Xm} a canonical set with $m$ elements. We will say that an element $a$ of a finite Boolean algebra $\\mathcal{P}(X_{m})$ is of order $k$ if it is a subset of $X_{m}$ with $k$ elements; it is clear that there are $\\binom{m}{0}=1$ elements of order $0$ (namely, $\\emptyset$, the bottom), $\\binom{m}{1}$ elements of order $1$, \\dots , and $\\binom{m}{m}=1$ elements of order $m$ (namely, $X_{m}$, the top). For what will follow, it is important to remember that the binomial coefficient is given by\n\\[\\binom{m}{n}=\\frac{m!}{n!(m-n)!},\\]\nfor $m, n\\in\\mathbb{N}$ and $n\\leq m$, and that the binomial theorem states \n\\[(x+y)^{m}=\\sum_{k=0}^{m}\\binom{m}{k}x^{k}y^{m-k},\\]\nfor $m\\in\\mathbb{N}$, and $x$ and $y$ any elements of a commutative ring with unity.\n\n\\begin{lemma}\\label{finding solutions of a given order}\nFor an element $a$ of $\\mathcal{P}(X_{m})$ of order $k$, there are $\\binom{k}{p}2^{m-k}$ elements $b$ such that $a\\wedge b$ has order $p\\leq k$, and $\\binom{m-k}{q}2^{k}$ elements $c$ such that $a\\vee c$ has order $q\\geq k$.\n\\end{lemma}\n\n\\begin{proof}\nIf $a\\wedge b$ has order $p$, this means $b$ has $p$ elements in common with $a$, and since there are $k$ elements in $a$, we obtain $\\binom{k}{p}$ possible values for $a\\cap b$; but $a^{c}\\cap b$ can be any subset of $a^{c}$, of which there exist $2^{m-k}$ given $a^{c}$ has $m-k$ elements. Combining the two values, we obtain $\\binom{k}{p}2^{m-k}$ solutions.\n\nNow, if $a\\vee c$ has order $q$, $a^{c}\\cap c$ has $q-k$ elements that $a$ doesn't, and since there are $m-k$ elements in $a^{c}$, this gives us $\\binom{m-k}{q}$ values for $a^{c}\\cap c$; but $a\\cap c$ can be any subset of $a$, giving us $2^{k}$ values for $a\\cap c$ since $a$ has $k$ elements. This leads to the total $\\binom{m-k}{q}2^{k}$ solutions.\n\\end{proof}\n\nNotice that, by adapting the proof of the previous lemma, we may show that, for an $a$ of order $k$, there are $\\binom{k}{p}$ elements $b$ such that $a\\vee b=1$ and $a\\wedge b$ has order $p\\leq k$. This is because: $a^{c}\\cap b$ must equal $a^{c}$, so that $a\\vee b=1$; and $a\\cap b$ must have $p$ elements, and since $a$ has $k$ elements, this gives us $\\binom{k}{p}$ solutions.\n\n\\begin{lemma}\\label{a case of the binomial theorem}\nFor $m\\in\\mathbb{N}$ and $p\\leq m$,\n\\[\\sum_{j=p}^{m}\\binom{j}{p}\\binom{m}{j}x^{m-j}=\\binom{m}{p}(x+1)^{m-p}.\\]\n\\end{lemma}\n\n\\begin{proof}\nWe see that, from the definition of the binomial coefficients and the binomial theorem,\n\\[\\sum_{j=p}^{m}\\binom{j}{p}\\binom{m}{j}x^{m-j}=\\sum_{j=p}^{m}\\frac{j!}{p!(j-p)!}\\frac{m!}{j!(m-j)!}x^{m-j}=\\]\n\\[\\sum_{j=p}^{m}\\frac{m!}{p!(j-p)!(m-j)!}x^{m-j}=\\sum_{j=p}^{m}\\frac{1}{p!}\\frac{m!}{(j-p)!(m-j)!}x^{m-j}=\\]\n\\[\\sum_{j=p}^{m}\\frac{m!}{p!(m-p)!}\\frac{(m-p)!}{(j-p)!(m-j)!}x^{m-j}=\\]\n\\[\\binom{m}{p}\\sum_{i=0}^{m-p}\\frac{(m-p)!}{((i+p)-p)!(m-(i+p))!}x^{m-(i+p)}=\\]\n\\[\\binom{m}{p}\\sum_{i=0}^{m-p}\\frac{(m-p)!}{i!((m-p)-i)!}x^{(m-p)-i}=\\binom{m}{p}\\sum_{i=0}^{m-p}\\binom{m-p}{i}x^{(m-p)-i}1^{i}=\\]\n\\[\\binom{m}{p}(x+1)^{m-p}.\\]\n\\end{proof}\n\n\n\\begin{lemma}\\label{counting certain snapshots}\nFor any $n\\geq 0$, if $\\mathcal{B}$ has $2^{m}$ elements then $B_{n}^{\\mathcal{B}}$ has precisely $\\binom{m}{p}(n+1)^{m-p}$ elements $(a_{1}, \\dotsc  , a_{n+1})$ such that $\\bigwedge_{i=1}^{n+1}a_{i}$ has order $p\\leq m$.\n\\end{lemma}\n\n\\begin{proof}\nLet $(a,b)$ be a pair in $B_{1}^{\\mathcal{B}}$, meaning $a\\vee b=1$, such that $a\\wedge b$ has order $p$. Then, by the commentary after Lemma \\ref{finding solutions of a given order}, if $a$ has order $k\\geq p$, there are $\\binom{k}{p}$ possible values for $b$, and if $k\\leq p$ there are none. Since there are $\\binom{m}{k}$ elements $a$ with order $k$, this gives us the number of pairs satisfying $a\\vee b=1$ and $a\\wedge b$ having order $p$ to be\n\\[\\binom{p}{p}\\binom{m}{p}+\\binom{p+1}{p}\\binom{m}{p+1}+\\cdots+\\binom{m}{p}\\binom{m}{m},\\]\nwhich equals $\\binom{m}{p}2^{m-p}$ from Lemma \\ref{a case of the binomial theorem}, once we use $x=1$.\n\nNow, suppose the described conditions hold for $B_{n}^{\\mathcal{B}}$: there are, by induction hypothesis, $\\binom{m}{k}(n+1)^{m-k}$ elements $(a_{1}, \\dotsc  , a_{n+1})$ such that $\\bigwedge_{i=1}^{n+1}a_{i}$ is of order $k$, and then there are $\\binom{k}{p}$ possible values for $a_{n+3}$,  such that $(a_{1}, \\dotsc  , a_{n+2})$ is in $B_{n+1}^{\\mathcal{B}}$ (meaning $a_{n+2}\\vee\\bigwedge_{i=1}^{n+1}a_{i}=1$) and $\\bigwedge_{i=1}^{n+2}a_{i}$ has order $p$, what gives the total\n\\[\\binom{p}{p}\\binom{m}{p}(n+1)^{m-p}+\\cdots+\\binom{m}{p}\\binom{m}{m}(n+1)^{m-m},\\]\nwhich adds to $\\binom{m}{p}(n+2)^{m-p}$ once we apply $x=n+1$ to Lemma \\ref{a case of the binomial theorem}, ending our proof.\n\\end{proof}\n\n\\begin{theorem}\nIf $\\mathcal{B}$ has $2^{m}$ elements, $B_{n}^{\\mathcal{B}}$ has $(n+2)^{m}$ elements.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma \\ref{counting certain snapshots}, there are $\\binom{m}{0}(n+1)^{m-0}$ elements $(a_{1}, \\dotsc  , a_{n+1})$ in $B_{n}^{\\mathcal{B}}$ such that $\\bigwedge_{i=1}^{n+1}a_{i}$ has order $0$, $\\binom{m}{1}(n+1)^{m-1}$ elements such that $\\bigwedge_{i=1}^{n+1}a_{i}$ has order $1$ and so on, adding up to a total of\n\\[\\binom{m}{0}(n+1)^{m-0}+\\binom{m}{1}(n+1)^{m-1}+\\cdots+\\binom{m}{m}(n+1)^{m-m}=\\]\n\\[\\sum_{p=0}^{m}\\binom{m}{p}(n+1)^{m-p}1^{p}=((n+1)+1)^{m}=(n+2)^{m}.\\]\n\\end{proof}\n\nNotice, furthermore, that $D_{1}^{\\mathcal{B}}$ has $2^{m}$ elements whenever $\\mathcal{B}$ has $2^{m}$ elements: this happens since $(a, b)\\in D_{1}^{\\mathcal{B}}$ whenever $a=1$ and $a\\vee b=1$, meaning $b$ can assume any value in $\\mathcal{B}$; and when $n\\geq 1$,\n\\[D_{n+1}^{\\mathcal{B}}=\\{(1, a_{1}, \\dotsc  , a_{n+1})\\in B_{n+1}^{\\mathcal{B}}: (a_{1}, \\dotsc  , a_{n+1})\\in B_{n}^{\\mathcal{B}}\\},\\]\nimplying $D_{n+1}^{\\mathcal{B}}$ has as many elements as $B_{n}^{\\mathcal{B}}$.\n\n\\begin{theorem}\nIf $\\mathcal{B}$ has $2^{m}$ elements:\n\\begin{enumerate}\n\\item $D_{n}^{\\mathcal{B}}$ has $(n+1)^{m}$ elements;\n\\item $Boo_{n}^{\\mathcal{B}}$ has $2^{m}$ elements.\n\\end{enumerate}\n\\end{theorem}\n\nA final, relevant, note is on the case that $\\mathcal{B}$ is infinite, and of cardinality $\\kappa$: since $B_{n}^{\\mathcal{B}}$ contains the set of Boolean elements $Boo_{n}^{\\mathcal{B}}$, isomorphic to $\\mathcal{B}$ itself, we can be sure that $B_{n}^{\\mathcal{B}}$ has at least cardinality $\\kappa$; however, since $B_{n}^{\\mathcal{B}}$ is a subset of $|\\mathcal{B}|^{n+1}$, which is too of cardinality $\\kappa$ given the assumption this is an infinite cardinal, we obtain that $B_{n}^{\\mathcal{B}}$ has precisely $\\kappa$ elements.\n\nTo provide one example of $B_{n}^{\\mathcal{B}}$'s complexity, let us take the four-valued Boolean algebra $\\mathcal{B}_{4}$ as the field of subsets of $\\{a,b\\}$, containing the elements $\\emptyset$ (which we shall denote by $0$), $\\{a\\}$, $\\{b\\}$, and $\\{a,b\\}$ (which we will denote by $1$). Then, $B_{1}^{\\mathcal{B}_{4}}$ has $9$ snapshots.\n\\begin{enumerate}\n\\item Designated and Boolean: $(1, 0)$.\n\\item Designated and not Boolean: $(1, \\{a\\})$, $(1, \\{b\\})$ and $(1, 1)$.\n\\item Undesignated and Boolean: $(0,1)$, $(\\{a\\}, \\{b\\})$ and $(\\{b\\}, \\{a\\})$.\n\\item Undesignated and not Boolean: $(\\{a\\}, 1)$ and $(\\{b\\}, 1)$.\n\\end{enumerate}\n\nStill on the Boolean algebra with four elements, $B_{2}^{\\mathcal{B}_{4}}$ has $16$ snapshots.\n\\begin{enumerate}\n\\item Designated and Boolean: $(1, 0, 1)$.\n\\item Designated and not Boolean: $(1, 1, 0)$, $(1, 1, \\{a\\})$, $(1, 1, \\{b\\})$, $(1, 1, 1)$, $(1, \\{a\\}, \\{b\\})$,\\\\ $(1, \\{b\\}, \\{a\\})$, $(1, \\{a\\}, 1)$ and $(1, \\{b\\}, 1)$.\n\\item Undesignated and Boolean: $(0, 1, 1)$, $(\\{a\\}, \\{b\\}, 1)$ and $(\\{b\\}, \\{a\\}, 1)$.\n\\item Undesignated and not Boolean: $(\\{a\\}, 1, 1)$, $(\\{a\\}, 1, \\{b\\})$, $(\\{b\\}, 1, 1)$ and $(\\{b\\}, 1, \\{a\\})$.\n\n\\end{enumerate}\n\nNow, consider the eight-valued Boolean algebra $\\mathcal{B}_{8}$, which we will take to be the Boolean algebra over the powerset of $\\{a, b, c\\}$. In that case, $B_{1}^{\\mathcal{B}_{8}}$ has $27$ elements:\n\n\\begin{enumerate}\n\\item Designated and Boolean: $(1, 0)$.\n\\item Designated and not Boolean: $(1, \\{a\\})$, $(1, \\{b\\})$, $(1, \\{c\\})$, $(1, \\{b, c\\})$, $(1, \\{a, c\\})$, $(1, \\{a, b\\})$ and $(1, 1)$.\n\\item Undesignated and Boolean: $(0, 1)$, $(\\{a\\}, \\{b, c\\})$, $(\\{b\\}, \\{a, c\\})$, $(\\{c\\}, \\{a, b\\})$, $(\\{b, c\\}, \\{a\\})$, $(\\{a, c\\}, \\{b\\})$ and $(\\{a, b\\}, \\{c\\})$.\n\\item Undesignated and not Boolean: $(\\{a\\}, 1)$, $(\\{b\\}, 1)$, $(\\{c\\}, 1)$, $(\\{b, c\\}, \\{a, c\\})$, $(\\{b, c\\}, \\{a, b\\})$, $(\\{b, c\\}, 1)$, $(\\{a, c\\}, \\{b, c\\})$, $(\\{a, c\\}, \\{a, b\\})$, $(\\{a, c\\}, 1)$, $(\\{a, b\\}, \\{b, c\\})$, $(\\{a, b\\}, \\{a, c\\})$\\\\ and $(\\{a, b\\}, 1)$.\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Category of restricted swap structures for $C_{n}$}\n\nGiven a class $C$ of RNmatrices, how to make it into a category $\\mathcal{C}$? We believe there may exist more than one natural definition of what a morphism on $\\mathcal{C}$ should be, the applications desired for such an object dictating the best ones, but at least one definition seems to be more or less universal. To start defining it, remember: an RNmatrix is a triple $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$, for $\\mathcal{A}$ a $\\Sigma$-multialgebra, $D$ a subset of its universe and $\\mathcal{F}$ a set of homomorphisms (of multialgebras) $\\nu:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$. So a morphism between RNmatrices $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ and $\\mathcal{M}^{\\prime}=(\\mathcal{A}^{\\prime}, D^{\\prime}, \\mathcal{F}^{\\prime})$, over the same signature $\\Sigma$, should be, at a minimum:\n\\begin{enumerate}\n\\item a homomorphism of multialgebras $h:\\mathcal{A}\\rightarrow\\mathcal{A}^{\\prime}$;\n\\item which preserves designated elements, meaning that $h(D)\\subseteq D^{\\prime}$;\n\\item and is absorbed by restricted valuations, meaning that for every $\\nu\\in\\mathcal{F}$, $h\\circ\\nu\\in\\mathcal{F}^{\\prime}$.\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n    \\mathcal{M} \\arrow{rr}{h}  &  & \\mathcal{M}^{\\prime}\\\\\n    & \\textbf{F}(\\Sigma, \\mathcal{V}) \\arrow{ul}{\\nu} \\arrow{ur}{h\\circ\\nu} & \n  \\end{tikzcd}\n  \\caption*{If $\\nu\\in\\mathcal{F}$, $h\\circ\\nu$ must be in $\\mathcal{F}^{\\prime}$}\n\\end{figure}\n\\end{enumerate}\n\n\\begin{theorem}\\label{category of RNmatrices}\nA class of RNmatrices $C$, endowed with the morphisms defined above, becomes a category $\\mathcal{C}$.\n\\end{theorem}\n\n\\begin{proof}\nTake morphisms $g:\\mathcal{M}\\rightarrow\\mathcal{M}^{\\prime}$ and $h:\\mathcal{M}^{\\prime}\\rightarrow\\mathcal{M}^{\\prime\\prime}$ as described above, and first of all we will show that their composition, as a composition of functions, returns again such a morphism.\n\\begin{enumerate}\n\\item Since $g$ and $h$ are both $\\Sigma$-homomorphisms of multialgebras, respectively from $\\mathcal{A}$ to $\\mathcal{A}^{\\prime}$ and from $\\mathcal{A}^{\\prime}$ to $\\mathcal{A}^{\\prime\\prime}$, it is clear that $h\\circ g$ is a $\\Sigma$-homomorphism from $\\mathcal{A}$ to $\\mathcal{A}^{\\prime\\prime}$.\n\\item Given that $g(D)\\subseteq D^{\\prime}$ and $h(D^{\\prime})\\subseteq D^{\\prime\\prime}$, we have that $h\\circ g(D)=h(g(D))\\subseteq h(D^{\\prime})\\subseteq D^{\\prime\\prime}$, meaning that $h\\circ g$ preserves designated elements.\n\\item Finally, we have that for any $\\nu\\in\\mathcal{F}$ and any $\\nu^{\\prime}\\in\\mathcal{F}^{\\prime}$, $g\\circ\\nu\\in\\mathcal{F}^{\\prime}$ and $h\\circ\\nu^{\\prime}\\in\\mathcal{F}^{\\prime\\prime}$; thus, for any $\\nu\\in\\mathcal{F}$, $g\\circ \\nu\\in\\mathcal{F}^{\\prime}$ and so $(h\\circ g)\\circ \\nu=h\\circ(g\\circ \\nu)\\in\\mathcal{F}^{\\prime\\prime}$, proving $h\\circ g$ is absorbed by restricted valuations.\n\\end{enumerate}\n\nThe associativity of the composition of these morphisms comes from the associativity of the composition of functions. The identity morphisms are precisely the identity functions, meaning that given $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$, the identity morphism in $\\mathcal{M}$ is the identity function on the universe of $\\mathcal{A}$, easily seem to be a homomorphism of multialgebras from $\\mathcal{A}$ to itself; that preserves designated elements; and is absorbed by restricted valuations.\n\\end{proof}\n\nNow, for a fixed da Costa's logic $C_{n}$, we take the class of RNmatrices $\\mathcal{RM}^{\\mathcal{B}}_{C_{n}}$ for $\\mathcal{B}$ a non-trivial Boolean algebra and construct its corresponding category $\\textbf{RSwap}_{C_{n}}$\\label{RSwapCn} as it was done in Theorem \\ref{category of RNmatrices} just above. To spell out our definition in this very important case, $\\textbf{RSwap}_{C_{n}}$ is the category: \n\\begin{enumerate}\n\\item with the class of (full) restricted swap structures $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$, for all Boolean algebras $\\mathcal{B}$, as objects; \n\\item as morphisms between $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}$ and $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{2}}$, all functions $\\varphi:B_{n}^{\\mathcal{B}_{1}}\\rightarrow B_{n}^{\\mathcal{B}_{2}}$ such that\n\\begin{enumerate}\n\\item $\\varphi$ is a $\\Sigma_{C}$-homomorphism between $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ and $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$, seem as $\\Sigma_{C}$-multialgebras;\n\\item for all $d\\in D_{n}^{\\mathcal{B}_{1}}$, $\\varphi(d)$ is in $D_{n}^{\\mathcal{B}_{2}}$;\n\\item for any $\\nu:\\textbf{F}(\\Sigma_{C}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{1}}$, $\\varphi\\circ\\nu:\\textbf{F}(\\Sigma_{C}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ is in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{2}}$.\n\\end{enumerate}\n\\end{enumerate}\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n    \\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}} \\arrow{rr}{\\varphi}  &  & \\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}\\\\\n    & \\textbf{F}(\\Sigma_{C}, \\mathcal{V}) \\arrow{ul}{\\nu} \\arrow{ur}{\\varphi\\circ\\nu} & \n  \\end{tikzcd}\n  \\caption*{If $\\nu\\in\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{1}}$, $\\varphi\\circ\\nu$ must be in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{2}}$}\n\\end{figure}\n\nOne important point to make is that demanding that a morphism of $\\textbf{RSwap}_{C_{n}}$ preserves designated elements is actually superfluous: we could prove this property from the fact that the morphisms in this category are absorbed by restricted valuations. However, the proof of this fact is somewhat involved, so it is easier to assume the preservation of designated elements as one of the defining characteristics of our morphisms.\n\nA first question regarding the definition of $\\textbf{RSwap}_{C_{n}}$ could be whether there are truly morphisms $\\varphi$ in it other than the identity ones.\n\n\\begin{proposition}\\label{homomorphisms of boolean algebras lead to morphisms}\nFor Boolean algebras $\\mathcal{B}_{1}$ and $\\mathcal{B}_{2}$ and a homomorphism $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ of Boolean algebras, $\\varphi:B_{n}^{\\mathcal{B}_{1}}\\rightarrow B_{n}^{\\mathcal{B}_{2}}$ such that, for every $z=(z_{1}, \\dotsc  , z_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}$,\n\\[\\varphi(z)_{i}=\\psi(z_{i}),\\quad\\text{for every}\\quad 1\\leq i\\leq n+1,\\]\nwhat we may write as $\\varphi(z)=(\\psi(z_{1}), \\dotsc  , \\psi(z_{n+1}))$, is a morphism between $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}$ and $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{2}}$ in $\\textbf{RSwap}_{C_{n}}$.\n\\end{proposition}\n\n\\begin{proof}\nNotice that if $z$ is a Boolean element of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$, then it is of the form $(a, {\\sim}a, 1, \\dotsc  , 1)$, and $\\varphi(z)=(\\psi(a), \\psi({\\sim}a), \\psi(1), \\dotsc  , \\psi(1))=(\\psi(a), {\\sim}\\psi(a), 1, \\dotsc  , 1)$, given $\\psi$ is a homomorphism; and since $\\psi(a)$ is an element of $\\mathcal{B}_{2}$, we see $\\varphi(z)$ is a Boolean element of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$, meaning $\\varphi$ preserves Boolean elements. \n\nTake elements $w$ and $z$ in $B_{n}^{\\mathcal{B}_{1}}$. First, suppose $w$ and $z$ are not both Boolean elements and $u\\in w\\tilde{\\#}z$, meaning that $u_{1}=w_{1}\\# z_{1}$: then \n\\[\\varphi(u)=(\\psi(u_{1}), \\psi(u_{2}), \\dotsc  , \\psi(u_{n+1}))\\quad\\text{equals}\\quad(\\psi(w_{1})\\#\\psi(z_{1}), \\psi(u_{2}), \\dotsc  , \\psi(u_{n+1})),\\]\nsince $\\psi$ is a homomorphism, and since $\\varphi(w)_{1}=\\psi(w_{1})$ and $\\varphi(z)_{1}=\\psi(z_{1})$, we obtain that $\\varphi(u)\\in \\varphi(w)\\tilde{\\#}\\varphi(z)$. Now, if $w$ and $z$ are both Boolean elements and $u\\in w\\tilde{\\#}z$, then $u_{1}=w_{1}\\# z_{1}$ and $u$ is a Boolean element, and from what we saw above we find that $\\varphi(u)_{1}=\\varphi(w)_{1}\\#\\varphi(z)_{1}$ and that $\\varphi(w)$, $\\varphi(z)$ and $\\varphi(u)$ are all Boolean elements, meaning $\\varphi(u)\\in\\varphi(w)\\tilde{\\#}\\varphi(z)$.\n\nNow, for any $z\\in B_{n}^{\\mathcal{B}_{1}}$ and $w\\in\\tilde{\\neg}z$, $w_{1}=z_{2}$ and $w_{2}\\leq z_{1}$. Then $\\varphi(w)=(\\psi(w_{1}), \\dotsc  , \\psi(w_{n+1}))$, and analogously for $\\varphi(z)$, meaning $\\varphi(w)_{1}=\\psi(w_{1})=\\psi(z_{2})=\\varphi(z)_{2}$ and $\\varphi(w)_{2}=\\psi(w_{2})\\leq \\psi(z_{1})=\\varphi(z)_{1}$,\\footnote{One should remember that homomorphisms of Boolean algebras preserve order: first of all, in a Boolean algebra $a\\leq b$ iff $a=a\\wedge b$ iff $b=a\\vee b$; given a homomorphism $\\psi:\\mathcal{B}_{1}\\rightarrow \\mathcal{B}_{2}$, if $a\\leq b$, $b=a\\vee b$ and so $\\psi(b)=\\psi(a\\vee b)=\\psi(a)\\vee\\psi(b)$, meaning that $\\psi(a)\\leq \\psi(b)$.}and therefore $\\varphi(w)\\in\\tilde{\\neg}\\varphi(z)$. With this, $\\varphi$ is a $\\Sigma_{\\textbf{C}}$-homomorphism between $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ and $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$.\n\nGiven a designated element $z=(1, z_{2}, \\dotsc  , z_{n+1})$ of $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}$, we get that $\\varphi(z)=(\\psi(1), \\psi(z_{2}), \\dotsc  , \\psi(z_{n+1}))=(1, \\psi(z_{2}), \\dotsc  , \\psi(z_{n+1}))$ is also designated, meaning that $\\varphi$ preserves designated elements.\n\nNow, a $\\Sigma_{\\textbf{C}}$-homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ is in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{1}}$ when, for all formulas $\\alpha$ and $\\beta$:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\wedge\\neg\\alpha)_{2}=\\nu(\\alpha)_{3}$;\n\\item $\\nu(\\alpha^{1})=(\\nu(\\alpha)_{3}, \\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}, \\nu(\\alpha)_{4}, \\dotsc  , \\nu(\\alpha)_{n+1}, {\\sim}(\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}))$;\n\\item and $\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})\\in D_{n}^{\\mathcal{B}_{1}}$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$.\n\\end{enumerate}\n\nQuite clearly $\\varphi\\circ\\nu:\\textbf{F}(\\Sigma_{\\textbf{C}}, \\mathcal{V})\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ is still a $\\Sigma_{\\textbf{C}}$-homomorphism.\n\n\\begin{enumerate}\n\\item We have $\\varphi(\\nu(\\alpha\\wedge\\neg\\alpha))_{2}=\\psi(\\nu(\\alpha\\wedge\\neg\\alpha)_{2})$, which equals, given $\\nu$ is in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{1}}$, $\\psi(\\nu(\\alpha)_{3})=\\varphi(\\nu(\\alpha))_{3}$.\n\\item See that $\\varphi(\\nu(\\alpha^{1}))$ equals\n\\[\\big(\\psi(\\nu(\\alpha)_{3}), \\psi(\\nu(\\alpha)_{1}\\wedge\\nu(\\alpha)_{2}), \\psi(\\nu(\\alpha)_{4}), \\dotsc  , \\psi(\\nu(\\alpha)_{n+1}), \\psi({\\sim}(\\bigwedge_{i=1}^{n+1}\\nu(\\alpha)_{i}))\\big)=\\]\n\\[\\big(\\psi(\\nu(\\alpha)_{3}), \\psi(\\nu(\\alpha)_{1})\\wedge\\psi(\\nu(\\alpha)_{2}), \\psi(\\nu(\\alpha)_{4}), \\dotsc  , \\psi(\\nu(\\alpha)_{n+1}), {\\sim}(\\bigwedge_{i=1}^{n+1}\\psi(\\nu(\\alpha)_{i}))\\big)=\\]\n\\[\\big(\\varphi(\\nu(\\alpha))_{3}, \\varphi(\\nu(\\alpha))_{1}\\wedge\\varphi(\\nu(\\alpha))_{2}, \\varphi(\\nu(\\alpha))_{4}, \\dotsc  , \\varphi(\\nu(\\alpha))_{n+1}, {\\sim}(\\bigwedge_{i=1}^{n+1}\\varphi(\\nu(\\alpha))_{i})\\big).\\]\n\\item Since, as we already proved, $\\varphi$ preserves designated elements, $\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)})\\in D_{n}^{\\mathcal{B}_{1}}$, for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, implies that $\\varphi(\\nu((\\alpha^{(n)}\\wedge\\beta^{(n)})\\rightarrow(\\alpha\\#\\beta)^{(n)}))\\in D_{n}^{\\mathcal{B}_{2}}$ for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$.\n\\end{enumerate}\nThis finishes proving that $\\varphi\\circ\\nu$ lies in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{2}}$.\n\n\\end{proof}\n\nNotice that, in the spirit of the last proposition, the identity morphism $Id_{\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}}$ of $\\textbf{RSwap}_{C_{n}}$ on $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ can be written, for an arbitrary $z=(z_{1}, \\dotsc  , z_{n+1})$ in $B_{n}^{\\mathcal{B}}$, as\n\\[Id_{\\mathcal{A}_{C_{n}}^{\\mathcal{B}}}(z)=(z_{1}, \\dotsc  , z_{n+1})=(Id_{\\mathcal{B}}(z_{1}), \\dotsc  , Id_{\\mathcal{B}}(z_{n+1})),\\]\nfor $Id_{\\mathcal{B}}:\\mathcal{B}\\rightarrow\\mathcal{B}$ the identity homomorphism on $\\mathcal{B}$, and therefore the identity morphisms in $\\textbf{RSwap}_{C_{n}}$ are, too, of the form described in the previous proposition.\n\nA second question regarding the definition of $\\textbf{RSwap}_{C_{n}}$ is whether there are morphisms in it other than those of the form described in Proposition \\ref{homomorphisms of boolean algebras lead to morphisms}. The answer, in this case, is no, and in the following section we prove exactly that.\n\n\n\n\n\n\n\n\n\n\\subsection{Morphisms of $\\textbf{RSwap}_{C_{n}}$}\n\nFor any function $\\varphi:B_{n}^{\\mathcal{B}_{1}}\\rightarrow B_{n}^{\\mathcal{B}_{2}}$ we will write, for any $z\\in B_{n}^{\\mathcal{B}_{1}}$, $\\varphi(z)$ as $(\\varphi_{1}(z), \\dotsc  , \\varphi_{n+1}(z))$: here, $\\varphi_{i}$, for any $1\\leq i\\leq n+1$, is a function from $B_{n}^{\\mathcal{B}_{1}}$ to $\\mathcal{B}_{2}$ obtained by composing the $i$-th projection $\\pi_{i}$\\label{pii} of $B_{n}^{\\mathcal{B}_{1}}$ with $\\varphi$.\n\nWith this, we can say that $\\varphi$ is a $\\Sigma_{\\textbf{C}}$-homomorphism from $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ to $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ if, and only if, for all snapshots $w, z\\in B_{n}^{\\mathcal{B}_{1}}$, and $u\\in w\\tilde{\\#}z$ (for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$) and $v\\in\\tilde{\\neg}z$, \n\\[\\varphi_{1}(u)=\\varphi_{1}(w)\\#\\varphi_{1}(z),\\quad\\text{and}\\quad\\varphi_{1}(v)=\\varphi_{2}(z)\\quad\\text{and}\\quad\\varphi_{2}(v)\\leq\\varphi_{1}(z),\\]\nwhat is equivalent to $\\varphi(u)\\in \\varphi(w)\\tilde{\\#}\\varphi(z)$ and $\\varphi(v)\\in\\tilde{\\neg}\\varphi(z)$ once one considers the definitions of $\\tilde{\\#}$ and $\\tilde{\\neg}$.\n\nSo, we would like to study the function $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$, given by $\\psi(a)=$\\\\$\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))$ for any $a$ in $\\mathcal{B}_{1}$, in the case that $\\varphi$ is actually a $\\Sigma_{\\textbf{C}}$-homomorphism. Take a snapshot $z=(z_{1}, \\dotsc  , z_{n+1})$ of $B_{n}^{\\mathcal{B}_{1}}$ and consider $z^{*}=(z_{1}, {\\sim}z_{1}, 1, \\dotsc  , 1)$: then $\\psi(z_{1})=\\varphi(z^{*})$ by definition of $\\psi$. Furthermore, recalling that $t^{n}_{0}=(1, 1, 0, 1, \\dotsc  , 1)$ is a non-Boolean snapshot of $B_{n}^{\\mathcal{B}}$, for any non-trivial Boolean algebra $\\mathcal{B}$, we find that\n\\[z\\tilde{\\wedge}t^{n}_{0}=z^{*}\\tilde{\\wedge}t^{n}_{0}=\\{w=(w_{1}, \\dotsc  , w_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}: w_{1}=z_{1}\\},\\]\nand thus $z, z^{*}\\in z\\tilde{\\wedge}t^{n}_{0}=z^{*}\\tilde{\\wedge}t^{n}_{0}$, since the first coordinates of both are precisely $z_{1}$; given $\\varphi$ is a homomorphism, $z\\in z\\tilde{\\wedge}t^{n}_{0}$ implies that $\\varphi_{1}(z)=\\varphi_{1}(z)\\wedge\\varphi_{1}(t^{n}_{0})$, while $z^{*}\\in z\\tilde{\\wedge}t^{n}_{0}$ implies that $\\varphi_{1}(z^{*})=\\varphi_{1}(z)\\wedge\\varphi_{1}(t^{n}_{0})$, leading us to $\\varphi_{1}(z)=\\varphi_{1}(z^{*})$ and then to $\\varphi_{1}(z)=\\psi(z_{1})$. In other words, the function $\\varphi_{1}$ depends exclusively on the first coordinate of a snapshot.\n\nFurthermore, for any $a, b\\in\\mathcal{B}_{1}$, it is true for the snapshots $(a, {\\sim}a, 1, \\dotsc  , 1)$ and $(b, {\\sim}b, 1, \\dotsc  , 1)$ in $B_{n}^{\\mathcal{B}_{1}}$ that \n\\[(a, {\\sim}a, 1, \\dotsc  , 1)\\tilde{\\#}(b, {\\sim}b, 1, \\dotsc  , 1)=\\{(a\\#b, {\\sim}a\\#b, 1, \\dotsc  , 1)\\},\\]\nfor any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$. This way, taking into consideration that $\\varphi$ is a homomorphism and $\\varphi_{1}(z)=\\psi(z_{1})$, $\\psi(a\\#b)=\\psi(a)\\#\\psi(b)$. This means that $\\psi$ is almost a homomorphism, and we will prove further ahead that it actually is a homomorphism of Boolean algebras.\n\nFor now,  let us also define the function $\\theta:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ by $\\theta(a)=\\varphi_{2}(({\\sim}a, a, 1, \\dotsc  , 1))$. For an arbitrary snapshot $z=(z_{1}, \\dotsc  , z_{n+1})$ in $B_{n}^{\\mathcal{B}_{1}}$, consider \n\\[z^{*}=({\\sim}z_{2}, z_{2}, 1, \\dotsc  , 1)\\quad\\text{and}\\quad z^{\\prime}=(z_{2}, {\\sim}z_{2}, 1, \\dotsc  , 1),\\]\nand by definition of $\\theta$ we have that $\\varphi_{2}(z^{*})=\\theta(z_{2})$. Since $z$ is in $B_{n}^{\\mathcal{B}_{1}}$, we have that $z_{1}\\vee z_{2}=1$ and so ${\\sim}z_{2}\\leq z_{1}$, implying that $z^{\\prime}\\in\\tilde{\\neg}z$ and $z^{\\prime}\\in\\tilde{\\neg}z^{*}$. \\footnote{Actually $\\tilde{\\neg}z^{*}=\\{z^{\\prime}\\}$ and vice-versa, since both are Boolean snapshots with complementary first-coordinates.} Using, once again, that $\\varphi$ is a homomorphism, $\\varphi(z^{\\prime})\\in \\tilde{\\neg}\\varphi(z)$ and $\\varphi(z^{\\prime})\\in\\tilde{\\neg}\\varphi(z^{*})$, leading to $\\varphi_{1}(z^{\\prime})=\\varphi_{2}(z)$ and $\\varphi_{1}(z^{\\prime})=\\varphi_{2}(z^{*})$, \\textit{id est} $\\varphi_{2}(z)=\\varphi_{2}(z^{*})$. This leads us to $\\varphi_{2}(z)=\\theta(z_{2})$, meaning $\\varphi_{2}$ also depends exclusively of one coordinate, in this case the second.\n\nFinally, we may prove, with our only hypothesis being that $\\varphi$ is a homomorphism, that $\\psi=\\theta$. To see this, take any $a\\in\\mathcal{B}_{1}$, and the snapshots $z=(a, {\\sim}a, 1, \\dotsc  , 1)$ and $z^{*}=({\\sim}a, a, 1, \\dotsc  , 1)$ of $B_{n}^{\\mathcal{B}_{1}}$, with the property that $\\tilde{\\neg}z=\\{z^{*}\\}$ and $\\tilde{\\neg}z^{*}=\\{z\\}$. Then $\\varphi(z)\\in \\tilde{\\neg}\\varphi(z^{*})$, so $\\varphi_{1}(z)=\\varphi_{2}(z^{*})$ (and also $\\varphi_{2}(z)\\leq \\varphi_{2}(z^{*})$) and therefore $\\psi(a)=\\theta(a)$, using that $\\varphi_{1}(z)=\\psi(a)$ and $\\varphi_{2}(z^{*})=\\theta(a)$. We may summarize the results so far in the following theorem.\n\n\\begin{theorem}\nIf $\\varphi:\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ is a $\\Sigma_{\\textbf{C}}$-homomorphism, there exists a function $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ such that \n\\[\\varphi_{1}(z)=\\psi(z_{1})\\quad\\text{and}\\quad\\varphi_{2}(z)=\\psi(z_{2}),\\]\nfor all snapshots $z=(z_{1}, \\dotsc  , z_{n+1})$ of $B_{n}^{\\mathcal{B}_{1}}$, and which satisfies, for all $a, b\\in \\mathcal{B}_{1}$ and $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\psi(a\\#b)=\\psi(a)\\#\\psi(b)$.\n\\end{theorem}\n\nNow we will demand that $\\varphi:B_{n}^{\\mathcal{B}_{1}}\\rightarrow B_{n}^{\\mathcal{B}_{2}}$ be, not only a homomorphism from $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ to $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$, but a morphism of the category $\\textbf{RSwap}_{C_{n}}$ from $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}$ to $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{2}}$, so it also preserves designated elements and is absorbed by restricted valuations.\n\nSo, take a snapshot $z=(1, z_{2}, \\dotsc  , z_{n+1})\\in D_{n}^{\\mathcal{B}_{1}}$: since we must have, by hypothesis over $\\varphi$, $\\varphi(z)\\in D_{n}^{\\mathcal{B}_{2}}$, and \n\\[\\varphi(z)=(\\varphi_{1}(z), \\dotsc  , \\varphi_{n+1}(z))=(\\psi(1), \\psi(z_{2}), \\varphi_{3}(z), \\dotsc  , \\varphi_{n+1}(z)),\\]\nwe obtain that $\\psi(1)=1$. Furthermore, for any formula $\\alpha$ in the language of $C_{n}$ and valuation $\\nu$ in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$, Proposition \\ref{Value of strong negation} shows that $\\nu(\\alpha\\wedge\\neg\\alpha\\wedge\\alpha^{(n)})=F_{n}$; since $\\varphi$ is absorbed by restricted valuations, for any $\\nu\\in\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{1}}$ we have $\\varphi\\circ\\nu\\in\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{2}}$ and so, for any formula $\\alpha$ of $C_{n}$\n\\[F_{n}=\\varphi\\circ\\nu(\\alpha\\wedge\\neg\\alpha\\wedge\\alpha^{(n)})=\\varphi(F_{n})=(\\psi(0), \\psi(1), \\varphi_{3}(F_{n}), \\dotsc  , \\varphi_{n+1}(F_{n})),\\]\nmeaning that we also have $\\psi(0)=0$. Since, for any $a, b\\in\\mathcal{B}_{1}$ and $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\psi(a\\#b)=\\psi(a)\\#\\psi(b)$, $\\psi(1)=1$ and $\\psi(0)=0$, we actually have that $\\psi$ is a homomorphism of Boolean algebras, as we had previously promised: given that ${\\sim}a=a\\rightarrow 0$, we have \n\\[\\psi({\\sim}a)=\\psi(a\\rightarrow 0)=\\psi(a)\\rightarrow\\psi(0)={\\sim}\\psi(a).\\]\n\nBut the fact that $\\varphi$ is absorbed by restricted valuations allows to prove something even stronger: remember that, for any $\\nu\\in\\mathcal{F}^{\\mathcal{B}}_{C_{n}}$ and formula $\\alpha$ of $C_{n}$, $\\nu(\\alpha^{k})_{1}=\\nu(\\alpha)_{k+2}$ for $1\\leq k\\leq n-1$, what we proved back in Lemma \\ref{Finding the coordinates} for $2\\leq k\\leq n-1$ and is obvious by the definition of $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ for $k=1$. Given a $z$ in $B_{n}^{\\mathcal{B}_{1}}$, for any propositional variable $p$ we can find a restricted valuation in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{1}}$ with $\\nu(p)=z$, by proceeding as in Section \\ref{RM is not trivial} if necessary; so $\\nu(p^{k})_{1}=z_{k+2}$.\n\nSince $\\varphi\\circ\\nu$ must be in $\\mathcal{F}_{C_{n}}^{\\mathcal{B}_{2}}$, we have that\n\\[\\varphi\\circ\\nu(p^{k})_{1}=\\varphi\\circ\\nu(p)_{k+2}=\\varphi(z)_{k+2}=\\varphi_{k+2}(z),\\]\nand at the same time,\n\\[\\varphi\\circ\\nu(p^{k})_{1}=\\psi(\\nu(p^{k})_{1})=\\psi(z_{k+2}),\\]\nimplying that $\\varphi_{k+2}(z)=\\psi(z_{k+2})$, for every $1\\leq k\\leq n-1$. We get then the following theorem.\n\n\\begin{theorem}\\label{Classifying morphisms in RSwap}\nIf $\\varphi:\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}\\rightarrow\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{2}}$ is a morphism of $\\textbf{RSwap}_{C_{n}}$, there exists a homomorphism of Boolean algebras $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ such that, for all snapshots $z\\in B_{n}^{\\mathcal{B}_{1}}$,\n\\[\\varphi_{i}(z)=\\psi(z_{i}),\\quad\\text{for all}\\quad1\\leq i\\leq n+1.\\]\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\textbf{BA}$ and $\\textbf{RSwap}_{C_{n}}$ are isomorphic}\\label{BA and RSwap}\n\n\\begin{proposition}\nThe set $Boo_{n}^{\\mathcal{B}}$ of Boolean elements of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ is a Boolean algebra isomorphic to $\\mathcal{B}$ when we define, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, and $a$ and $b$ elements of $\\mathcal{B}$,\n\\[(a, {\\sim}a, 1, \\dotsc  , 1)\\#(b, {\\sim}b, 1, \\dotsc  , 1)=(a\\#b, {\\sim}(a\\#b), 1, \\dotsc  , 1),\\]\n\\[{\\sim}(a, {\\sim}a, 1, \\dotsc  , 1)=({\\sim}a, a, 1, \\dotsc  , 1),\\]\n$\\top=(1, 0, 1, \\dotsc  , 1)$ and $\\bot=(0, 1, 1, \\dotsc  , 1)$\n\\end{proposition}\n\n\\begin{proof}\nConsider the map $\\rho:B\\rightarrow Boo_{n}^{\\mathcal{B}}$, for $B$ the universe of $\\mathcal{B}$, given by $\\rho(a)=$\\\\$(a, {\\sim}a, 1, \\dotsc  , 1)$. Then, for $a, b\\in B$:\n\\begin{enumerate}\n\\item for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, \n\\[\\rho(a\\#b)=(a\\#b, {\\sim}(a\\#b), 1, \\dotsc  , 1)=(a, {\\sim}a, 1, \\dotsc  , 1)\\#(b, {\\sim}b, 1, \\dotsc  , 1)=\\rho(a)\\#\\rho(b);\\]\n\\item $\\rho({\\sim}a)=({\\sim}a, {\\sim}{\\sim}a, 1, \\dotsc  , 1)=({\\sim}a, a, 1, \\dotsc  , 1)={\\sim}(a, {\\sim}a, 1, \\dotsc  , 1)={\\sim}\\rho(a)$;\n\\item $\\rho(\\top)=(1, 0, 1, \\dotsc  , 1)=\\top$;\n\\item $\\rho(\\bot)=(0, 1, 1, \\dotsc  , 1)=\\bot$.\n\\end{enumerate}\nFurthermore, $\\rho$ is clearly both injective and surjective, being therefore an isomorphism.\n\\end{proof}\n\nNotice that the operations we give to $Boo_{n}^{\\mathcal{B}}$ are the natural choice of Boolean algebra structure for this set, taking into consideration the multioperations on $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$, since for elements $a$ and $b$ of $\\mathcal{B}$, and $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, \n\\[\\rho(a)\\tilde{\\#}\\rho(b)=\\{\\rho(a)\\#\\rho(b)\\}\\quad\\text{and}\\quad\\tilde{\\neg}\\rho(a)=\\{{\\sim}\\rho(a)\\}.\\]\n\nConsider now the category $\\textbf{BA}$\\label{BA} of non-degenerate Boolean algebras\\index{Boolean algebra, Non-degenerate} (that is, Boolean algebras where $0\\neq 1$), with homomorphisms of Boolean algebras as morphisms, and the functors $\\mathcal{A}_{n}:\\textbf{BA}\\rightarrow\\textbf{RSwap}_{C_{n}}$\\label{An}, which takes\n\\begin{enumerate}\n\\item a Boolean algebra $\\mathcal{B}$ to $\\mathcal{A}_{n}\\mathcal{B}=\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$;\n\\item a homomorphism of Boolean algebras $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ to the morphism $\\mathcal{A}_{n}\\psi:\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ such that, for $z=(z_{1}, \\dotsc  , z_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}$, $\\mathcal{A}_{n}\\psi(z)_{i}=\\psi(z_{i})$, for every $i\\in\\{1, \\dotsc  , n+1\\}$;\n\\end{enumerate}\nand $\\textbf{Boo}_{n}$\\label{tBoon}, which takes\n\\begin{enumerate}\n\\item the restricted swap structure $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ to the Boolean algebra $\\mathcal{B}$ (or, what we saw to be equivalent, $Boo_{n}^{\\mathcal{B}}$ with its natural Boolean algebra structure);\n\\item a morphism $\\varphi:\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ to the homomorphism of Boolean algebras $\\textbf{Boo}_{n}\\varphi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ such that, for $a$ an element of $\\mathcal{B}_{1}$, $\\textbf{Boo}_{n}\\varphi(a)=\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))_{1}$.\n\\end{enumerate}\n\n\\begin{proposition}\nAs described, $\\mathcal{A}_{n}$ is, indeed, a functor.\n\\end{proposition}\n\n\\begin{proof}\nAs we proved in Theorem \\ref{Classifying morphisms in RSwap}, for a homomorphism of Boolean algebras $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$, $\\mathcal{A}_{n}\\psi:B_{n}^{\\mathcal{B}_{1}}\\rightarrow B_{n}^{\\mathcal{B}_{2}}$ such that, for an arbitrary $z=(z_{1}, \\dotsc  , z_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}$, $\\mathcal{A}_{n}\\psi(z)_{i}=\\psi(z_{i})$, for $1\\leq i\\leq n+1$, is indeed a morphism in $\\textbf{RSwap}_{C_{n}}$.\n\nNow, for a second homomorphism of Boolean algebras $\\theta:\\mathcal{B}_{2}\\rightarrow\\mathcal{B}_{3}$, one has\n\\[\\mathcal{A}_{n}(\\theta\\circ\\psi)(z)=(\\theta\\circ\\psi(z_{1}), \\dotsc  , \\theta\\circ\\psi(z_{n+1}))=\\mathcal{A}_{n}\\theta((\\psi(z_{1}), \\dotsc  , \\psi(z_{n+1}))=\\]\n\\[\\mathcal{A}_{n}\\theta\\circ\\mathcal{A}_{n}\\psi(z);\\]\nfurthermore, for the identity $Id_{\\mathcal{B}}:\\mathcal{B}\\rightarrow\\mathcal{B}$ of $\\mathcal{B}$, $\\mathcal{A}_{n}Id_{\\mathcal{B}}:\\mathcal{A}_{C_{n}}^{\\mathcal{B}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ satisfies that, for $z\\in B_{n}^{\\mathcal{B}}$, \n\\[\\mathcal{A}_{n}Id_{\\mathcal{B}}(z)=(Id_{\\mathcal{B}}(z_{1}), \\dotsc  , Id_{\\mathcal{B}}(z_{n+1}))=(z_{1}, \\dotsc  , z_{n+1})=z,\\]\nand therefore is precisely the identity on $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$.\n\\end{proof}\n\n\\begin{proposition}\nAs described, $\\textbf{Boo}_{n}$ is, indeed, a functor.\n\\end{proposition}\n\n\\begin{proof}\nSince, for any morphism $\\varphi:\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$, there exists a homomorphism $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ such that, for any $z=(z_{1}, \\dotsc  , z_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}$, $\\varphi(z)_{i}=\\psi(z_{i})$, for any $1\\leq i\\leq n+1$, one sees that \n\\[\\textbf{Boo}_{n}\\varphi(a)=\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))_{1}=\\psi(a),\\]\nand so $\\textbf{Boo}_{n}\\varphi$ is indeed a homomorphism of Boolean algebras.\n\nFor a second morphism $\\eta:\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}\\rightarrow \\mathcal{A}_{C_{n}}^{\\mathcal{B}_{3}}$ and a homomorphism $\\theta:\\mathcal{B}_{2}\\rightarrow\\mathcal{B}_{3}$ such that, for every $w\\in B_{n}^{\\mathcal{B}_{2}}$, $\\eta(w)_{i}=\\theta(w_{i})$, consider an element $a$ of $\\mathcal{B}_{1}$: then\n\\[\\textbf{Boo}_{n}(\\eta\\circ\\varphi)(a)=\\eta(\\varphi((a, {\\sim}a, 1, \\dotsc  , 1)))_{1}=\\eta_{1}(\\varphi((a, {\\sim}a, 1, \\dotsc  , 1)))=\\]\n\\[\\theta(\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))_{1})=\\theta(\\varphi_{1}((a, {\\sim}a, 1, \\dotsc  , 1)))=\\theta(\\psi(a))=\\theta(\\textbf{Boo}_{n}\\varphi(a))=\\]\n\\[\\textbf{Boo}_{n}\\eta\\circ\\textbf{Boo}_{n}\\varphi(a).\\]\n\nFinally, consider the identity homomorphism $Id_{\\mathcal{A}_{C_{n}}^{\\mathcal{B}}}:\\mathcal{A}_{C_{n}}^{\\mathcal{B}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$: for an $a$ in $\\mathcal{B}$,\n\\[\\textbf{Boo}_{n}Id_{\\mathcal{A}_{C_{n}}^{\\mathcal{B}}}(a)=Id_{\\mathcal{A}_{C_{n}}^{\\mathcal{B}}}((a, {\\sim}a, 1, \\dotsc  , 1))_{1}=a,\\]\nproving $\\textbf{Boo}_{n}Id_{\\mathcal{A}_{C_{n}}^{\\mathcal{B}}}$ is the identity on $\\mathcal{B}$.\n\\end{proof}\n\n\\begin{theorem}\n$\\textbf{Boo}_{n}\\circ\\mathcal{A}_{n}=Id_{\\textbf{BA}}$.\n\\end{theorem}\n\n\\begin{proof}\nA Boolean algebra $\\mathcal{B}$ is taken by $\\mathcal{A}_{n}$ into $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$, and $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ is taken to $\\mathcal{B}$ again by $\\textbf{Boo}_{n}$. This means $\\textbf{Boo}_{n}\\circ\\mathcal{A}_{n}$ is the identity on objects.\n\nNow, for Boolean algebras $\\mathcal{B}_{1}$ and $\\mathcal{B}_{2}$, a homomorphism $\\psi:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$, and an element $a$ of $\\mathcal{B}_{1}$, let $\\varphi=\\mathcal{A}_{n}\\psi$:\n\\[(\\textbf{Boo}_{n}\\circ\\mathcal{A}_{n})\\psi(a)=\\textbf{Boo}_{n}\\varphi(a)=\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))_{1}=\\psi(a),\\]\nimplying $\\textbf{Boo}_{n}\\circ\\mathcal{A}_{n}$ is also identical when applied to morphisms.\n\\end{proof}\n\n\\begin{theorem}\n$\\mathcal{A}_{n}\\circ\\textbf{Boo}_{n}=Id_{\\textbf{RSwap}_{C_{n}}}$.\n\\end{theorem}\n\n\\begin{proof}\n$\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ is taken by $\\textbf{Boo}_{n}$ to $\\mathcal{B}$, which is taken back to $\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ by $\\mathcal{A}_{n}$, giving us the identity on objects.\n\nNow, for restricted swap structures $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ and $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$, a morphism $\\varphi:\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}\\rightarrow\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$ and an element $z=(z_{1}, \\dotsc  , z_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}$, let us denote $\\textbf{Boo}_{n}\\varphi$ by $\\psi$: \n\\[\\mathcal{A}_{n}\\circ\\textbf{Boo}_{n}\\varphi(z)=\\mathcal{A}_{n}\\psi(z)=(\\psi(z_{1}), \\dotsc  , \\psi(z_{n+1}));\\]\nsince $\\textbf{Boo}_{n}\\varphi=\\psi$, for any $a$ in $\\mathcal{B}_{1}$, $\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))_{1}=\\psi(a)$. We also know that there exists a homomorphism $\\theta:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ such that, for any $z\\in B_{n}^{\\mathcal{B}_{1}}$, $\\varphi(z)_{i}=\\theta(z_{i})$ for $1\\leq i\\leq n+1$.\n\nFrom the fact that $\\theta(a)=\\varphi((a, {\\sim}a, 1, \\dotsc  , 1))_{1}=\\psi(a)$, we obtain $\\theta=\\psi$, and so\n\\[\\varphi(z)=(\\psi(z_{1}), \\dotsc  , \\psi(z_{n+1}))=\\mathcal{A}_{n}\\circ\\textbf{Boo}_{n}\\varphi(z),\\]\nimplying finally that $\\mathcal{A}_{n}\\circ\\textbf{Boo}_{n}$ maintains morphisms fixed as well.\n\\end{proof}\n\nNow that we have $\\textbf{RSwap}_{C_{n}}$ is isomorphic to $\\textbf{BA}$, we may use this isomorphism to bring some results from the latter category into the former: of course, $\\textbf{BA}$ is one of the better known categories, so there are many such results. Let us start by reminding that every atomic, complete Boolean algebra is isomorphic to some power of the two-valued Boolean algebra $\\textbf{2}$; in particular, an atomic, complete Boolean algebra with $\\kappa$ atoms is isomorphic to precisely $\\textbf{2}^{\\kappa}$, where $\\kappa$ may be a finite or infinite cardinal.\n\nSince every finite Boolean algebra is forcibly both atomic and complete, and from the results of Section \\ref{Counting snapshots} $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ is finite if and only if $\\mathcal{B}$ is finite, we find a first representation result in $\\textbf{RSwap}_{C_{n}}$.\n\n\\begin{corollary}\nEvery finite $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ is isomorphic to a power of $\\mathcal{RM}_{C_{n}}$.\n\\end{corollary}\n\nAnother representation theorem on Boolean algebras, a stronger one, states that every Boolean algebra is isomorphic to a field of sets, that is, a Boolean subalgebra of the powerset Boolean algebra of a given set; since every powerset is a complete and atomic Boolean algebra, and is therefore isomorphic to some power $\\textbf{2}^{\\kappa}$ of $\\textbf{2}$, it follows that every Boolean algebra is isomorphic to a subalgebra of a power of $\\textbf{2}$.\n\nNow, we still haven't defined what it means for an RNmatrix to be a \"subRNmatrix\" of another, although we could merely translate what this means from $\\textbf{BA}$ into $\\textbf{RSwap}_{C_{n}}$; we chose to give a more general approach instead. So, given RNmatrices $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ and $\\mathcal{M}^{\\prime}=(\\mathcal{A}^{\\prime}, D^{\\prime}, \\mathcal{F}^{\\prime})$ over the same signature, we say that $\\mathcal{M}^{\\prime}$ is a subRNmatrix\\index{SubRNmatrix} of $\\mathcal{M}$ if:\n\\begin{enumerate}\n\\item $\\mathcal{A}^{\\prime}$ is a submultialgebra of $\\mathcal{A}$;\n\\item $D^{\\prime}$ is a subset of $D$, and\n\\item $\\{j\\circ\\nu: \\nu\\in\\mathcal{F}^{\\prime}\\}\\subseteq \\mathcal{F}$, for $j$ the inclusion of the universe of $\\mathcal{A}^{\\prime}$ into the universe of $\\mathcal{A}$.\n\\end{enumerate}\nIn other words, $\\mathcal{M}^{\\prime}$ is a subRNmatrix of $\\mathcal{M}$ if $j$ is a morphism on the underlying category of RNmatrices.\n\nNow, we state that in $\\textbf{RSwap}_{C_{n}}$ the concept just described of a subRNmatrix corresponds exactly to what one would obtain by translating the notion of being a subalgebra from $\\textbf{BA}$, that is, $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}$ is a subRNmatrix of $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{2}}$ if and only if $\\mathcal{B}_{1}$ is a subalgebra of $\\mathcal{B}_{2}$.\n\nIn one direction, by assuming that $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{1}}$ is a subRNmatrix of $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}_{2}}$, we have that $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ is a submultialgebra of $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$, so for every $a\\in\\mathcal{B}_{1}$, given that $(a, {\\sim}a, 1, \\dotsc  , 1)$ is an element of $B_{n}^{\\mathcal{B}_{1}}$, we have that it must also be an element of $B_{n}^{\\mathcal{B}_{2}}$, and therefore $a\\in\\mathcal{B}_{2}$. Since, for any $a, b\\in\\mathcal{B}_{1}$ and $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, \n\\[(a, {\\sim}a, 1, \\dotsc  , 1)\\tilde{\\#}(b, {\\sim}b, 1, \\dotsc  , 1)=\\{(a\\#b, {\\sim}(a\\#b), 1, \\dotsc  , 1)\\]\nand $\\tilde{\\neg}(a, {\\sim}a, 1, \\dotsc  , 1)=\\{({\\sim}a, a, 1, \\dotsc  , 1)\\}$, and given that the operations $\\tilde{\\#}$ and $\\tilde{\\neg}$ in $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{1}}$ are the same as those in $\\mathcal{A}_{C_{n}}^{\\mathcal{B}_{2}}$, we obtain that the operations in $\\mathcal{B}_{1}$ are the same as those in $\\mathcal{B}_{2}$, proving the former is a Boolean subalgebra of the latter.\n\nReciprocally, suppose $\\mathcal{B}_{1}$ is a Boolean subalgebra of $\\mathcal{B}_{2}$: given a snapshot $z=(z_{1}, \\dotsc  , z_{n+1})\\in B_{n}^{\\mathcal{B}_{1}}$, we have that, fist of all, $z_{1}, \\dotsc  , z_{n+1}\\in \\mathcal{B}_{1}$, implying that $z_{1}, \\dotsc  , z_{n+1}\\in \\mathcal{B}_{2}$; second, since $(\\bigwedge_{i=1}^{k}z_{i})\\vee z_{k+1}=1$, for all $1\\leq k\\leq n$, in $\\mathcal{B}_{1}$, and since the operations in $\\mathcal{B}_{2}$, when restricted to the universe of $\\mathcal{B}_{1}$, coincide to the operations in $\\mathcal{B}_{2}$, we obtain that $(\\bigwedge_{i=1}^{k}z_{i})\\vee z_{k+1}=1$, now in $\\mathcal{B}_{2}$. With this, $z$ is a snapshot of $B_{n}^{\\mathcal{B}_{2}}$.\n\nSo we can consider the inclusion $j:B_{n}^{\\mathcal{B}_{1}}\\rightarrow B_{n}^{\\mathcal{B}_{2}}$, and it is easy to prove that it is a morphism of $\\textbf{RSwap}_{C_{n}}$: after all, it can be written as $j(z)=(i(z_{1}), \\dotsc  , i(z_{n+1}))$ for $i:\\mathcal{B}_{1}\\rightarrow\\mathcal{B}_{2}$ the inclusion homomorphism, and $z=(z_{1}, \\dotsc  , z_{n+1})$ an arbitrary snapshot in $B_{n}^{\\mathcal{B}_{1}}$.\n\n\\begin{corollary}\nEvery $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ is a subRNmatrix of a power of $\\mathcal{RM}_{C_{n}}$.\n\\end{corollary}\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{6}\n\\chapter{Logics of formal incompatibility}\\label{Chapter7}\\label{Chapter 7}\n\nIn classical logics, a formula and its negation are not compatible, in the sense that having both $\\alpha$ and ${\\sim} \\alpha$ to be true trivialize whatever argument we are working over.\n\nWhen dealing with logics of formal inconsistency, this is no longer true: we can have $\\alpha$ and its negation $\\neg\\alpha$ without trivializing our logic, as long as $\\alpha$ is inconsistent, that is, $\\circ\\alpha$ is not true, when we have at our disposal the consistency connective \"$\\circ$\".\n\nTo formalize such a notion of incompatibility, we will consider a binary connective that, when connecting formulas $\\alpha$ and $\\beta$, will stand intuitively for \"$\\alpha$ is incompatible with $\\beta$\". When choosing a symbol for such connective, one somewhat natural choice is the Sheffer's stroke\\index{Sheffer's stroke}: in classical propositional logic, the Sheffer's stroke may be defined from the usual connectives as\n\\[\\alpha\\Uparrow\\beta={\\sim}(\\alpha\\wedge\\beta),\\]\nand of course having $\\alpha\\Uparrow\\beta$ to hold, along with $\\alpha$ and $\\beta$, trivializes an argument. The basic axiom we will expect a system for incompatibility to satisfy will be\\index{Incompatibility}\\label{uparrow}\n\\[(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\gamma)),\\]\nfor any formula $\\gamma$, or more generally, if we don't have a deduction meta-theorem,\n\\[\\alpha\\uparrow\\beta, \\alpha, \\beta\\vdash_{\\mathscr{L}}\\gamma;\\]\nin words, that means that having $\\alpha$ and $\\beta$ to be true while having $\\alpha$ and $\\beta$ to be incompatible imply that our logic is trivial.\n\nReferring back to paraconsistency, one sees consistency may be characterized as incompatibility: $\\alpha$ is consistent if, and only if, is incompatible with $\\neg\\alpha$. But, given a logic dealing with incompatibility, we are also tempted to say that any formula $\\beta$ which is incompatible with a given $\\alpha$ is a negation of $\\alpha$, which leads to a notion of consistency and back again to logics having inconsistency in their scope.\n\nBefore we go any further, it is important to formally define with which structures we are working. We define the signature $\\Sigma_{\\bI}$\\label{SigmabI} by: $(\\Sigma_{\\bI})_{2}=\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$, and $(\\Sigma_{\\bI})_{n}=\\emptyset$ for $n\\neq 2$.\n\n\\begin{definition}\\label{Logics of formal incompatibility}\nA logic $\\mathcal{L}$ is said to be a logic of formal incompatibility\\index{Logic of formal incompatibility}, or in short $\\textbf{LIp}$\\label{LIp}, if it contains a family of formulas $\\Uuparrow(p,q)$, exactly on the variables $p$ and $q$, such that there exist formulas $\\varphi$, $\\phi$ and $\\psi$ for which\n\\[\\varphi, \\phi\\not\\vdash\\psi;\\]\nformulas $\\alpha$, $\\beta$ and $\\gamma$ such that:\n\\begin{enumerate}\n\\item $\\Uuparrow(\\alpha, \\beta), \\alpha\\not\\vdash\\gamma$ and\n\\item $\\Uuparrow(\\alpha, \\beta), \\beta\\not\\vdash\\gamma$,\n\\end{enumerate}\nwhere $\\Uuparrow(\\alpha, \\beta)$ is the set obtained by replacing, in each formula of $\\Uuparrow(p,q)$, $p$ by $\\alpha$ and $q$ by $\\beta$; and, for all formulas $\\gamma$, $\\theta$ and $\\omega$,\n\\[\\Uuparrow(\\gamma, \\theta), \\gamma, \\theta\\vdash\\omega.\\]\n\\end{definition}\n\nMore often than not, we want $\\Uuparrow(p,q)$ to be composed of only one formula, given by a primitive, binary connective evaluated on $p$ and $q$ for which we will use the infix notation, that is, $p\\uparrow q$.\n\nNow, one can ask oneself when incompatibility has an interpretation in natural logics, and perhaps the most canonical example would be the one of limited resources\\index{Limited resources}. Suppose we are given a basis of true statements, $\\Gamma$, and formulas $\\varphi$ that must be tested against $\\Gamma$ in the most resource-efficient way possible, in regards to both time and amount of information recorded.\n\nNow, if $\\Gamma\\vdash {\\sim} \\varphi$, we must discard $\\varphi$: but, if the proof of this implication is unbelievably long, we waste precious time. One could, to avoid spending time unnecessarily, have a vast table pre-establishing if $\\varphi$ follows from $\\Gamma$ or not, but this would, of course, be memory-consuming, and in a way also time-consuming: without a strong algorithm to check the list, this could prove to be a very long search to identify $\\varphi$ or ${\\sim} \\varphi$ on said list.\n\nThe solution could lie halfway between those two approaches: one could have a small list of statements that do not follow from $\\Gamma$, which are incompatible with $\\Gamma$, and, given a $\\varphi$, attempt to prove or disprove the given formula, knowing a few shortcuts given by our list of incompatible statements.\n\nThis is very common not only in computer science, but also in science in general, mathematics, and logic: one does not completely prove a statement starting from the axioms, but rather accept a few results, derived from previous work, as true, and therefore accept their negations as incompatible with whatever result is being searched for, and proceed from there; it is also useful, in this context, to consider compatibility\\index{Compatibility} of two formulas $\\alpha$ and $\\beta$ rather than incompatibility, defined trivially as $\\alpha\\downarrow\\beta={\\sim}(\\alpha\\uparrow\\beta)$\\label{downarrow}.\n\nAnother useful application of incompatibility is when dealing with partial information\\index{Partial information}: suppose one must test $\\varphi$ against a basis of true statements $\\Delta$, without having direct access to $\\Delta$ but rather $\\Gamma\\subset \\Delta$; in this case, knowing some key incompatibilities between $\\varphi$ and $\\Delta$ or $\\varphi$ and the complement of $\\Delta$ may help deriving $\\Delta\\vdash \\varphi$ or $\\Delta\\vdash{\\sim} \\varphi$ while only using directly $\\Gamma$.\n\nPerhaps more importantly, incompatibility has an obvious connection to probability, specifically independence (for what is to come, any reference in probability logic is sufficient, such as \\cite{ProbabilityLogic}): consider a sample space $X$, that is, the set of all things that may occur in ones analysis; if, to give one example, we were to look at the outcome of flipping a coin, our sample space could be $X=\\{heads, tails\\}$. The next step, when dealing with probabilities, is taking a $\\sigma$-algebra\\footnote{A $\\sigma$-algebra $\\mathcal{A}$ on a set $X$ is a collection of subsets of $X$ containing $X$ itself and closed under complements and countable unions, meaning that if $A\\in \\mathcal{A}$ and $\\{A_{n}\\}_{n\\in\\mathbb{N}}\\subseteq\\mathcal{A}$, then $X\\setminus A$ and $\\bigcup_{n\\in\\mathbb{N}}A_{n}$ are both in $\\mathcal{A}$.} $\\mathcal{A}$ of subsets of $X$, most commonly the whole powerset of $X$. Finally, we also need a probability measure $P$ on $\\mathcal{A}$ to obtain a probability space $(X, \\mathcal{A}, P)$, that is, a function from $\\mathcal{A}$ into the interval $[0,1]$ satisfying $P(X)=1$ and \n\\[P(\\bigcup_{n\\in\\mathbb{N}}A_{n})=\\sum_{n\\in\\mathbb{N}}P(A_{n}),\\]\nfor $\\{A_{n}\\}_{n\\in\\mathbb{N}}$ a family of pairwise disjoint sets of $\\mathcal{A}$. Intuitively, $P$ tells us the probability of an event (\\textit{i. e.} element of $\\mathcal{A}$) happening, and $\\mathcal{A}$ tells us to which events a probability can actually be assigned. We say two events $A$ and $B$ are mutually exclusive whenever \n\\[\\text{$P(A\\cap B)=0$ or, what is equivalent, $P(A\\cup B)=P(A)+P(B)$;}\\]\nintuitively, mutually exclusive events are those events that cannot both occur in an experiment. Then, if a propositional variable $p$ is associated to an event $A$, and a second propositional variable is associated to an event $B$, it is very natural to interpret the incompatibility $p\\uparrow q$ of $p$ and $q$ as the mutual exclusivity of the events $A$ and $B$, in which case the axiom $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow (\\beta\\rightarrow\\gamma))$ clearly models the fact that both events $A$ and $B$ can not simultaneously occur if the two are independent.\n\nThis seems like a specially fruitful approach to apply to probability logics, systems used in the formal study of probability theory and statistics, as well as in Bayesian perspectives on epistemology of science. But, still on the field of probability theory, we can think of an alternative interpretation of incompatibility: two events $A$ and $B$ are independent if \n\\[P(A\\cap B)=P(A)P(B);\\]\nindependent events naturally arise  from the fact that one can usually only derive more information about the probability of a complex event by knowing that its constituent events are pairwise independent, sometimes an even stronger independence condition being necessary. One could, then, for variables $p$ and $q$ standing for, respectively, events $A$ and $B$, interpret $p\\uparrow q$ as the independence of $A$ and $B$, what makes sense at first glance due to the binary nature of the two concepts. However, in this interpretation, the axiom $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow (\\beta\\rightarrow\\gamma))$ is no longer the most desirable one, at least in a naive interpretation of implication, given that two independent events can, and often do, simultaneously happen.\n\nMost of the research developed in this chapter can be found as a preprint in \\cite{Frominconsistency}.\n\n\n\n\n\\section{The logic $\\bI$}\\label{Defining bI}\n\nOur simplest $\\textbf{LIp}$, which we shall denote by $\\bI$\\label{bI}, satisfies the axiom schemata of the positive fragment of classical propositional logic,\n\n\\begin{enumerate}\n\\item[\\textbf{Ax\\: 1}] $\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$;\n\\item[\\textbf{Ax\\: 2}] $\\big(\\alpha\\rightarrow (\\beta\\rightarrow \\gamma)\\big)\\rightarrow\\big((\\alpha\\rightarrow\\beta)\\rightarrow(\\alpha\\rightarrow\\gamma)\\big)$;\n\\item[\\textbf{Ax\\: 3}] $\\alpha\\rightarrow\\big(\\beta\\rightarrow(\\alpha\\wedge\\beta)\\big)$;\n\\item[\\textbf{Ax\\: 4}] $(\\alpha\\wedge\\beta)\\rightarrow \\alpha$;\n\\item[\\textbf{Ax\\: 5}] $(\\alpha\\wedge\\beta)\\rightarrow \\beta$;\n\\item[\\textbf{Ax\\: 6}] $\\alpha\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 7}] $\\beta\\rightarrow(\\alpha\\vee\\beta)$;\n\\item[\\textbf{Ax\\: 8}] $(\\alpha\\rightarrow\\gamma)\\rightarrow\\Big((\\beta\\rightarrow\\gamma)\\rightarrow \\big((\\alpha\\vee\\beta)\\rightarrow\\gamma\\big)\\Big)$;\n\\item[$\\textbf{Ax\\: 9}^{*}$] $(\\alpha\\rightarrow \\beta)\\vee\\alpha$,\n\\end{enumerate}\nplus\\label{Ip}\n\\[\\tag{\\textbf{Ip}}(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow \\gamma))\\]\nand\\label{Comm}\n\\[\\tag{\\textbf{Comm}}(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha),\\]\nand follows the inference rule of Modus Ponens.\n\nCompare it with the classical definition of $\\textbf{mbC}$ and its extensions: they are logics over $\\Sigma_{\\textbf{LFI}}$, which is simply $\\Sigma_{\\bI}$ once we exchange $\\uparrow$ for $\\circ$ and add a negation, whose Hilbert calculus consists at least of the axiom schemata for the positive fragment of propositional logic, excluded middle and the schema\n\\[\\tag{\\textbf{bc1}}\\circ\\alpha\\rightarrow(\\alpha\\rightarrow({\\sim} \\alpha\\rightarrow \\beta)),\\]\nwith Modus Ponens as inference rule. The main difference here is the presence of $\\textbf{Comm}$, standing for the commutativity\\index{Commutativity of incompatibility} of the connective $\\uparrow$. For convenience, we will denote the logic $\\bI$ without $\\textbf{Comm}$ by $\\bI^{-}$\\label{bI-}.\n\n\\begin{example}\nWe now present a model of $\\bI$: take the signature $\\Sigma_{\\bI}$ and let $\\textbf{CPL}$ be the classical propositional logic; we make $\\textbf{CPL}$ into a model $\\textbf{CPL}_{\\uparrow}$ of $\\bI$ by defining \n\\[\\alpha\\uparrow\\beta\\quad\\text{to be}\\quad\\alpha\\Uparrow\\beta={\\sim}(\\alpha\\wedge\\beta),\\]\nthat is, $\\uparrow$ is the classically defined Sheffer's stroke. Clearly $\\textbf{CPL}_{\\uparrow}$ satisfies the axioms of the positive fragment of classical propositional logic and Modus Ponens, remaining for us to show that it also validates $\\textbf{Ip}$ and $\\textbf{Comm}$.\n\nThis is easy since, if we have $\\alpha\\uparrow\\beta$, $\\alpha$ and $\\beta$, and $\\alpha\\uparrow\\beta$ means that ${\\sim}(\\alpha\\wedge\\beta)$, we have, by use of the deduction meta-theorem for classical propositional logic, that $(\\alpha\\wedge\\beta)\\wedge{\\sim}(\\alpha\\wedge\\beta)$, which by the explosivity of negation in $\\textbf{CPL}$ implies any $\\gamma$.\n\nAnd since the conjunction is commutative in $\\textbf{CPL}$, ${\\sim}(\\alpha\\wedge\\beta)\\rightarrow{\\sim}(\\beta\\wedge\\alpha)$ or, what is equivalent, $(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha)$, proving $\\textbf{Comm}$ is valid in $\\textbf{CPL}_{\\uparrow}$.\n\\end{example}\n\n\n\n\\subsection{Bottom and top elements, and classical negation}\n\nBefore anything else, we wish to show that $\\bI$ has a formula equivalent to a bottom, and how we can use this to define a classical negation on it. So, for any two formulas $\\alpha$ and $\\beta$ in the language of $\\bI$, consider\n\\[\\bot_{\\alpha\\beta}=\\alpha\\wedge(\\beta\\wedge(\\alpha\\uparrow\\beta)).\\]\n\n\\begin{lemma}\\label{Deduction meta-theorem for bI}\n\\begin{enumerate}\n\\item For a set $\\Gamma\\cup\\{\\alpha, \\beta\\}$ of formulas in $\\bI$, we have that $\\Gamma, \\alpha\\vdash_{\\bI}\\beta$ if and only if $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ (this is known as the deduction meta-theorem).\n\\item If $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ and $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\gamma$, then $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\gamma$.\n\\item For a set $\\Gamma\\cup\\{\\alpha, \\beta, \\varphi\\}$ of formulas in $\\bI$, we have that, if $\\Gamma, \\alpha\\vdash_{\\bI}\\varphi$ and $\\Gamma, \\beta\\vdash_{\\bI}\\varphi$, then $\\Gamma, \\alpha\\vee\\beta\\vdash_{\\bI}\\varphi$ (this is know as a proof by cases).\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item The proof of this result for the positive fragment of $\\textbf{CPL}$ can be found in \\cite{Men-IntLog} (notice the proof does not use the negation), Proposition $1.9$, and the result extends to $\\bI$ since this last logic has the axiom schemata of the previous one; but we make a point of proving this theorem given its importance.\n\nSuppose first that $\\Gamma, \\alpha\\vdash_{\\bI}\\beta$ and let $\\varphi_{1}, \\dotsc , \\varphi_{n}=\\beta$ be a demonstration of $\\beta$ from $\\Gamma\\cup\\{\\alpha\\}$; we show by induction that, for every $1\\leq i\\leq n$, $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{i}$, and therefore $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ when we take $i=n$. \n\nThe case $i=1$ follows from the general case, being an instance of an axiom or a premise, so let us assume that the result holds for $\\varphi_{1}$ through $\\varphi_{i}$, and prove it for $\\varphi_{i+1}$; then $\\varphi_{i+1}$ can be one of the following.\n\n\\begin{enumerate}\n\\item An instance of an axiom: in this case, since $\\varphi_{i+1}$ and $\\varphi_{i+1}\\rightarrow(\\alpha\\rightarrow \\varphi_{i+1})$ are instances of axioms, the second formula of $\\textbf{Ax\\: 1}$, we obtain by Modus Ponens that $\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{i+1}$, and so $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{i+1}$ since $\\bI$ is tarskian.\n\\item A premise: $\\varphi_{i+1}\\rightarrow(\\alpha\\rightarrow \\varphi_{i+1})$ certainly remains an instance of $\\textbf{Ax\\: 1}$, and through Modus Ponens $\\varphi_{i+1}\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{i+1}$, thus leaving us with $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{i+1}$.\n\\item A conclusion of an inference rule: since we have only Modus Ponens as inference rule, this means that there exist $1\\leq j<k\\leq i$ with either $\\varphi_{j}=\\varphi_{k}\\rightarrow\\varphi_{i+1}$ or $\\varphi_{k}=\\varphi_{j}\\rightarrow\\varphi_{i+1}$, and we may assume without loss of generality the first case. \n\nFrom the induction hypothesis, $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{j}$ (and so $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow(\\varphi_{k}\\rightarrow\\varphi_{i+1})$) and $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{k}$, and from the instance\n\\[\\big(\\alpha\\rightarrow(\\varphi_{k}\\rightarrow\\varphi_{i+1})\\big)\\rightarrow\\big((\\alpha\\rightarrow\\varphi_{k})\\rightarrow(\\alpha\\rightarrow\\varphi_{i+1})\\big)\\]\nof $\\textbf{Ax\\: 2}$ and two applications of Modus Ponens, we find $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi_{i+1}$, as we needed to show.\n\\end{enumerate}\n\nReciprocally, if $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$, take a demonstration $\\psi_{1}, \\dotsc , \\psi_{m}=\\alpha\\rightarrow\\beta$ of $\\alpha\\rightarrow\\beta$ from $\\Gamma$, and we state that $\\psi_{1},\\dotsc , \\psi_{m}, \\psi_{m+1}, \\psi_{m+2}$ is a demonstration of $\\beta$ from $\\Gamma\\cup\\{\\alpha\\}$, where $\\psi_{m+1}=\\alpha$ and $\\psi_{m+2}=\\beta$. Of course, we have that $\\psi_{1}, \\dotsc , \\psi_{m}$ remains a demonstration (of $\\alpha\\rightarrow\\beta$) from $\\Gamma\\cup\\{\\alpha\\}$, since this is a larger set of premises, and then:\n\\begin{enumerate}\n\\item $\\psi_{m+1}=\\alpha$ is a premise of $\\Gamma\\cup\\{\\alpha\\}$;\n\\item and $\\psi_{m+2}=\\beta$ is the conclusion of an inference rule, with $\\psi_{m}=\\alpha\\rightarrow\\beta=$\\\\$\\psi_{m+1}\\rightarrow\\psi_{m+2}$, meaning we are done.\n\\end{enumerate}\n\n\\item From $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ we get that $\\Gamma, \\alpha\\vdash_{\\bI}\\beta$, and from $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\gamma$, the fact that $\\bI$ is tarskian and one application of Modus Ponens, we obtain that $\\Gamma, \\alpha\\vdash_{\\bI}\\gamma$. So $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\gamma$.\n\\item If $\\Gamma, \\alpha\\vdash_{\\bI}\\varphi$ and $\\Gamma, \\beta\\vdash_{\\bI}\\varphi$, by the previous results of the lemma $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\varphi$ and $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\varphi$; from the instance\n\\[(\\alpha\\rightarrow\\varphi)\\rightarrow((\\beta\\rightarrow\\varphi)\\rightarrow(\\alpha\\vee\\beta)\\rightarrow\\varphi))\\]\nof $\\textbf{Ax\\: 8}$ and two consecutive applications of Modus Ponens, $\\Gamma\\vdash_{\\bI}(\\alpha\\vee\\beta)\\rightarrow\\varphi$. Again by the above results, this means $\\Gamma, \\alpha\\vee\\beta\\vdash_{\\bI}\\varphi$.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{proposition}\nFor any formulas $\\alpha$, $\\beta$, $\\varphi$ and $\\psi$ in $\\bI$, $\\bot_{\\alpha\\beta}$ and $\\bot_{\\varphi\\psi}$ are equivalent, meaning $\\vdash_{\\bI}\\bot_{\\alpha\\beta}\\rightarrow\\bot_{\\varphi\\psi}$ and $\\vdash_{\\bI}\\bot_{\\varphi\\psi}\\rightarrow\\bot_{\\alpha\\beta}$.\n\\end{proposition}\n\n\\begin{proof}\nFrom the deduction meta-theorem, we have both statements are equivalent to, respectively, $\\bot_{\\alpha\\beta}\\vdash_{\\bI}\\bot_{\\varphi\\psi}$ and $\\bot_{\\varphi\\psi}\\vdash_{\\bI}\\bot_{\\alpha\\beta}$, and from $\\textbf{Ax\\: 4}$, $\\textbf{Ax\\: 5}$ and $\\textbf{Ax\\: 3}$, those are in turn equivalent to $\\alpha, \\beta, \\alpha\\uparrow\\beta\\vdash_{\\bI}\\bot_{\\varphi\\psi}$ and $\\varphi, \\psi, \\varphi\\uparrow\\psi\\vdash_{\\bI}\\top_{\\alpha\\beta}$, both obviously true from $\\textbf{Ip}$.\n\\end{proof}\n\nIt is also clear from this proof that each $\\bot_{\\alpha\\beta}$ behaves like a bottom, since for any three formulas $\\alpha$, $\\beta$ and $\\gamma$ we have that $\\vdash_{\\bI}\\bot_{\\alpha\\beta}\\rightarrow\\gamma$.\n\nNow, we define ${\\sim}\\alpha$ as the formula\n\\[\\alpha\\rightarrow\\bot_{\\alpha\\alpha},\\]\nand state that is satisfies all the three axioms commonly assigned to classical negation, at least in $\\textbf{CPL}$.\n\n\\begin{enumerate}\n\\item \\mbox{}\\vspace{-\\baselineskip}\\[(\\alpha\\rightarrow\\beta)\\rightarrow((\\alpha\\rightarrow{\\sim} \\beta)\\rightarrow{\\sim} \\alpha)\\]\nUsing several applications of the deduction meta-theorem, it is clear proving the validity of this statement is equivalent to proving that $\\alpha\\rightarrow\\beta, \\alpha\\rightarrow{\\sim} \\beta\\vdash_{\\bI}{\\sim} \\alpha$. Now, $\\alpha\\rightarrow{\\sim} \\beta$ is simply $\\alpha\\rightarrow(\\beta\\rightarrow\\bot_{\\beta\\beta})$, which by $\\textbf{Ax\\: 2}$ implies $(\\alpha\\rightarrow\\beta)\\rightarrow(\\alpha\\rightarrow\\bot_{\\beta\\beta})$, and therefore \n\\[\\alpha\\rightarrow\\beta, \\alpha\\rightarrow{\\sim} \\beta\\vdash_{\\bI}\\alpha\\rightarrow\\bot_{\\beta\\beta};\\]\nsince $\\vdash_{\\textbf{mbC}}\\bot_{\\beta\\beta}\\rightarrow\\bot_{\\alpha\\alpha}$, and the fact implication is transitive in $\\bI$ from Lemma \\ref{Deduction meta-theorem for bI}, we arrive at the desired result.\n\n\\item \\mbox{}\\vspace{-\\baselineskip} \\[\\alpha\\rightarrow({\\sim} \\alpha\\rightarrow\\beta)\\]\nAgain by applying the deduction meta-theorem several times, we know it is enough to prove that $\\alpha, {\\sim} \\alpha\\vdash_{\\textbf{mbC}}\\beta$, and since ${\\sim} \\alpha=\\alpha\\rightarrow\\bot_{\\alpha\\alpha}$, this is equivalent, by Modus Ponens, to $\\bot_{\\alpha\\alpha}\\vdash_{\\textbf{mbC}}\\beta$, which is true given all elements of the form $\\bot_{\\alpha\\alpha}$ behave like bottoms.\n\n\\item \\mbox{}\\vspace{-\\baselineskip} \\[\\alpha\\vee{\\sim} \\alpha\\]\nSince ${\\sim} \\alpha=\\alpha\\rightarrow\\bot_{\\alpha\\alpha}$, this is merely an instance of $\\textbf{Ax\\: 9}^{*}$.\n\\end{enumerate}\n\nBy defining, for a formula $\\alpha$ in $\\bI$,\n\\[\\top_{\\alpha}=\\alpha\\rightarrow\\alpha,\\]\nwe also have top elements, all equivalent to one another. To see those behave like top elements, for any two formulas $\\alpha$ and $\\beta$ we must show $\\beta\\rightarrow\\top_{\\alpha}$, or what is the same, that $\\beta\\rightarrow(\\alpha\\rightarrow\\alpha)$. By the deduction meta-theorem, this is equivalent to $\\beta\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$: it is clear how $\\alpha\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$, from the instance \n\\[\\alpha\\rightarrow(\\alpha\\rightarrow\\alpha)\\]\nof $\\textbf{Ax\\: 1}$ and Modus Ponens; and trivially $\\alpha\\rightarrow\\alpha\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$, which implies by a proof by cases that $(\\alpha\\rightarrow\\alpha)\\vee\\alpha\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$. Since $(\\alpha\\rightarrow\\alpha)\\vee\\alpha$ is an instance of $\\textbf{Ax\\: 9}^{*}$, we find that $\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$, and since $\\bI$ is tarskian, $\\beta\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$ as we wanted to prove.\n\nAnd from this remark, the fact that all such top elements are equivalent is almost trivial: for any two formulas $\\alpha$ and $\\beta$ we have both that $\\vdash_{\\bI}\\top_{\\alpha}\\rightarrow\\top_{\\beta}$ and $\\vdash_{\\bI}\\top_{\\beta}\\rightarrow\\top_{\\alpha}$, since both $\\top_{\\alpha}$ and $\\top_{\\beta}$ behave like top elements, what means $\\top_{\\alpha}$ and $\\top_{\\beta}$ are equivalent.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Bivaluations}\\label{bivaluations for bI}\n\nA \\index{Bivaluation for $\\bI$}bivaluation for $\\bI$ is a map $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow \\{0,1\\}$ such that:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\vee\\beta)=1$ if and only if $\\nu(\\alpha)=1$ or $\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\wedge\\beta)=1$ if and only if $\\nu(\\alpha)=\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\rightarrow\\beta)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$;\n\\item if $\\nu(\\alpha\\uparrow\\beta)=1$ and $\\nu(\\alpha)=1$, then $\\nu(\\beta)=0$;\n\\item $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$.\n\\end{enumerate}\n\nBivaluations for $\\bI^{-}$ are simply bivaluations for $\\bI$ that do not necessarily satisfy the condition that $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$, and the result that $\\Gamma\\vdash_{\\bI^{-}}\\varphi$ if and only if $\\Gamma\\vDash_{\\bI^{-}}\\varphi$ follows quite the same steps as those of the same result for $\\bI$ instead.\n\nGiven a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bI$, we say that $\\Gamma$ proves $\\varphi$ semantically, and write $\\Gamma\\vDash_{\\bI}\\varphi$\\label{vDashbI}, if for every bivaluation $\\nu$ for $\\bI$, $\\nu(\\Gamma)\\subseteq\\{1\\}$ implies that $\\nu(\\varphi)=1$.\n\n\n\n\n\n\n\n\n\n\\subsubsection{Soundness}\n\nWe notice, first of all, that \"$\\vDash_{\\bI}$\" emulates Modus Ponens, meaning that if $\\Gamma\\vDash_{\\bI}\\alpha$ and $\\Gamma\\vDash_{\\bI}\\alpha\\rightarrow\\beta$, then $\\Gamma\\vDash_{\\bI}\\beta$: this is because, if $\\nu$ is a bivaluation for $\\bI$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$, by hypothesis $\\nu(\\alpha)=\\nu(\\alpha\\rightarrow\\beta)=1$; since $\\nu(\\alpha\\rightarrow\\beta)=1$ implies that either $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$ and we already know $\\nu(\\alpha)=1$, we find that $\\nu(\\beta)=1$.\n\nThen, it is possible to see that, if $\\varphi$ is an instance of an axiom of classical propositional logic, then $\\vDash_{\\bI}\\varphi$ (meaning that $\\emptyset\\vDash_{\\bI}\\varphi$); to give one example of how a proof of that would go, take the instance of axiom $\\varphi=\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$: if $\\nu(\\varphi)=0$, we must have that $\\nu(\\alpha)=1$ and $\\nu(\\beta\\rightarrow\\alpha)=0$, which in turn implies that $\\nu(\\beta)=1$ and $\\nu(\\alpha)=0$, making us reach a contradiction. To give another example, take an instance $\\psi=(\\alpha\\rightarrow\\beta)\\vee\\alpha$ of $\\textbf{Ax\\: 9}^{*}$: if $\\nu(\\psi)=0$, we must have $\\nu(\\alpha\\rightarrow\\beta)=\\nu(\\alpha)=0$; from $\\nu(\\alpha\\rightarrow\\beta)=0$ one gets in turn that $\\nu(\\alpha)=1$ and $\\nu(\\beta)=0$, what constitutes a contradiction. The proof for the other axioms follows analogously.\n\nFinally, we state that if $\\varphi$ is an instance of $\\textbf{Ip}$ or $\\textbf{Comm}$, then once again we have that $\\vDash_{\\bI}\\varphi$: to see that, suppose $\\nu$ is a bivaluation for which, by contradiction, $\\nu(\\varphi)=0$, implying that $\\nu(\\alpha\\uparrow\\beta)=1$ but\n\\[\\nu(\\alpha\\rightarrow(\\beta\\rightarrow\\gamma))=0;\\]\nthis last equality implies that $\\nu(\\alpha)=1$ and $\\nu(\\beta\\rightarrow\\gamma)=0$, and thus $\\nu(\\beta)=1$ and $\\nu(\\gamma)=0$. But, at the same time, $\\nu(\\alpha\\uparrow\\beta)=1$ and $\\nu(\\alpha)=1$ imply together that $\\nu(\\beta)=0$, which contradicts our previous observations. One must have then that $\\nu(\\varphi)=1$, for any bivaluation $\\nu$, and therefore $\\vDash_{\\bI}\\varphi$.\n\nNow, for $\\textbf{Comm}$, if $\\nu(\\alpha\\uparrow\\beta)=0$, clearly we already have that $\\nu((\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha))=1$; if, otherwise, $\\nu(\\alpha\\uparrow \\beta)=1$, then by the fact $\\nu$ is a bivaluation for $\\bI$ we get $\\nu(\\beta\\uparrow\\alpha)=1$, and once again $\\nu((\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha))=1$.\n\n\\begin{theorem}\\label{Proof of soundness for bI}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bI$, if $\\Gamma\\vdash_{\\bI}\\varphi$ then $\\Gamma\\vDash_{\\bI}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nIf $\\Gamma\\vdash_{\\bI}\\varphi$, there exists a demonstration $\\alpha_{1}, \\dotsc , \\alpha_{n}$ of $\\varphi$ from $\\Gamma$, with $\\alpha_{n}=\\varphi$.\n\nLet $\\nu$ be a bivaluation for $\\bI$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$: we want to prove that, in this case, $\\nu(\\varphi)=1$; so we prove, by induction, that $\\alpha_{1}$ through $\\alpha_{n}$ have image $1$ under $\\nu$, and therefore $\\nu(\\varphi)=\\nu(\\alpha_{n})=1$.\n\nThe formula $\\alpha_{1}$ is either an axiom, when $\\nu(\\alpha_{1})=1$ since all instances of axioms have image $1$ through any bivaluations, or $\\alpha_{1}$ is a premise, that is, an element of $\\Gamma$, and since $\\nu(\\Gamma)\\subseteq\\{1\\}$ we have that $\\nu(\\alpha_{1})=1$.\n\nSuppose then that $\\nu(\\alpha_{1})=\\cdots=\\nu(\\alpha_{i-1})=1$, and we have three cases to consider:\n\\begin{enumerate}\n\\item if $\\alpha_{i}$ is an instance of an axiom, as we commented above $\\nu(\\alpha_{i})=1$;\n\\item if $\\alpha_{i}$ is a premise, $\\nu(\\alpha_{i})=1$ since $\\alpha_{i}\\in\\Gamma$ and $\\nu(\\Gamma)\\subseteq\\{1\\}$;\n\\item if there are $\\alpha_{j}$ and $\\alpha_{k}$ with $1<j<k<i$ such that $\\alpha_{j}=\\alpha_{k}\\rightarrow\\alpha_{i}$ or $\\alpha_{k}=\\alpha_{j}\\rightarrow\\alpha_{i}$, since $\\nu(\\alpha_{j})=\\nu(\\alpha_{k})=1$ we find in both cases that $\\nu(\\alpha_{i})=1$, what ends the proof.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\\subsubsection{Completeness}\n\nNow, for a non-trivial, closed set of formulas $\\Gamma$ maximal with respect to not proving $\\varphi$, we want to prove that the function $\\nu$, from the formulas of $\\bI$ to $\\{0,1\\}$ and such that $\\nu(\\gamma)=1$ if and only if $\\gamma\\in\\Gamma$, is a bivaluation.\n\n\\begin{enumerate}\n\\item If $\\nu(\\alpha\\vee\\beta)=1$, $\\alpha\\vee\\beta\\in\\Gamma$: suppose, by contradiction, that $\\alpha, \\beta\\notin\\Gamma$; then $\\Gamma, \\alpha\\vdash_{\\bI}\\varphi$ and $\\Gamma, \\beta\\vdash_{\\bI}\\varphi$. So, by a proof by cases, $\\Gamma, \\alpha\\vee\\beta\\vdash_{\\bI}\\varphi$, and since $\\alpha\\vee\\beta\\in\\Gamma$ we find that $\\Gamma\\vdash_{\\bI}\\varphi$, which is absurd. This means either $\\alpha\\in\\Gamma$, and then $\\nu(\\alpha)=1$, or $\\beta\\in\\Gamma$, when $\\nu(\\beta)=1$.\n\nReciprocally, if $\\nu(\\alpha)=1$, since $\\alpha\\rightarrow(\\alpha\\vee\\beta)$ is an instance of an axiom and is therefore in $\\Gamma$, we find that $\\alpha\\vee\\beta\\in\\Gamma$, since this set is closed; the same occurs if $\\beta\\in\\Gamma$, forcing us to conclude $\\nu(\\alpha\\vee\\beta)=1$.\n\n\\item Now, assume $\\nu(\\alpha\\wedge\\beta)=1$; being $(\\alpha\\wedge\\beta)\\rightarrow\\alpha$ an instance of an axiom, and $\\Gamma$ closed, $(\\alpha\\wedge\\beta)\\rightarrow\\alpha\\in\\Gamma$ and then $\\alpha\\in\\Gamma$, and the same line of thought informs us that $\\beta\\in\\Gamma$, meaning $\\nu(\\alpha)=\\nu(\\beta)=1$.\n\nReciprocally, if $\\nu(\\alpha)=\\nu(\\beta)=1$, $\\alpha, \\beta\\in\\Gamma$ and, given the instance $\\alpha\\rightarrow(\\beta\\rightarrow(\\alpha\\wedge\\beta))$ of axiom $\\textbf{Ax\\: 3}$, by two consecutive applications of Modus Ponens we get that $\\alpha\\wedge\\beta\\in\\Gamma$, meaning $\\nu(\\alpha\\wedge\\beta)=1$.\n\n\\item For implication, assume $\\nu(\\alpha\\rightarrow\\beta)=1$: then, either $\\nu(\\alpha)=1$, meaning $\\alpha\\in\\Gamma$ and, since $\\Gamma$ is closed and also contains $\\alpha\\rightarrow\\beta$, $\\beta\\in\\Gamma$ and therefore $\\nu(\\beta)=1$; or $\\nu(\\alpha)=0$.\n\nReciprocally, if $\\nu(\\beta)=1$ we have that $\\beta\\in\\Gamma$, and since $\\beta\\rightarrow(\\alpha\\rightarrow\\beta)$ is an instance of axiom $\\textbf{Ax\\: 1}$ and $\\Gamma$ is closed, we obtain that $\\alpha\\rightarrow\\beta\\in\\Gamma$.\n\nIf $\\nu(\\alpha)=0$, from the maximality of $\\Gamma$ with respect to not proving $\\varphi$ we find $\\Gamma, \\alpha\\vdash_{\\bI}\\varphi$; suppose, by contradiction, that $\\alpha\\rightarrow\\beta\\notin\\Gamma$, and again by the maximality of $\\Gamma$ we discover $\\Gamma, \\alpha\\rightarrow\\beta\\vdash_{\\bI}\\varphi$, and from a proof by cases\n\\[\\Gamma, (\\alpha\\rightarrow\\beta)\\vee\\alpha\\vdash_{\\bI}\\varphi.\\]\nHowever, by $\\textbf{Ax\\: 9}^{*}$, $(\\alpha\\rightarrow\\beta)\\vee\\alpha$ is an instance of an axiom, and we get that $\\Gamma\\vdash_{\\bI}\\varphi$, which is absurd. It follows that, if $\\nu(\\alpha)=0$, then $\\nu(\\alpha\\rightarrow\\beta)=1$.\n\n\\item Suppose $\\nu(\\alpha\\uparrow\\beta)=1$ and $\\nu(\\alpha)=1$, meaning that $\\alpha\\uparrow\\beta, \\alpha\\in \\Gamma$; by Modus Ponens and the correct instance of the axiom $\\textbf{Ip}$,\n\\[(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\varphi)),\\]\nwe derive that $\\alpha\\rightarrow(\\beta\\rightarrow \\varphi)\\in\\Gamma$ since this set is closed; using the closedness of $\\Gamma$ and Modus Ponens again, $\\beta\\rightarrow\\varphi\\in\\Gamma$, meaning, from what we saw before, that either $\\beta\\notin \\Gamma$ or $\\varphi\\in\\Gamma$. Since $\\Gamma$ does not prove $\\varphi$, we cannot have $\\varphi\\in\\Gamma$, and therefore $\\beta\\notin\\Gamma$, that is, $\\nu(\\beta)=0$.\n\n\\item Finally, if $\\nu(\\alpha\\uparrow\\beta)=0$, $\\alpha\\uparrow\\beta\\notin \\Gamma$; if we had $\\beta\\uparrow\\alpha\\in \\Gamma$, given that $\\Gamma$ is closed and \n\\[(\\beta\\uparrow\\alpha)\\rightarrow(\\alpha\\uparrow\\beta)\\]\nis an instance of $\\textbf{Comm}$, we would have $\\alpha\\uparrow\\beta\\in \\Gamma$, which is a contradiction; so we must have $\\beta\\uparrow\\alpha\\notin\\Gamma$ and therefore $\\nu(\\beta\\uparrow\\alpha)=0$.\n\nIf we have instead that $\\nu(\\alpha\\uparrow\\beta)=1$, $\\alpha\\uparrow\\beta\\in\\Gamma$, and since $\\Gamma$ is closed and $(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha)$ is an instance of axiom $\\textbf{Comm}$, we get that $\\beta\\uparrow\\alpha\\in \\Gamma$, meaning $\\nu(\\beta\\uparrow\\alpha)=1$.\n\\end{enumerate}\n\nTherefore, the function $\\nu$ is a bivaluation for $\\bI$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ in $\\bI$, \n\\[\\Gamma\\vdash_{\\bI}\\varphi\\quad\\text{if and only if}\\quad\\Gamma\\vDash_{\\bI}\\varphi.\\]\n\\end{theorem}\n\n\\begin{proof}\nWe have already proved in Theorem \\ref{Proof of soundness for bI} that, if $\\Gamma\\vdash_{\\bI}\\varphi$, then $\\Gamma\\vDash_{\\bI}\\varphi$.\n\nReciprocally, suppose by contradiction $\\Gamma\\not\\vdash_{\\bI}\\varphi$: then there exists a closed, non-trivial extension $\\Delta$ of $\\Gamma$ maximal with respect to not proving $\\varphi$, and the function $\\nu$ from the formulas of $\\bI$ to $\\{0,1\\}$ such that $\\nu(\\delta)=1$ if and only if $\\delta\\in\\Delta$ is a bivaluation.\n\nBut then $\\nu$ is a bivaluation such that $\\nu(\\Gamma)\\subseteq\\{1\\}$ and $\\nu(\\varphi)=0$, contradicting our assumption that $\\Gamma\\vDash_{\\bI}\\varphi$.\n\n\\end{proof}\n\n\n\nNow, with the aid of bivaluations, we can prove that $\\bI$ is a logic of formal incompatibility, or $\\textbf{LIp}$, as in Definition \\ref{Logics of formal incompatibility}, with respect to the set $\\Uuparrow(p, q)=\\{p\\uparrow q\\}$. Take any three propositional variables $p$, $q$ and $r$.\n\\begin{enumerate}\n\\item Taking a bivaluation $\\nu$ with $\\nu(p)=\\nu(q)=1$ and $\\nu(r)=0$, we then have that $p, q\\not\\vdash_{\\bI}r$.\n\\item Taking a bivaluation $\\nu$ with $\\nu(p)=\\nu(p\\uparrow q)=1$ (and so $\\nu(q)=0$ and $\\nu(q\\uparrow p)=1$) and $\\nu(r)=0$, it becomes clear that $p\\uparrow q, p\\not\\vdash_{\\bI}r$.\n\\item Taking a bivaluation $\\nu$ with $\\nu(q)=\\nu(p\\uparrow q)=1$ (and so $\\nu(p)=0$ and $\\nu(q\\uparrow p)=1$) and $\\nu(r)=0$, it becomes clear that $p\\uparrow q, q\\not\\vdash_{\\bI}r$.\n\\item For any formula $\\alpha$, it is true that $p\\uparrow q, p, q\\vdash_{\\bI}\\alpha$ from $\\textbf{Ip}$ and Modus Ponens.\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fidel structures}\\label{Fidel structures for bI}\n\nFidel structures are usually seem as generalized Boolean algebras, generalization originally achieved by adding relations to these algebraic structures. Designed to treat non-classical logics, their first occurrences in literature appear in the work of Fidel, as in \\cite{Fidel3, Fidel, Fidel2}. They were first created to deal with certain paraconsistent logics, specifically da Costa's $C_{n}$, and in such a context they are probably more intuitively defined, but they will prove themselves very adequate for dealing semantically with logics of incompatibility, when correctly presented as multialgebras. When we focus our study on logics of formal inconsistency we will present a more classical approach to Fidel structures, as it can be found in Definition $6.1.2$ of \\cite{ParLog}, but for now we will simply present the Fidel structures best tailored for $\\bI$. \n\nHere, it becomes necessary to clarify the use of the nomenclature ``Fidel structures'': why are we using this name? These semantics were originally defined as certain implicative lattices\\footnote{That, in the case of $C_{n}$, were proven to actually be Boolean algebras.} equipped with some unary relations to treat da Costa's hierarchy plus $C_{\\omega}$. Then, as more non-classical logics needed semantical characterizations, all Boolean-like algebras (including implicative lattices, Heyting algebras and others), equipped with unary relations (usually standing for negation and consistency), could be called Fidel structures; the earliest examples of Fidel structures over algebras that are not Boolean are also found in Fidel's work, for da Costa's $C_{\\omega}$ and Nelson's logics, in respectively \\cite{Fidel3} and \\cite{FidelNelson}. Here, our reasoning is the following: Fidel structures should be seem as RNmatrices with an underlying Boolean-like (deterministic) component, and multioperations (now of any arity) arising from what was previously presented as relations. Perhaps it is too much to classify under the title of ``Fidel structures'', but since the logics at hand are still quite similar, in spirit, to da Costa's hierarchy, the name seemed a fitting tribute to Fidel's breakthrough semantics.\n\nFirstly, we expand the signature $\\Sigma_{\\bI}$ by adding a bottom, a top and a classical negation, making the signature $\\Sigma_{\\bI}^{\\textbf{CPL}}$\\label{SigmabICPL} such that: $(\\Sigma_{\\bI}^{\\textbf{CPL}})_{0}=\\{\\bot, \\top\\}$, $(\\Sigma_{\\bI}^{\\textbf{CPL}})_{1}=\\{{\\sim}\\}$, $(\\Sigma_{\\bI}^{\\textbf{CPL}})_{2}=\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$, and $(\\Sigma_{\\bI}^{\\textbf{CPL}})_{n}=\\emptyset$ for $n>2$. This allows us to define a \\index{Fidel structure for $\\bI$}Fidel structure, presented as a $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra, for $\\bI$ to be any $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\bI}^{\\textbf{CPL}}})$ such that:\n\\begin{enumerate}\n\\item $(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra;\\footnote{Here, a distinction is important: the operations corresponding to the symbols $\\bot$, $\\top$, $\\sim$, $\\vee$, $\\wedge$ and $\\rightarrow$ are deterministic, and will be treated as such; meanwhile, $\\uparrow$ will be, at most times, non-deterministic.}\n\\item for all $a, b\\in A$ and $c\\in\\uparrow_{\\mathcal{A}}(a,b)$, \n\\[\\wedge_{\\mathcal{A}}(a, \\wedge_{\\mathcal{A}}(b,c))=\\bot_{\\mathcal{A}};\\]\n\\item for all $a, b\\in A$, $\\uparrow_{\\mathcal{A}}(a, b)=\\uparrow_{\\mathcal{A}}(b,a)$.\\footnote{Technically, this last condition could be replaced by, for all $a, b\\in A$, $\\uparrow_{\\mathcal{A}}(a,b)\\cap\\uparrow_{\\mathcal{A}}(b,a)\\neq\\emptyset$, or simply erased from our definition of Fidel structures for $\\bI$; it only guarantees that, for every such Fidel structure, any function from the variables $\\mathcal{V}$ into $A$ may be extended to a restricted valuation. However, given the condition $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$ we will ask of these restricted valuations, dropping $\\uparrow_{\\mathcal{A}}(a, b)=\\uparrow_{\\mathcal{A}}(b,a)$ from the definition of a Fidel structure would only mean that some Fidel structure would not help in establishing the validity of an argument; the class of all Fidel structures, however, would still be sound and complete.}\n\\end{enumerate}\n\nFor simplicity, we will drop the indexes $\\mathcal{A}$ and use the standard infix notation.\n\nGiven a Fidel structure $\\mathcal{A}$, presented as a $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra, for $\\bI$, a valuation over $\\mathcal{A}$ is a function $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow A$ such that:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\#\\beta)=\\nu(\\alpha)\\#\\nu(\\beta)$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$;\n\\item $\\nu(\\alpha\\uparrow\\beta)\\in\\nu(\\alpha)\\uparrow\\nu(\\beta)$.\n\\end{enumerate}\n\nNotice that $\\nu$ is a valuation for $\\mathcal{A}$ if and only if it is a $\\Sigma_{\\bI}$-homomorphism, between $\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})$ and $(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\bI}})$. \n\nFor every Fidel structure $\\mathcal{A}$ for $\\bI$, we will consider the restricted Nmatrix\\\\ $(\\mathcal{A}, \\{\\top\\}, \\mathcal{F}_{\\mathcal{A}})$, where $\\mathcal{F}_{\\mathcal{A}}$ is the set of valuations $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\mathcal{A}$ such that \n\\[\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha),\\]\nfor any two formulas $\\alpha$ and $\\beta$ in $F(\\Sigma_{\\bI}, \\mathcal{V})$. If $\\Gamma$ proves $\\varphi$ according to such restricted Nmatrices, we will write $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bI}\\varphi$\\label{VdashFbI}: as one could suspect, $\\Gamma\\vdash_{\\bI}\\varphi$ if and only if $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bI}\\varphi$.\n\n\\begin{proposition}\nEach restricted Nmatrix $(\\mathcal{A}, \\{\\top\\}, \\mathcal{F}_{\\mathcal{A}})$, as described above, is structural.\n\\end{proposition}\n\n\\begin{proof}\nTake any $\\Sigma_{\\bI}$-homomorphism $\\sigma:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})$ and an element $\\nu\\in\\mathcal{F}_{\\mathcal{A}}$, which is by definition of $\\mathcal{F}_{\\mathcal{A}}$ a $\\Sigma_{\\bI}$-homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow \\mathcal{A}$ such that, for any two formulas $\\alpha$ and $\\beta$, $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$.\n\nClearly $\\nu\\circ\\sigma:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow \\mathcal{A}$ is a $\\Sigma_{\\bI}$-homomorphism, so we only need to show that, for any two formulas $\\alpha$ and $\\beta$, $\\nu\\circ\\sigma(\\alpha\\uparrow\\beta)=\\nu\\circ\\sigma(\\beta\\uparrow\\alpha)$; and since \n\\[\\nu\\circ\\sigma(\\alpha\\uparrow\\beta)=\\nu(\\sigma(\\alpha\\uparrow\\beta))=\\nu(\\sigma(\\alpha)\\uparrow\\sigma(\\beta))=\\nu(\\sigma(\\beta)\\uparrow\\sigma(\\alpha))=\\nu(\\sigma(\\beta\\uparrow\\alpha))=\\nu\\circ\\sigma(\\beta\\uparrow\\alpha),\\]\nwe finish the proof.\n\\end{proof}\n\nIf we simply consider the class $\\mathbb{M}$ of Nmatrices $(\\mathcal{A},\\{\\top\\})$, where $\\mathcal{A}$ is a Fidel structure for $\\bI$, it is not hard to prove that $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$ if and only if $\\Gamma\\vdash_{\\bI^{-}}\\varphi$.\n\n\n\\subsubsection{Soundness}\n\nFirst of all, we state that for any instance of an axiom $\\alpha$ of the positive fragment of classical propositional logic in $\\bI$, $\\Vdash_{\\mathcal{F}}^{\\bI}\\alpha$ (meaning that $\\emptyset\\Vdash_{\\mathcal{F}}^{\\bI}\\alpha$): this is true because Boolean algebras model classical propositional logic. To see one example, take an instance $\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$ of $\\textbf{Ax\\: 1}$: we have that\n\\[\\nu(\\alpha\\rightarrow(\\beta\\rightarrow\\alpha))=\\nu(\\alpha)\\rightarrow(\\nu(\\beta)\\rightarrow\\nu(\\alpha)),\\]\nand remembering that in a Boolean algebra, $x\\rightarrow y={\\sim}x\\vee y$, this equals ${\\sim}\\nu(\\alpha)\\vee({\\sim}\\nu(\\beta)\\vee\\nu(\\alpha))$; using that $\\vee$ is commutative and associative, in this order, this equals\n\\[({\\sim}\\nu(\\alpha)\\vee\\nu(\\alpha))\\vee{\\sim}\\nu(\\beta)=\\top\\vee{\\sim}\\nu(\\beta)=\\top,\\]\nand since for any valuation $\\nu$ we have that $\\nu(\\alpha\\rightarrow(\\beta\\rightarrow\\alpha))=\\top$, we find $\\Vdash_{\\mathcal{F}}^{\\bI}\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)$.\n\nNow we take an instance $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\gamma))$ of axiom $\\textbf{Ip}$; using again that $x\\rightarrow y={\\sim}x\\vee y$, the image under a valuation $\\nu$ of this formula is \n\\[{\\sim}\\nu(\\alpha\\uparrow\\beta)\\vee({\\sim}\\nu(\\alpha)\\vee({\\sim}\\nu(\\beta)\\vee\\nu(\\gamma)))=({\\sim}\\nu(\\alpha\\uparrow\\beta)\\vee({\\sim}\\nu(\\alpha)\\vee{\\sim}\\nu(\\beta)))\\vee\\nu(\\gamma);\\]\nusing that ${\\sim}x\\vee{\\sim}y={\\sim}(x\\wedge y)$ in a Boolean algebra, one of the De Morgan's laws, we get this expression equals\n\\[{\\sim}(\\nu(\\alpha\\uparrow\\beta)\\wedge(\\nu(\\alpha)\\wedge\\nu(\\beta)))\\vee\\nu(\\gamma),\\]\nand from the requirements made on the multioperation $\\uparrow$ in the definition of Fidel structures for $\\bI$, we get that $\\nu(\\alpha\\uparrow\\beta)\\wedge(\\nu(\\alpha)\\wedge\\nu(\\beta))=\\bot$, and therefore the whole expression simplifies to\n\\[{\\sim}\\bot\\vee\\nu(\\gamma)=\\top\\vee\\nu(\\gamma)=\\top.\\]\n\nFinally, take an instance $(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha)$ of $\\textbf{Comm}$, and for any restricted Nmatrix $(\\mathcal{A},\\{\\top\\},\\mathcal{F}_{\\mathcal{A}})$ for $\\bI$, its image under a $\\nu\\in\\mathcal{F}_{\\mathcal{A}}$ is $\\nu(\\alpha\\uparrow\\beta)\\rightarrow\\nu(\\beta\\uparrow\\alpha)$. Since $\\nu\\in\\mathcal{F}_{\\mathcal{A}}$, we have that $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$, and therefore $\\nu((\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha))$ equals $\\nu(\\alpha\\uparrow\\beta)\\rightarrow\\nu(\\alpha\\uparrow\\beta)$, which is always $\\top$, meaning \"$\\Vdash_{\\mathcal{F}}^{\\bI}$\" models all axioms of $\\bI$.\n\n\\begin{theorem}\\label{Soundness of Fidel structures for bI}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bI$, if $\\Gamma\\vdash_{\\bI}\\varphi$ then $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bI}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\alpha_{1}, \\dotsc , \\alpha_{n}$ be a demonstration of $\\varphi$ from $\\Gamma$, with $\\alpha_{n}=\\varphi$, and let $\\mathcal{A}$ be a Fidel structure, represented as a $\\Sigma_{\\bI}$-multialgebra, for $\\bI$, with $\\nu$ a valuation for $\\bI$ over $\\mathcal{A}$ in $\\mathcal{F}_{\\mathcal{A}}$.\n\nWe aim to show that, if $\\nu(\\Gamma)\\subseteq\\{\\top\\}$, then $\\nu(\\alpha_{i})=\\top$ for every $i\\in\\{1, \\dotsc , n\\}$. We will proceed by induction, assuming that the result already holds for all formulas prior to $\\alpha_{i}$ (notice that $\\alpha_{0}$ is either an axiom or a premise). Then we have that $\\alpha_{i}$ is either:\n\\begin{enumerate}\n\\item an instance of an axiom, when we already proved $\\Vdash_{\\mathcal{F}}^{\\bI}\\alpha_{i}$, and therefore $\\nu(\\alpha_{i})=\\top$;\n\\item a premise, and by our hypothesis that $\\nu(\\Gamma)\\subseteq\\{\\top\\}$ we find $\\nu(\\alpha_{i})=\\top$;\n\\item there exist $j,k<i$ such that either $\\alpha_{j}=\\alpha_{k}\\rightarrow\\alpha_{i}$ or $\\alpha_{k}=\\alpha_{j}\\rightarrow\\alpha_{i}$, where we will assume, without loss of generality, that the first case holds;\n\nby induction hypothesis, $\\nu(\\alpha_{j})=\\nu(\\alpha_{k}\\rightarrow\\alpha_{i})=\\top$, and therefore ${\\sim}\\nu(\\alpha_{k})\\vee\\nu(\\alpha_{i})=\\top$, and since $\\nu(\\alpha_{k})=\\top$, meaning ${\\sim}\\nu(\\alpha_{k})=\\bot$, we must have that $\\nu(\\alpha_{i})=\\top$, what finishes the proof.\n\\end{enumerate}\n\\end{proof}\n\n\n\\subsubsection{Completeness}\\label{Completeness fo Fidel structures for bI}\n\nNow, we want to show the reciprocal of Theorem \\ref{Soundness of Fidel structures for bI} is also true: if $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bI}\\varphi$, then $\\Gamma\\vdash_{\\bI}\\varphi$.\n\nFor a set of formulas $\\Gamma$ of $\\bI$, we define a relation \"$\\equiv_{\\Gamma}^{\\bI}$\"\\label{equivbI} between formulas of $\\bI$ such that\n\\[\\alpha\\equiv_{\\Gamma}^{\\bI}\\beta\\quad\\text{if and only if}\\quad\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta\\quad\\text{and}\\quad\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha.\\]\n\n\\begin{proposition}\nFor any $\\Gamma$, $\\equiv_{\\Gamma}^{\\bI}$ is an equivalence relation.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item For any formula $\\alpha$, we trivially have that $\\alpha\\vdash_{\\bI}\\alpha$, and therefore $\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$; since $\\bI$ is tarskian, we find $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\alpha$, and therefore $\\alpha\\equiv_{\\Gamma}^{\\bI}\\alpha$.\n\n\\item If $\\alpha\\equiv_{\\Gamma}^{\\bI}\\beta$, then $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ and $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha$; therefore $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha$ and $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ or, what is equivalent, $\\beta\\equiv_{\\Gamma}^{\\bI}\\alpha$.\n\n\\item If $\\alpha\\equiv_{\\Gamma}^{\\bI}\\beta$ and $\\beta\\equiv_{\\Gamma}^{\\bI}\\gamma$, we have that $\\Gamma$ demonstrates $\\alpha\\rightarrow\\beta$, $\\beta\\rightarrow\\alpha$, $\\beta\\rightarrow\\gamma$ and $\\gamma\\rightarrow\\beta$ in $\\bI$; from $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ we get that $\\Gamma, \\alpha\\vdash_{\\bI}\\beta$, and then from $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\gamma$ and Modus Ponens we obtain that $\\Gamma, \\alpha\\vdash_{\\bI}\\gamma$, or what is equivalent, that $\\Gamma_{\\bI}\\alpha\\rightarrow\\gamma$. From $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha$ and $\\Gamma\\vdash_{\\bI}\\gamma\\rightarrow\\beta$ we find that $\\Gamma\\vdash_{\\bI}\\gamma\\rightarrow\\alpha$, meaning $\\alpha\\equiv_{\\Gamma}^{\\bI}\\gamma$.\n\\end{enumerate}\n\\end{proof}\n\n\n\\begin{theorem}\nFor any $\\Gamma$, $\\equiv_{\\Gamma}^{\\bI}$ is a congruence with relation to the operators in $\\{\\vee, \\wedge, \\rightarrow\\}$, meaning that for any operator $\\#$ in this set, and any formulas $\\alpha_{1}$, $\\alpha_{2}$, $\\beta_{1}$ and $\\beta_{2}$ in $\\bI$ such that $\\alpha_{1}\\equiv_{\\Gamma}^{\\bI}\\beta_{1}$ and $\\alpha_{2}\\equiv_{\\Gamma}^{\\bI}\\beta_{2}$,\n\\[\\alpha_{1}\\#\\beta_{1}\\equiv_{\\Gamma}^{\\bI}\\alpha_{2}\\#\\beta_{2}.\\]\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\alpha\\equiv_{\\Gamma}^{\\bI}\\beta$ and $\\varphi\\equiv_{\\Gamma}^{\\bI}\\psi$, meaning $\\Gamma$ proves $\\alpha\\rightarrow\\beta$, $\\beta\\rightarrow\\alpha$, $\\varphi\\rightarrow\\psi$ and $\\psi\\rightarrow\\varphi$. \n\n\\begin{enumerate}\n\\item We have that $\\Gamma, \\alpha\\vdash_{\\bI}\\beta$, and by $\\textbf{Ax\\: 6}$ we find $\\Gamma, \\alpha\\vdash_{\\bI}\\beta\\vee\\psi$; analogously, we use $\\Gamma, \\varphi\\vdash_{\\bI}\\psi$ to show that $\\Gamma, \\varphi\\vdash_{\\bI}\\beta\\vee\\psi$, meaning through a proof by cases that\n\\[\\Gamma, \\alpha\\vee\\varphi\\vdash_{\\bI}\\beta\\vee\\psi\\]\nand that, therefore, $\\Gamma\\vdash_{\\bI}(\\alpha\\vee\\varphi)\\rightarrow(\\beta\\vee\\psi)$. Using that $\\Gamma$ proves $\\beta\\rightarrow\\alpha$ and $\\psi\\rightarrow\\varphi$, we get that $\\Gamma\\vdash_{\\bI}(\\beta\\vee\\psi)\\rightarrow(\\alpha\\vee\\varphi)$, and therefore $\\alpha\\vee\\varphi\\equiv_{\\Gamma}^{\\bI}\\beta\\vee\\psi$.\n\n\\item From $\\Gamma, \\alpha\\vdash_{\\bI}\\beta$ and $\\Gamma, \\varphi\\vdash_{\\bI}\\psi$, the instance $\\beta\\rightarrow(\\psi\\rightarrow(\\beta\\wedge \\psi))$ of $\\textbf{Ax\\: 3}$ and two applications of Modus Ponens, we obtain that $\\Gamma, \\alpha, \\varphi\\vdash_{\\bI}\\beta\\wedge\\psi$.\n\nSince $\\alpha\\wedge\\varphi\\vdash_{\\bI}\\alpha, \\varphi$, and by the fact that $\\bI$ is tarskian, we obtain that $\\Gamma, \\alpha\\wedge\\varphi\\vdash_{\\bI}\\beta\\wedge\\psi$, that is, $\\Gamma\\vdash_{\\bI}(\\alpha\\wedge\\varphi)\\rightarrow(\\beta\\wedge\\psi)$. Using $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha$ and $\\Gamma\\vdash_{\\bI}\\psi\\rightarrow\\varphi$, we obtain analogously $\\Gamma\\vdash_{\\bI}(\\beta\\wedge\\psi)\\rightarrow(\\alpha\\wedge\\varphi)$, and that implies $\\alpha\\wedge\\varphi\\equiv_{\\Gamma}^{\\bI}\\beta\\wedge\\psi$.\n\n\\item From the fact that $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha$ and $\\Gamma\\vdash_{\\bI}\\varphi\\rightarrow\\psi$, applying twice that implication is transitive we find\n\\[\\Gamma, \\alpha\\rightarrow\\varphi\\vdash_{\\bI}\\beta\\rightarrow\\psi,\\]\nand so $\\Gamma\\vdash_{\\bI}(\\alpha\\rightarrow\\varphi)\\rightarrow(\\beta\\rightarrow\\psi)$. From the fact that $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta, \\psi\\rightarrow\\varphi$ we obtain, in a similar way, that $\\Gamma\\vdash_{\\bI}(\\beta\\rightarrow\\psi)\\rightarrow(\\alpha\\rightarrow\\varphi)$, meaning $\\alpha\\rightarrow\\varphi\\equiv_{\\Gamma}^{\\bI}\\beta\\rightarrow\\psi$.\n\n\\end{enumerate}\n\\end{proof}\n\nSo the quotient set $A^{\\bI}_{\\Gamma}=F(\\Sigma_{\\bI}, \\mathcal{V})\/\\equiv_{\\Gamma}^{\\bI}$\\label{AbIGamma} is a well defined set, where we will denote the quotient class of the formula $\\alpha$ by $[\\alpha]$. More than that, from the fact that $\\equiv_{\\Gamma}^{\\bI}$ is a congruence with respect to the connectives in $\\{\\vee, \\wedge, \\rightarrow\\}$ we get that, by making\n\\[[\\alpha]\\#_{\\mathcal{A}}[\\beta]=[\\alpha\\#\\beta],\\quad\\text{for}\\quad\\#\\in\\{\\vee, \\wedge, \\rightarrow\\},\\]\nsuch operations are well-defined. To see that, suppose both $\\alpha$ and $\\varphi$ are in the class $[\\alpha]$ (meaning $\\alpha\\equiv_{\\Gamma}^{\\bI}\\varphi$) and both $\\beta$ and $\\psi$ are in $[\\beta]$: we get that $[\\alpha]\\#_{\\mathcal{A}}[\\beta]=[\\alpha\\#\\beta]$ and $[\\varphi]\\#_{\\mathcal{A}}[\\psi]=[\\varphi\\#\\psi]$, which equals $[\\alpha\\#\\beta]$ since $\\equiv_{\\Gamma}^{\\bI}$ is a congruence for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$.\n\nWe then define, as it would be expected from our commentaries on the definable bottoms, tops and classical negation in $\\bI$, $\\bot_{\\mathcal{A}}=[\\bot_{\\alpha\\beta}]$ and $\\top_{\\mathcal{A}}=[\\top_{\\alpha}]$ for any formulas $\\alpha$ in $\\beta$ in $\\bI$, what is possible since those bottom and top elements are equivalent to one another, and \n\\[{\\sim}_{\\mathcal{A}}([\\alpha])=[\\alpha\\rightarrow\\bot_{\\alpha\\alpha}].\\]\nTo see that the classical negation is well-defined, take $\\beta\\in[\\alpha]$, and we wish to show that ${\\sim}_{\\mathcal{A}}([\\alpha])={\\sim}_{\\mathcal{A}}([\\beta])$, meaning $\\alpha\\rightarrow\\bot_{\\alpha\\alpha}\\equiv^{\\bI}_{\\Gamma}\\beta\\rightarrow\\bot_{\\beta\\beta}$. This is equivalent to showing that, by applying the deduction meta-theorem,\n\\[\\Gamma, \\alpha\\rightarrow\\bot_{\\alpha\\alpha}\\vdash_{\\bI}\\beta\\rightarrow\\bot_{\\beta\\beta}\\quad\\text{and}\\quad\\Gamma, \\beta\\rightarrow\\bot_{\\beta\\beta}\\vdash_{\\bI}\\alpha\\rightarrow\\bot_{\\alpha\\alpha}.\\]\nBut since $\\beta\\in[\\alpha]$, we have $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ and $\\Gamma\\vdash_{\\bI}\\beta\\rightarrow\\alpha$, while $\\bot_{\\alpha\\alpha}\\rightarrow\\bot_{\\beta\\beta}$ and $\\bot_{\\beta\\beta}\\rightarrow\\bot_{\\alpha\\alpha}$ are tautologies, which implies the desired result through the transitive property of implication.\n\n\\begin{theorem}\n$\\mathcal{A}=(A^{\\bI}_{\\Gamma}, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra.\n\\end{theorem}\n\n\\begin{proof}\nFor simplicity, we will drop the index $\\mathcal{A}$. So, it must be proven that, for any formulas $\\alpha$, $\\beta$ and $\\gamma$ in $\\bI$,\n\\begin{enumerate}\n\\item $[\\alpha]\\vee\\bot=[\\alpha]$ and $[\\alpha]\\wedge\\top=[\\alpha]$;\n\\item $[\\alpha]\\vee[\\beta]=[\\beta]\\vee[\\alpha]$ and $[\\alpha]\\wedge[\\beta]=[\\beta]\\wedge[\\alpha]$;\n\\item $[\\alpha]\\vee([\\beta]\\wedge[\\gamma])=([\\alpha]\\vee[\\beta])\\wedge([\\alpha]\\vee[\\gamma])$ and $[\\alpha]\\wedge([\\beta]\\vee[\\gamma])=([\\alpha]\\wedge[\\beta])\\vee([\\alpha]\\wedge[\\gamma])$;\n\\item $[\\alpha]\\vee{\\sim} [\\alpha]=\\top$ and $[\\alpha]\\wedge{\\sim} [\\alpha]=\\bot$;\n\\item $[\\alpha]\\rightarrow[\\beta]={\\sim} [\\alpha]\\vee[\\beta]$.\n\\end{enumerate}\nFor those items containing more than one assertion, we shall only prove the first, being the second completely analogous.\n\\begin{enumerate}\n\\item We must prove that $\\alpha\\vee\\bot_{\\alpha\\alpha}\\equiv_{\\Gamma}^{\\bI}\\alpha$. Clearly $\\Gamma\\vdash_{\\bI}\\alpha\\rightarrow\\alpha\\vee\\bot_{\\alpha\\alpha}$, since this formula is an instance of $\\textbf{Ax\\: 6}$.\n\nWe also know that $\\bot_{\\alpha\\alpha}$ behaves like a bottom element, i.e., for any formula $\\beta$, $\\bot_{\\alpha\\alpha}\\vdash_{\\bI}\\beta$, and therefore $\\Gamma, \\bot_{\\alpha\\alpha}\\vdash_{\\bI}\\alpha$; since $\\alpha\\rightarrow\\alpha$ is a tautology, $\\Gamma, \\alpha\\vdash_{\\bI}\\alpha$, and therefore $\\Gamma, \\alpha\\vee\\bot_{\\alpha\\alpha}\\vdash_{\\bI}\\alpha$, implying $\\Gamma\\vdash_{\\bI}\\alpha\\vee\\bot_{\\alpha\\alpha}\\rightarrow\\alpha$.\n\n\\item The statement is equivalent to $\\alpha\\vee\\beta\\equiv_{\\Gamma}^{\\bI}\\beta\\vee\\alpha$: from the instance $\\beta\\rightarrow(\\alpha\\vee\\beta)$ of $\\textbf{Ax\\: 7}$ and the instance $\\alpha\\rightarrow(\\alpha\\vee\\beta)$ of $\\textbf{Ax\\: 6}$, we get that $\\Gamma,\\beta\\vdash_{\\bI}\\alpha\\vee\\beta$ and $\\Gamma, \\alpha\\vdash_{\\bI}\\alpha\\vee\\beta$, and therefore $\\Gamma, \\beta\\vee\\alpha\\vdash_{\\bI}\\alpha\\vee\\beta$, meaning $\\Gamma\\vdash_{\\bI}(\\beta\\vee\\alpha)\\rightarrow(\\alpha\\vee\\beta)$.\n\nUsing the instances of axioms $\\alpha\\rightarrow(\\beta\\vee\\alpha)$ and $\\beta\\rightarrow(\\beta\\vee\\alpha)$ we get the reciprocal.\n\n\\item We must prove that $\\alpha\\vee(\\beta\\wedge\\gamma)\\equiv_{\\Gamma}^{\\bI}(\\alpha\\vee\\beta)\\wedge(\\alpha\\vee\\gamma)$: $(\\beta\\wedge\\gamma)\\rightarrow\\beta$ and $(\\beta\\wedge\\gamma)\\rightarrow\\gamma$ are instances of $\\textbf{Ax\\: 4}$ and $\\textbf{Ax\\: 5}$, and $\\beta\\rightarrow(\\alpha\\vee\\beta)$ and $\\gamma\\rightarrow(\\alpha\\vee\\gamma)$ are instances of $\\textbf{Ax\\: 7}$, meaning $\\Gamma, (\\beta\\wedge\\gamma)\\vdash_{\\bI}\\alpha\\vee\\beta$ and $\\Gamma, (\\beta\\wedge\\gamma)\\vdash_{\\bI}\\alpha\\vee\\gamma$.\n\nSince $\\alpha\\rightarrow(\\alpha\\vee\\beta)$ and $\\alpha\\rightarrow(\\alpha\\vee\\gamma)$ are instances of $\\textbf{Ax\\: 6}$, $\\Gamma, \\alpha\\vee(\\beta\\wedge\\gamma)\\vdash_{\\bI}\\alpha\\vee\\beta$ and $\\Gamma, \\alpha\\vee(\\beta\\wedge\\gamma)\\vdash_{\\bI}\\alpha\\vee\\gamma$, and from the instance\n\\[\\alpha\\vee\\beta\\rightarrow(\\alpha\\vee\\gamma\\rightarrow((\\alpha\\vee\\beta)\\wedge(\\alpha\\vee\\gamma)))\\]\nof $\\textbf{Ax\\: 3}$ and Modus Ponens, we get that $\\Gamma\\vdash_{\\bI}[\\alpha\\vee(\\beta\\wedge\\gamma)]\\rightarrow[(\\alpha\\vee\\beta)\\wedge(\\alpha\\vee\\gamma)]$.\n\nReciprocally, is enough to show, by $\\textbf{Ax\\: 4}$ and $\\textbf{Ax\\: 5}$, that $\\Gamma, \\alpha\\vee\\beta, \\alpha\\vee\\gamma\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$. Then we have four cases:\n\\begin{enumerate}\n\\item $\\Gamma, \\alpha\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$, from the instance $\\alpha\\rightarrow[\\alpha\\vee(\\beta\\wedge\\gamma)]$ of $\\textbf{Ax\\: 6}$;\n\\item $\\Gamma, \\alpha, \\beta\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$ for the same reason as above;\n\\item $\\Gamma, \\alpha, \\gamma\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$, again for the same reason;\n\\item $\\Gamma, \\beta, \\gamma\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$, from the instance $\\beta\\rightarrow(\\gamma\\rightarrow(\\beta\\wedge\\gamma))$ of $\\textbf{Ax\\: 3}$ and two applications of Modus Ponens that give us $\\Gamma, \\beta, \\gamma\\vdash_{\\bI}\\beta\\wedge\\gamma$, and then the use of $\\textbf{Ax\\: 7}$.\n\\end{enumerate}\nSo, from the first two items, $\\Gamma, \\alpha, \\alpha\\vee\\beta\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$, and from the last two $\\Gamma, \\alpha\\vee\\beta, \\gamma\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma)$, meaning\n\\[\\Gamma, \\alpha\\vee\\beta, \\alpha\\vee\\gamma\\vdash_{\\bI}\\alpha\\vee(\\beta\\wedge\\gamma).\\]\n\n\\item To prove $\\alpha\\vee(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\equiv_{\\Gamma}^{\\bI}\\top_{\\alpha}$, we notice first that $[\\alpha\\vee(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})]\\rightarrow\\top_{\\alpha}$ follows from the fact that $\\top_{\\alpha}$ behaves like a top element, meaning $\\Gamma\\vdash_{\\bI}[\\alpha\\vee(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})]\\rightarrow\\top_{\\alpha}$.\n\nReciprocally, $\\alpha\\vee(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})$ is an instance of an axiom, and for this reason $\\vdash_{\\bI}\\alpha\\vee(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})$, meaning $\\Gamma\\vdash_{\\bI}\\top_{\\alpha}\\rightarrow[\\alpha\\vee(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})]$.\n\n\n\\item Finally, we need now only to prove that $\\alpha\\rightarrow\\beta\\equiv_{\\Gamma}^{\\bI}(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta$, that is, by applying the deduction meta-theorem, $\\Gamma, \\alpha\\rightarrow\\beta\\vdash_{\\bI}(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta$ and $\\Gamma, (\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta\\vdash_{\\bI}\\alpha\\rightarrow\\beta$. \n\nClearly $\\alpha, \\alpha\\rightarrow\\beta\\vdash_{\\bI}(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta$, by using Modus Ponens and the instance $\\beta\\rightarrow(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta$ of $\\textbf{Ax\\: 7}$; meanwhile, $\\alpha\\rightarrow\\bot_{\\alpha\\alpha}, \\alpha\\rightarrow\\beta\\vdash_{\\bI}(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta$ by the instance \n\\[(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\rightarrow(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta\\]\nof $\\textbf{Ax\\: 6}$. From a proof by cases, $(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\alpha, \\alpha\\rightarrow\\beta\\vdash_{\\bI}(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta$, and since $(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\alpha$ is an instance of $\\textbf{Ax\\: 9}^{*}$ and $\\bI$ is tarskian, we obtain the desired result.\n\nReciprocally, $\\beta\\vdash_{\\bI}\\alpha\\rightarrow\\beta$ by $\\textbf{Ax\\: 1}$, and since $\\bot_{\\alpha\\alpha}\\rightarrow\\beta$ is a tautology, $\\alpha\\rightarrow\\bot_{\\alpha\\alpha}\\vdash_{\\bI}\\alpha\\rightarrow\\beta$, by use of the transitivity of implication. By a proof by cases, $(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\vee\\beta\\vdash_{\\bI}\\alpha\\rightarrow\\beta$.\n\\end{enumerate}\n\n\\end{proof}\n\nNow, for a given $\\Gamma$, we can make $\\mathcal{A}=(A^{\\bI}_{\\Gamma}, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ into a Fidel structure, presented as a $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra, for $\\bI$: we simply define\n\\[[\\alpha]\\uparrow[\\beta]=\\{[\\varphi\\uparrow\\psi]\\ :\\  \\varphi\\in [\\alpha], \\psi\\in[\\beta]\\};\\]\nfor $[\\gamma]\\in[\\alpha]\\uparrow[\\beta]$, we must show that $[\\alpha]\\wedge([\\beta]\\wedge[\\gamma])=\\bot$; taking $\\varphi\\in[\\alpha]$ and $\\psi\\in[\\beta]$ such that $\\gamma\\in[\\varphi\\uparrow\\psi]$, this means\n\\[\\varphi\\wedge(\\psi\\wedge\\gamma)\\equiv_{\\Gamma}^{\\bI}\\bot_{\\varphi\\psi}.\\]\nClearly $\\Gamma\\vdash_{\\bI}\\bot_{\\varphi\\psi}\\rightarrow[\\varphi\\wedge(\\psi\\wedge\\gamma)]$, so it remains to be shown that $\\Gamma\\vdash_{\\bI}[\\varphi\\wedge(\\psi\\wedge\\gamma)]\\rightarrow\\bot_{\\varphi\\psi}$. This is equivalent to $\\Gamma, \\varphi, \\psi, \\gamma\\vdash_{\\bI}\\bot_{\\varphi\\psi}$, from $\\textbf{Ax\\: 4}$, and since $\\Gamma\\vdash_{\\bI}\\gamma\\rightarrow(\\varphi\\uparrow\\psi)$, $\\Gamma\\vdash_{\\bI}(\\varphi\\uparrow\\psi)\\rightarrow\\gamma$ and $\\bot_{\\varphi\\psi}=\\varphi\\wedge(\\psi\\wedge(\\varphi\\uparrow\\psi))$, by Modus Ponens this equals\n\\[\\Gamma, \\varphi, \\psi, \\varphi\\uparrow\\psi\\vdash_{\\bI}\\varphi\\wedge(\\psi\\wedge(\\varphi\\uparrow\\psi)),\\]\nbut this is obviously true. We can also prove that if $\\varphi\\in[\\alpha]$ and $\\psi\\in[\\beta]$, $[\\psi\\uparrow\\varphi]\\in[\\alpha]\\uparrow[\\beta]$. Very clearly $[\\varphi\\uparrow\\psi]\\in [\\alpha]\\uparrow[\\beta]$, so it is enough to prove that $[\\psi\\uparrow\\varphi]=[\\varphi\\uparrow\\psi]$, or in other terms, that $\\psi\\uparrow\\varphi\\equiv_{\\Gamma}^{\\bI}\\varphi\\uparrow\\psi$, meaning\n\\[\\Gamma\\vdash_{\\bI}(\\psi\\uparrow\\varphi)\\rightarrow(\\varphi\\uparrow\\psi)\\quad\\text{and}\\quad\\Gamma\\vdash_{\\bI}(\\varphi\\uparrow\\psi)\\rightarrow(\\psi\\uparrow\\varphi);\\]\nand this is clearly true, since both implications are instances of $\\textbf{Comm}$. This shows that $[\\alpha]\\uparrow[\\beta]=[\\beta]\\uparrow[\\alpha]$, since the elements of both are the same.\n\nThe $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra $\\mathcal{A}^{\\bI}_{\\Gamma}$\\label{mAbIGamma} just described, with universe $A^{\\bI}_{\\Gamma}$, is called a Lindenbaum-Tarski multialgebra of $\\bI$, specifically the one associated to $\\Gamma$. \n\n\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bI$, if $\\Gamma\\Vdash^{\\bI}_{\\mathcal{F}}\\varphi$ then $\\Gamma\\vdash_{\\bI}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that $\\Gamma\\not\\vdash_{\\bI}\\varphi$, and we know there exists a closed set of formulas $\\Delta$ of $\\bI$ that contains $\\Gamma$ and is maximal with respect to not proving $\\varphi$. We take the Lindenbaum-Tarski multialgebra $\\mathcal{A}^{\\bI}_{\\Delta}$ of $\\bI$ associated to $\\Delta$ and its associated RNmatrix, and consider the map $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow A^{\\bI}_{\\Delta}$ such that $\\nu(\\alpha)=[\\alpha]$. We have that:\n\\begin{enumerate}\n\\item for any $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\nu(\\alpha\\#\\beta)=[\\alpha\\#\\beta]=[\\alpha]\\#[\\beta]=\\nu(\\alpha)\\#\\nu(\\beta)$;\n\\item $\\nu(\\alpha\\uparrow\\beta)=[\\alpha\\uparrow\\beta]$, which is in $\\{[\\varphi\\uparrow\\psi]\\ :\\  \\varphi\\in [\\alpha], \\psi\\in[\\beta]\\}=[\\alpha]\\uparrow[\\beta]$;\n\\item $\\nu(\\alpha\\uparrow\\beta)=[\\alpha\\uparrow\\beta]=[\\beta\\uparrow\\alpha]=\\nu(\\beta\\uparrow\\alpha)$.\n\\end{enumerate}\nSo $\\nu$ is a valuation over $\\mathcal{A}^{\\bI}_{\\Delta}$ in $\\mathcal{F}_{\\mathcal{A}^{\\bI}_{\\Delta}}$. Furthermore, $\\nu(\\alpha)=\\top$ if and only if $\\Delta\\vdash_{\\bI}\\alpha$; given that $\\Delta\\not\\vdash_{\\bI}\\varphi$, we obtain $\\nu(\\Delta)\\subseteq \\{\\top\\}$ and $\\nu(\\varphi)\\neq\\top$, meaning that $\\Delta\\not\\Vdash^{\\bI}_{\\mathcal{F}}\\varphi$ and therefore $\\Gamma\\not\\Vdash^{\\bI}_{\\mathcal{F}}\\varphi$.\n\\end{proof}\n\n\n\n\\subsection{Decision method}\\label{Decision Method for bI}\n\nFirst of all, we remember the set $\\{0,1\\}$ can be made into a Boolean algebra $\\textbf{2}$ when we define $\\top=1$, $\\bot=0$, and the other operations according to the following tables.\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{2cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n& ${\\sim} $\\\\ \\hline\n$0$ & $1$\\\\ \\hline\n$1$ & $0$\\\\\\hline\n\\end{tabular}\n\\caption*{Negation}\n\\end{minipage}\n\\hspace{1cm}\n\\centering\n\\begin{minipage}[t]{2.5cm}\n\\centering\n\\begin{tabular}{|l|c|r|}\n\\hline\n$\\vee$ & $0$ & $1$\\\\ \\hline\n$0$ & $0$ & $1$\\\\ \\hline\n$1$ & $1$ & $1$\\\\\\hline\n\\end{tabular}\n\\caption*{Disjunction}\n\\end{minipage}\n\\hspace{1cm}\n\\centering\n\\begin{minipage}[t]{2.5cm}\n\\centering\n\\begin{tabular}{|l|c|r|}\n\\hline\n$\\wedge$ & $0$ & $1$\\\\ \\hline\n         $0$ & $0$ & $0$\\\\ \\hline\n         $1$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\caption*{Conjunction}\n\\end{minipage}\n\\hspace{1cm}\n\\centering\n\\begin{minipage}[t]{2.5cm}\n\\centering\n\\begin{tabular}{|l|c|r|}\n\\hline\n$\\rightarrow$ & $0$ & $1$\\\\ \\hline\n             $0$ & $1$ & $1$\\\\ \\hline\n             $1$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\caption*{Implication}\n\\end{minipage}\n\\end{figure}\n\nNotice that, to give one example, technically $\\bot=\\{0\\}$, since we are in the environment of multialgebras; however, given the operations associated to the elements in $\\Sigma^{\\textbf{CPL}}$ are supposed to be single-valued, since $(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra, we will identify a singleton to the element it contains, as is usual.\n\nWe make $\\textbf{2}$ into a Fidel structure $\\textbf{2}_{\\bI}$\\label{2bI}, presented as a $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra, for $\\bI$ by defining $\\uparrow_{\\textbf{2}_{\\bI}}(1,1)=\\{0\\}$ and $\\uparrow_{\\textbf{2}_{\\bI}}(x,y)=\\{0,1\\}$ otherwise. For simplicity, we will denote $\\uparrow_{\\textbf{2}_{\\bI}}$ simply by $\\uparrow$ and use the infix notation.\n\n\\begin{figure}[H]\n\\centering\n\\captionsetup{justification=centering}\n\\begin{tabular}{|l|c|r|}\n\\hline\n$\\uparrow$ & $0$ & $1$\\\\ \\hline\n$0$ & $\\{0,1\\}$ & $\\{0,1\\}$\\\\ \\hline\n$1$ & $\\{0,1\\}$ & $\\{0\\}$\\\\\\hline\n\\end{tabular}\n\\caption*{Table for $\\uparrow$ in $\\textbf{2}_{\\bI}$}\n\\end{figure}\n\nThis is very clearly a Fidel structure for $\\bI$ since, for $z\\in x\\uparrow y$, we have that $x\\wedge(y\\wedge z)=0$: in the case where neither $x$ nor $y$ equals $0$, the only possible value for $z$ is exactly $0$. Furthermore, one always has $x\\uparrow y=y\\uparrow x$.\n\n\\begin{theorem}\n$\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\{0,1\\}$ is a bivaluation for $\\bI$ if, and only if, it is a $\\Sigma_{\\bI}$-homomorphism from $\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})$ to $\\textbf{2}_{\\bI}$ in $\\mathcal{F}_{\\textbf{2}_{\\bI}}$.\n\\end{theorem}\n\n\\begin{proof}\nAssume first that $\\nu$ is a bivaluation. Then:\n\\begin{enumerate}\n\\item if $\\nu(\\alpha)=1$ or $\\nu(\\beta)=1$, we have that $\\nu(\\alpha)\\vee\\nu(\\beta)=1=\\nu(\\alpha\\vee\\beta)$; otherwise, when both $\\nu(\\alpha)$ and $\\nu(\\beta)$ are $0$, we have that $\\nu(\\alpha)\\vee\\nu(\\beta)=0=\\nu(\\alpha\\vee\\beta)$;\n\\item if both $\\nu(\\alpha)$ and $\\nu(\\beta)$ are $1$, $\\nu(\\alpha)\\wedge\\nu(\\beta)=1=\\nu(\\alpha\\wedge\\beta)$; otherwise, $\\nu(\\alpha)\\wedge\\nu(\\beta)=0$, which is also the value of $\\nu(\\alpha\\wedge\\beta)$;\n\\item if either $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$, $\\nu(\\alpha)\\rightarrow\\nu(\\beta)=1=\\nu(\\alpha\\rightarrow\\beta)$; in the case that $\\nu(\\alpha)=1$ and $\\nu(\\beta)=0$, $\\nu(\\alpha)\\rightarrow\\nu(\\beta)=0=\\nu(\\alpha\\rightarrow\\beta)$;\n\\item for any value of $\\nu(\\alpha\\uparrow\\beta)$ such that either $\\nu(\\alpha)$ or $\\nu(\\beta)$ is not $1$, it is clear $\\nu(\\alpha\\uparrow\\beta)\\in\\nu(\\alpha)\\uparrow\\nu(\\beta)$, since in those cases $\\nu(\\alpha)\\uparrow\\nu(\\beta)$ is the whole universe of $\\textbf{2}_{\\bI}$, so assume $\\nu(\\alpha)=\\nu(\\beta)=1$; if we had $\\nu(\\alpha\\uparrow\\beta)=1$, by the fact $\\nu(\\alpha\\uparrow\\beta)=1$ and $\\nu(\\alpha)=1$ we would get that $\\nu(\\beta)=0$, which is a contradiction, forcing us to have\n\\[\\nu(\\alpha\\uparrow\\beta)=0\\in\\{0\\}=1\\uparrow1=\\nu(\\alpha)\\uparrow\\nu(\\beta);\\]\n\\item since $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$ and, from the previous items, $\\nu$ is a $\\Sigma_{\\bI}$-homomorphism, we have that $\\nu$ lies in $\\mathcal{F}_{\\textbf{2}_{\\bI}}$\n\\end{enumerate}\nTo summarize, $\\nu$ is a $\\Sigma_{\\bI}$-homomorphism in $\\mathcal{F}_{\\textbf{2}_{\\bI}}$.\n\nReciprocally, assume $\\nu$ is a $\\Sigma_{\\bI}$-homomorphism which lies in $\\mathcal{F}_{\\textbf{2}_{\\bI}}$.\n\\begin{enumerate}\n\\item since $\\nu(\\alpha\\vee\\beta)=\\nu(\\alpha)\\vee\\nu(\\beta)$, looking at the table for disjunction we find that $\\nu(\\alpha\\vee\\beta)=1$ if and only if $\\nu(\\alpha)=1$ or $\\nu(\\beta)=1$;\n\\item since $\\nu(\\alpha\\wedge\\beta)=\\nu(\\alpha)\\wedge\\nu(\\beta)$, $\\nu(\\alpha\\wedge\\beta)=1$ if and only if $\\nu(\\alpha)=\\nu(\\beta)=1$;\n\\item given $\\nu(\\alpha\\rightarrow\\beta)=\\nu(\\alpha)\\rightarrow\\nu(\\beta)$, $\\nu(\\alpha\\rightarrow\\beta)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$;\n\\item if $\\nu(\\alpha\\uparrow\\beta)=1$, from the table for $\\uparrow$ there are three possible cases: $\\nu(\\alpha)=\\nu(\\beta)=0$, $\\nu(\\alpha)=0$ and $\\nu(\\beta)=1$, and $\\nu(\\alpha)=1$ and $\\nu(\\beta)=0$; so if $\\nu(\\alpha\\uparrow\\beta)=1$ and $\\nu(\\alpha)=1$, we have that $\\nu(\\beta)=0$;\n\\item since $\\nu\\in\\mathcal{F}_{\\textbf{2}_{\\bI}}$, $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$.\n\\end{enumerate}\nThis finishes proving that $\\nu$ is then a bivaluation.\n\\end{proof}\n\nWe will denote the restricted Nmatrix $(\\textbf{2}_{\\bI}, \\{1\\}, \\mathcal{F}_{\\textbf{2}_{\\bI}})$ by $\\mathbb{2}_{\\bI}$.\n\n\\begin{theorem}\\label{Decision method for BI proof}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bI$, $\\Gamma\\vDash_{\\bI}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathbb{2}_{\\bI}}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is quite straightforward: suppose $\\Gamma\\vDash_{\\bI}\\varphi$; then, for any $\\Sigma_{\\bI}$-homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\textbf{2}_{\\bI}$ in $\\mathcal{F}_{\\textbf{2}_{\\bI}}$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$, from the previous theorem we have that $\\nu$ is a bivaluation, and since $\\nu(\\Gamma)\\subseteq\\{1\\}$ and $\\Gamma\\vDash_{\\bI}\\varphi$, we find that $\\nu(\\varphi)=1$. Therefore, $\\Gamma\\vDash_{\\textbf{2}_{\\bI}}\\varphi$.\n\nThe reciprocal is analogous.\n\\end{proof}\n\nAs it was discussed in depth in Section \\ref{Decision methods}, not all finite RNmatrices lead straightforwardly to decision methods by row-branching, row-eliminating truth tables, since there may not exist an algorithm to decide whether a given row is, or is not, a restricted valuation. As luck would have it, in the case of $\\bI$ the RNmatrix $\\mathbb{2}_{\\bI}$ does indeed offer a decision method; we will, however, just briefly discuss how it works, the complete proof being quite similar to the one found in Section \\ref{Row-branching,row-eliminating}.\n\nWe start as it is usual: for a given formula $\\varphi$, produce a list of its subformulas in non-decreasing order of complexity $\\varphi_{1}, \\dotsc , \\varphi_{n}=\\varphi$; the complexity, also known as order, of a formula $\\varphi$ of $\\bI$ is defined as $0$ if $\\varphi$ is a variable, and if $\\alpha$ and $\\beta$ have complexity $p$ and $q$, respectively, and $\\varphi=\\alpha\\#\\beta$ for a $\\#\\in\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$, then the complexity of $\\varphi$ is $\\max\\{p, q\\}+1$.\n\nWe produce a table with columns headed by the formulas $\\varphi_{i}$; we have some $1\\leq m<n$ such that all $\\varphi_{1}, \\dotsc , \\varphi_{m}$ are propositional variables, and so we write down $2^{m}$ initial rows, each with a possible combination of zeros and ones for these variables. Then we start filling in the rows for $\\varphi_{m+1},\\dotsc , \\varphi_{n}$ in the following way: if $\\varphi_{l}=\\varphi_{i}\\#\\varphi_{j}$ (where necessarily $i, j< l$), $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$ and, on a certain row, $\\varphi_{i}$ is given the value $a$ and $\\varphi_{j}$ is given the value $b$, then on this very row $\\varphi_{l}$ is given the value $a\\#b$. If $\\varphi_{l}=\\varphi_{i}\\uparrow\\varphi_{j}$ and both $\\varphi_{i}$ and $\\varphi_{j}$ are $1$ on a row, then $\\varphi_{l}$ is $0$; if, however, on a specific row $\\varphi_{i}$ or $\\varphi_{j}$ takes the value $0$, then this row is split in two, one where $\\varphi_{l}$ assumes the value $0$, the other where it assumes the value $1$.\n\nThe caveat of this last part is that undesired valuations need to be erased: if there is a formula $\\varphi_{k}=\\varphi_{j}\\uparrow\\varphi_{i}$, with $i, j<k<l$, that takes the value $0$ in the present row, then the row where $\\varphi_{l}=\\varphi_{i}\\uparrow\\varphi_{l}$ takes the value $1$ is erased, in order to $\\varphi_{i}\\uparrow\\varphi_{j}$ and $\\varphi_{j}\\uparrow\\varphi_{i}$ be given the same value; analogously, if in this row $\\varphi_{k}$ is $1$, the row where $\\varphi_{l}$ is $0$ is eliminated.\\footnote{Of course, one could write down the row-branching truth table for $\\varphi$, corresponding to the Nmatrix subjacent to $\\mathbb{2}_{\\bI}$, without simultaneously erasing the undesired rows, and only at the end select those rows where $\\varphi_{i}\\uparrow\\varphi_{j}$ and $\\varphi_{j}\\uparrow\\varphi_{i}$ are given different values to eliminate; this may be less efficient, but the result is the same.} At the end, the column headed by $\\varphi$ is filled with nothing but ones if, and only if, the formula is a tautology for $\\bI$.\n\nFor a finite set $\\Gamma=\\{\\gamma_{1}, \\dotsc , \\gamma_{n}\\}$, we can test whether $\\Gamma\\vdash_{\\bI}\\varphi$ by simply testing if $\\gamma_{1}\\rightarrow(\\cdots\\rightarrow(\\gamma_{n}\\rightarrow\\varphi)\\cdots)$ or $\\bigwedge_{i=1}^{n}\\gamma_{i}\\rightarrow\\varphi$ is a tautology in the same logic; of course, we can be more efficient and simply merge the tables for $\\varphi$ and all $\\gamma_{i}$ and check if, in all rows where all elements of $\\Gamma$ are designated, so is $\\varphi$.\n\nTo give one example of how our row-branching, row-eliminating truth tables work, consider the deduction $p, p\\uparrow q\\vdash_{\\bI}q\\rightarrow (q\\uparrow p)$; basically, it states that if $p$ is true and incompatible with $q$, then $q$ must not be true (notice that $q\\rightarrow(q\\uparrow p)$, in the presence of $p$, is equivalent to ${\\sim}q$).\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$p$ & $q$ & $p\\uparrow q$ & $q\\uparrow p$ & $q\\rightarrow(q\\uparrow p)$\\\\ \\hline\n$1$ & $1$ & $0$ & $0$ & $0$\\\\\\hline\n\\multirow{2}{*}{$1$} & \\multirow{2}{*}{$0$} & $1$ & $1$ & $1$\\\\ \\cline{3-5}\n& & $0$ & $0$ & $1$\\\\\\hline\n\\multirow{2}{*}{$0$} & \\multirow{2}{*}{$1$} & $1$ & $1$ & $1$\\\\ \\cline{3-5}\n& & $0$ & $0$ & $0$\\\\\\hline\n\\multirow{2}{*}{$0$} & \\multirow{2}{*}{$0$} & $1$ & $1$ & $1$\\\\ \\cline{3-5}\n& & $0$ & $0$ & $1$\\\\\\hline\n\\end{tabular}\n\\end{center}\n\nNotice that, since in every row where both $p$ and $p\\uparrow q$ are $1$ so is $q\\rightarrow(q\\uparrow p)$, the deduction is valid.\n\n\\begin{theorem}\nLet $\\mathbb{2}_{\\textbf{bI}^{-}}$ be the Nmatrix $(\\textbf{2}_{\\bI}, \\{1\\})$; for any formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\textbf{bI}^{-}$, $\\Gamma\\vDash_{\\textbf{bI}^{-}}\\varphi$ if and only if $\\Gamma\\vDash_{\\textbf{bI}^{-}}\\varphi$.\n\\end{theorem}\n\nThis result, whose proof is very similar to the one of Theorem \\ref{Decision method for BI proof}, shows a strange property of logics of incompatibility and their semantics which deserves to be further studied: while none of the logics $\\textbf{bI}^{-}$ and $\\textbf{CPL}$, respectively weaker and stronger than $\\textbf{bI}$, needs RNmatrices to be characterized, $\\textbf{bI}$, lying squarely in the middle, does indeed require RNmatrices, what is formally proved in Section \\ref{bI is not characterized by Nmatrices section}.\n\n\\subsection{Another decision method}\\label{Tableaux for bI}\n\nIn Section \\ref{Tableaux semantics for Cn} we showed how it is sometimes possible to extract from the tables of the operations of a finite RNmatrix tableau rules that, when paired with adequate closure conditions for branches, produce a tableau semantics for the logic characterized by said RNmatrix. This can again be done in $\\bI$, but the full proof that this works is left as an exercise: it follows the lines of the same result for $C_{n}$, being however much simpler.\n\n\\begin{definition}\nThe following are the tableau rules with labeled formulas, where $\\varphi$ and $\\psi$ are formulas of $\\bI$, and $\\textsf{0}$ and $\\textsf{1}$ are all possible labels, for the tableau calculus we denote by $\\mathbb{T}_{\\bI}$.\n\n$$\n\\begin{array}{cp{1.5cm}cp{1.5cm}c}\n\\displaystyle \\frac{\\textsf{0}(\\varphi\\vee\\psi)}{\\begin{array}{c}\\textsf{0}(\\varphi) \\\\ \\textsf{0}(\\psi)\\end{array}} & & \\displaystyle \\frac{\\textsf{0}(\\varphi\\wedge\\psi)}{\\textsf{0}(\\varphi)\\mid\\textsf{0}(\\psi)} & & \\displaystyle \\frac{\\textsf{0}(\\varphi\\rightarrow\\psi)}{\\begin{array}{c}\\textsf{1}(\\varphi) \\\\ \\textsf{0}(\\psi)\\end{array}}  \\\\[2mm]\n&&&&\\\\[2mm]\n \\displaystyle \\frac{\\textsf{1}(\\varphi\\vee\\psi)}{\\textsf{1}(\\varphi)\\mid\\textsf{1}(\\psi)} & & \\displaystyle \\frac{\\textsf{1}(\\varphi\\wedge\\psi)}{\\begin{array}{c}\\textsf{1}(\\varphi) \\\\ \\textsf{1}(\\psi)\\end{array}} & & \\displaystyle \\frac{\\textsf{1}(\\varphi\\rightarrow\\psi)}{\\textsf{0}(\\varphi)\\mid\\textsf{1}(\\psi)}\\\\[2mm]\n&&&&\\\\[2mm]\n\\end{array}\n$$\n\n\\[\\frac{\\textsf{1}(\\varphi\\uparrow\\psi)}{\\textsf{0}(\\varphi)\\mid\\textsf{0}(\\psi)}\\]\n\nBranches of tableaux in $\\mathbb{T}_{\\bI}$ are closed iff:\n\\begin{enumerate}\n\\item they contain labeled formulas $\\textsf{L}(\\varphi)$ and $\\textsf{L}^{*}(\\varphi)$ with $\\textsf{L}\\neq\\textsf{L}^{*}$;\n\\item or they contain labeled formulas $\\textsf{L}(\\varphi\\uparrow\\psi)$ and $\\textsf{L}^{*}(\\psi\\uparrow\\varphi)$ with $\\textsf{L}\\neq\\textsf{L}^{*}$.\n\\end{enumerate}\n\nA branch $\\theta$ is complete whenever it contains, for each $\\textsf{L}(\\gamma)$ in $\\theta$ not of the form $\\textsf{0}(\\varphi\\uparrow\\psi)$ and with $\\gamma$ not a variable, all the labeled formulas of one of the branches in the rule headed by $\\textsf{L}(\\gamma)$; complete branches are open if they are not closed.\n\nA tableau in $\\mathbb{T}_{\\bI}$ is: closed if all of its branches are closed; complete if all of its branches are either complete or closed; and open if it is complete but not closed.\n\\end{definition}\n\nOf course, the labels $\\textsf{0}$ and $\\textsf{1}$ correspond, respectively, to the values $0$ and $1$ of $\\mathbb{2}_{\\bI}$; notice, furthermore, that if we had included in the signature of $\\bI$ the classical negation that is definable in this logic, it wouldn't be necessary to use labels, but this is in no way mandatory for our tableaux to properly work.\n\nNotice that all the rules found in $\\mathbb{T}_{\\bI}$ are analytic, in the sense that the complexities of the formulas obtained from applying a rule are strictly smaller than the complexity of the formula that motivated the application of the rule. This means that all tableaux in $\\mathbb{T}_{\\bI}$ can be completed in a finite number of steps. A formula $\\varphi$ of $\\bI$ is provable according to tableaux in $\\mathbb{T}_{\\bI}$, when we write $\\vdash_{\\mathbb{T}_{\\bI}}\\varphi$, if there is a closed tableau in $\\mathbb{T}_{\\bI}$ starting from $\\textsf{0}(\\varphi)$; for a finite set of formulas $\\Gamma=\\{\\gamma_{1}, \\dotsc , \\gamma_{n}\\}$ of $\\bI$, we say $\\Gamma$ proves $\\varphi$ according to $\\mathbb{T}_{\\bI}$ if \n\\[\\vdash_{\\mathbb{T}_{\\bI}}\\bigwedge_{i=1}^{n}\\gamma_{n}\\rightarrow\\varphi,\\]\nor alternatively \n\\[\\vdash_{\\mathbb{T}_{\\bI}}\\gamma_{1}\\rightarrow (\\gamma_{2}\\rightarrow\\cdots (\\gamma_{n}\\rightarrow\\varphi)\\cdots).\\]\nThe following theorem is proved as in the case of da Costa's hierarchy, found in Section \\ref{Tableaux semantics for Cn}, and shows we indeed have a decision method.\n\n\n\n\\begin{theorem}\nFor any finite set of formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bI$, $\\Gamma\\vdash_{\\bI}\\varphi$ if, and only if, $\\Gamma\\vdash_{\\mathbb{T}_{\\bI}}\\varphi$.\n\\end{theorem}\n\n\n\n\\section{Other logics}\n\nOf course, when exploring a new logical environment, there are many possibilities that arise according to your philosophical interpretation of what the logic at hand should accomplish.\n\nThe most basic logic of formal incompatibility, $\\bI$, is just that, the most basic one: there are many, very natural, different logics of formal incompatibility that occur when dealing with this subject, from which we present a few in the next sections.\n\n\n\n\\subsection{The logic $\\bIpr$}\n\nThe more one analyzes the concept of incompatibility, the more it seems it should have some sort of interplay, of relationship with other logical objects such as disjunction and conjunction. In the logic we shall call $\\bIpr$\\label{bIpr} we add axioms relating $\\uparrow$ with $\\vee$ and $\\wedge$.\n\nIt seems that, in natural language, when $\\gamma$ is incompatible with $\\alpha$ and incompatible with $\\beta$, then it is incompatible with the disjunction of $\\alpha$ and $\\beta$ ; alternatively, when $\\gamma$ is incompatible with $\\alpha$ or incompatible with $\\beta$, then it is incompatible with the conjunction of $\\alpha$ and $\\beta$.\n\nTo see how $\\textbf{CPL}$ would approach this, we remember the usual interpretation of the Sheffer's stroke is $\\alpha\\Uparrow\\beta={\\sim}(\\alpha\\wedge\\beta)$; then, we see that\n\\[(\\alpha\\vee\\beta)\\Uparrow\\gamma={\\sim}\\big[(\\alpha\\vee\\beta)\\wedge\\gamma\\big],\\]\nand by using the distributivity of conjunction over disjunction this is equivalent to ${\\sim}[(\\alpha\\wedge\\gamma)\\vee(\\beta\\wedge\\gamma)]$, which by De Morgan's law is equivalent to\n\\[{\\sim}(\\alpha\\wedge\\gamma)\\wedge{\\sim}(\\beta\\wedge\\gamma)=(\\alpha\\Uparrow\\gamma)\\wedge(\\beta\\Uparrow\\gamma),\\]\nas we suggested before. Quite analogously, $(\\alpha\\wedge\\beta)\\Uparrow\\gamma$ is equivalent to $(\\alpha\\Uparrow\\gamma)\\vee(\\beta\\Uparrow\\gamma)$ in $\\textbf{CPL}$.\n\nTo make $\\bI$ more similar to the classical account of incompatibility, we add to it axiom schemata regarding the propagation of incompatibility\\label{prwedge}\\label{prvee}\n\\[\\tag{$\\textbf{pr}_{\\wedge}$}\\big[(\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma)\\big]\\rightarrow\\big[(\\alpha\\vee\\beta)\\uparrow\\gamma\\big]\\]\nand\n\\[\\tag{$\\textbf{pr}_{\\vee}$}\\big[(\\alpha\\uparrow\\gamma)\\vee(\\beta\\uparrow\\gamma)\\big]\\rightarrow\\big[(\\alpha\\wedge\\beta)\\uparrow\\gamma\\big],\\]\nwhat gives us the logic $\\bIpr$.\n\nWe can prove that $\\bIpr$ is strictly stronger than $\\bI$: take the instance $[(p_{1}\\uparrow p_{3})\\wedge(p_{2}\\uparrow p_{3})]\\rightarrow[(p_{1}\\vee p_{2})\\uparrow p_{3}]$ of $\\textbf{pr}_{\\wedge}$, which is clearly true in $\\bIpr$, and take a bivaluation $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\{0,1\\}$ for $\\bI$ such that $\\nu(p_{1})=\\nu(p_{2})=\\nu(p_{3})=0$,\n\\[\\nu((p_{1}\\vee p_{2})\\uparrow p_{3})=0,\\quad \\nu(p_{1}\\uparrow p_{3})=1\\quad\\text{and}\\quad\\nu(p_{2}\\uparrow p_{3})=1.\\]\nThen we have that \n\\[\\nu([(p_{1}\\uparrow p_{3})\\wedge(p_{2}\\uparrow p_{3})]\\rightarrow[(p_{1}\\vee p_{2})\\uparrow p_{3}])=\\nu((p_{1}\\uparrow p_{3})\\wedge(p_{2}\\uparrow p_{3}))\\rightarrow\\nu((p_{1}\\vee p_{2})\\uparrow p_{3})=\\]\n\\[[\\nu(p_{1}\\uparrow p_{3})\\wedge\\nu(p_{2}\\uparrow p_{3})]\\rightarrow0=(1\\wedge1)\\rightarrow0=1\\rightarrow0=0,\\]\nmeaning $\\bI$ can not actually prove some instances of $\\textbf{pr}_{\\wedge}$.\n\n\n\\subsubsection{Bivaluations}\n\n\\begin{definition}\\label{Bivaluation for bIpr}\nA bivaluation for $\\bIpr$ is a map $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\{0,1\\}$ that is a bivaluation for $\\bI$ and also satisfies that, for any formulas $\\alpha$, $\\beta$ and $\\gamma$,\n\\[\\text{if}\\quad \\nu(\\alpha\\uparrow\\gamma)=1\\quad\\text{and}\\quad\\nu(\\beta\\uparrow\\gamma)=1,\\quad\\text{then}\\quad\\nu((\\alpha\\vee\\beta)\\uparrow\\gamma)=1,\\]\nand\n\\[\\text{if}\\quad\\nu(\\alpha\\uparrow\\gamma)=1\\quad\\text{or}\\quad\\nu(\\beta\\uparrow \\gamma)=1,\\quad\\text{then}\\quad \\nu((\\alpha\\wedge\\beta)\\uparrow\\gamma)=1.\\]\n\\end{definition}\n\n\\begin{definition}\nGiven a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bIpr$, we say $\\Gamma$ proves $\\varphi$ according to bivaluations for $\\bIpr$, and write $\\Gamma\\vDash_{\\bIpr}\\varphi$\\label{vDashbIpr}, if for every bivaluation $\\nu$ for $\\bIpr$ we have that, if $\\nu(\\Gamma)\\subseteq\\{1\\}$, then $\\nu(\\varphi)=1$.\n\\end{definition}\n\nAs it would be expected, given instances $\\phi=[(\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma)]\\rightarrow[(\\alpha\\vee\\beta)\\uparrow\\gamma]$ of $\\textbf{pr}_{\\wedge}$ and $\\psi=[(\\alpha\\uparrow\\gamma)\\vee(\\beta\\uparrow\\gamma)]\\rightarrow[(\\alpha\\wedge\\beta)\\uparrow\\gamma]$ of $\\textbf{pr}_{\\vee}$, we have that $\\vDash_{\\bIpr}\\phi$ and $\\vDash_{\\bIpr}\\psi$, meaning that for any bivaluation $\\nu$ for $\\bIpr$, $\\nu(\\phi)=\\nu(\\psi)=1$.\n\nTo prove that, we will start with $\\phi$: we have $\\nu(\\phi)=1$ if and only if $\\nu((\\alpha\\vee\\beta)\\uparrow\\gamma)=1$, or \n\\[\\nu((\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma))=0.\\]\nIf $\\nu((\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma))=0$ there is nothing to be done, so let us assume $\\nu((\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma))=1$: in this case, since $\\nu$ is a bivaluation for $\\bI$, $\\nu(\\alpha\\uparrow\\gamma)=1$ and $\\nu(\\beta\\uparrow\\gamma)=1$, meaning $\\nu((\\alpha\\vee\\beta)\\uparrow\\gamma)=1$. Either way, $\\nu(\\phi)=1$.\n\nNow $\\nu(\\psi)=1$ if and only if $\\nu((\\alpha\\wedge\\beta)\\uparrow\\gamma)=1$ or \n\\[\\nu((\\alpha\\uparrow\\gamma)\\vee(\\beta\\uparrow\\gamma))=0;\\]\nas before, if $\\nu((\\alpha\\uparrow\\gamma)\\vee(\\beta\\uparrow\\gamma))=0$ there is nothing to prove, so let us assume $\\nu((\\alpha\\uparrow\\gamma)\\vee(\\beta\\uparrow\\gamma))=1$. In this case, $\\nu(\\alpha\\uparrow\\gamma)=1$ or $\\nu(\\beta\\uparrow\\gamma)=1$, implying $\\nu((\\alpha\\wedge\\beta)\\uparrow\\gamma)=1$. Once again, we derive that in any case one has $\\nu(\\psi)=1$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bIpr$, if $\\Gamma\\vdash_{\\bIpr}\\varphi$ then $\\Gamma\\vDash_{\\bIpr}\\varphi$.\n\\end{theorem}\n\nTo prove the reciprocal, we use the standard method of defining $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\{0,1\\}$, for a closed, non-trivial set of formulas $\\Gamma$ of $\\bIpr$ maximal with respect to not proving a given $\\varphi$, as $\\nu(\\gamma)=1$ if and only if $\\gamma\\in \\Gamma$.\n\nWe already know that $\\nu$ is a bivaluation for $\\bI$, since a set of formulas that is closed under $\\bIpr$ must be closed under $\\bI$, given $\\bIpr$ extends $\\bI$. So assume $\\nu(\\alpha\\uparrow\\gamma)=1$ and $\\nu(\\beta\\uparrow\\gamma)=1$, meaning that $\\alpha\\uparrow\\gamma, \\beta\\uparrow\\gamma\\in \\Gamma$; from the instance \n\\[\\big[(\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma)\\big]\\rightarrow\\big[(\\alpha\\vee\\beta)\\uparrow\\gamma\\big]\\]\nof $\\textbf{pr}_{\\wedge}$ and the fact that $\\Gamma$ is closed, we derive $(\\alpha\\vee\\beta)\\uparrow\\gamma\\in\\Gamma$, and therefore $\\nu((\\alpha\\vee\\beta)\\uparrow\\gamma)=1$, proving the first extra condition of Definition \\ref{Bivaluation for bIpr}. To prove the remaining property, assume $\\nu(\\alpha\\uparrow\\gamma)=1$ or $\\nu(\\beta\\uparrow\\gamma)=1$, meaning either $\\alpha\\uparrow\\gamma\\in\\Gamma$ or $\\beta\\uparrow\\gamma\\in\\Gamma$. From the instance \n\\[\\big[(\\alpha\\uparrow\\gamma)\\vee(\\beta\\uparrow\\gamma)\\big]\\rightarrow\\big[(\\alpha\\wedge\\beta)\\uparrow\\gamma\\big]\\]\nof $\\textbf{pr}_{\\vee}$ and the fact $\\Gamma$ is closed, we get that $(\\alpha\\wedge\\beta)\\uparrow\\gamma\\in \\Gamma$, and so $\\nu((\\alpha\\wedge\\beta)\\uparrow\\gamma)=1$; this finishes the proof.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bIpr$, if $\\Gamma\\vDash_{\\bIpr}\\varphi$ then $\\Gamma\\vdash_{\\bIpr}\\varphi$.\n\\end{theorem}\n\nNotice, and the importance of this fact shall become clearer when we arrive at Section \\ref{Collapsing axioms part 1}, that $\\textbf{bIpr}$ is not classical propositional logic, with $\\alpha\\uparrow\\beta$ standing for $\\alpha\\Uparrow\\beta={\\sim}(\\alpha\\wedge\\beta)$. To see this, take propositional variables $p$ and $q$ and a bivaluation $\\nu$ for $\\bIpr$ with $\\nu(p)=\\nu(q)=\\nu(p\\uparrow q)=0$: it is clear such a bivaluation is possible, and yet $\\nu(p\\Uparrow q)=1$, implying that $p\\Uparrow q\\not\\vdash_{\\bIpr}p\\uparrow q$.\n\n\n\n\\subsubsection{Fidel Structures}\n\n\nThe Fidel structures, as $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebras, for $\\bIpr$ will simply be the same as those for $\\bI$. For every such Fidel structure, presented as a $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra, we consider the restricted Nmatrix $(\\mathcal{A}, \\{\\top\\}, \\mathcal{F}_{\\mathcal{A}})$, where $\\mathcal{F}_{\\mathcal{A}}$ is the set of homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\mathcal{A}$ such that:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$, for any formulas $\\alpha$ and $\\beta$;\n\\item for any instance $\\phi$ of $\\textbf{pr}_{\\wedge}$, $\\nu(\\phi)=\\top$;\n\\item for any instance $\\psi$ of $\\textbf{pr}_{\\vee}$, $\\nu(\\psi)=\\top$.\n\\end{enumerate}\n\nIt is clear how such conditions keep the generalized Nmatrices in question structural: if $\\sigma$ is an endomorphism of $\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})$, and $\\phi$ is an instance of $\\textbf{pr}_{\\wedge}$ and $\\psi$ is an instance of $\\textbf{pr}_{\\vee}$, then $\\sigma(\\phi)$ and $\\sigma(\\psi)$ remain instances of, respectively, $\\textbf{pr}_{\\wedge}$ and $\\textbf{pr}_{\\vee}$.\n\nThe second and third conditions in the previous definition can be modified, if we remember a few simple properties of Boolean algebras: to give one example, take an instance \n\\[\\phi=\\big[(\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma)\\big]\\rightarrow\\big[(\\alpha\\vee\\beta)\\uparrow\\gamma\\big]\\]\nof $\\textbf{pr}_{\\wedge}$; applying to $\\phi$ a valuation $\\nu$ in $\\mathcal{F}_{\\mathcal{A}}$ gives us\n\\[{\\sim}\\nu(\\alpha\\uparrow\\gamma)\\vee{\\sim}\\nu(\\beta\\uparrow\\gamma)\\vee\\nu((\\alpha\\vee\\beta)\\uparrow\\gamma),\\]\nso, asking that $\\nu(\\phi)=\\top$ is equivalent to asking that $x\\vee{\\sim}y_{1}\\vee{\\sim}y_{2}=\\top$, where $\\nu((\\alpha\\vee\\beta)\\uparrow\\gamma)=x$, $\\nu(\\alpha\\uparrow\\gamma)=y_{1}$ and $\\nu(\\beta\\uparrow\\gamma)=y_{2}$.\n\nIf $\\Gamma$ proves $\\varphi$ according to the class of these restricted Nmatrices, we write $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bIpr}\\varphi$\\label{VdashFbIpr}.\n\nWe already know that the axioms and rules of inference of $\\bI$ are modeled by \"$\\Vdash_{\\mathcal{F}}^{\\bIpr}$\", remaining for us to show that this is also true for those axioms particular to $\\bIpr$, that is, $\\textbf{pr}_{\\wedge}$ and $\\textbf{pr}_{\\vee}$, meaning that for any instance $\\phi$ of $\\textbf{pr}_{\\wedge}$ and any instance $\\psi$ of $\\textbf{pr}_{\\vee}$, we wish to prove $\\Vdash_{\\mathcal{F}}^{\\bIpr}\\phi$ and $\\Vdash_{\\mathcal{F}}^{\\bIpr}\\psi$.\n\nThis translates to proving that, for any Fidel structure $\\mathcal{A}$ for $\\bI$ and any homomorphism $\\nu\\in\\mathcal{F}_{\\mathcal{A}}$, $\\nu(\\phi)=\\top$ and $\\nu(\\psi)=\\top$, which is trivially true given our definition of $\\mathcal{F}_{\\mathcal{A}}$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bIpr$, if $\\Gamma\\vdash_{\\bIpr}\\varphi$ then $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bIpr}\\varphi$.\n\\end{theorem}\n\nFor the reciprocal, once again we define an equivalence relation on the formulas of $\\bIpr$ such that $\\alpha\\equiv_{\\Gamma}^{\\bIpr}\\beta$ if and only if $\\Gamma\\vdash_{\\bIpr}\\alpha\\rightarrow\\beta$ and $\\Gamma\\vdash_{\\bIpr}\\beta\\rightarrow\\alpha$. As mentioned, this is indeed an equivalence relation, but also a congruence for the connectives in $\\{\\vee,\\wedge,\\rightarrow\\}$, allowing us to make $A^{\\bIpr}_{\\Gamma}=F(\\Sigma_{\\bI}, \\mathcal{V})\/\\equiv_{\\Gamma}^{\\bIpr}$ into a Boolean algebra as we did when studying $\\bI$ in Section \\ref{Fidel structures for bI}.\n\nWe define, for formulas $\\alpha$ and $\\beta$,\n\\[\\uparrow_{\\mathcal{A}^{\\bIpr}_{\\Gamma}}([\\alpha],[\\beta])=\\{[\\varphi\\uparrow\\psi]\\ :\\  \\varphi\\in[\\alpha], \\psi\\in[\\beta]\\}.\\]\n\nFor simplicity, let us drop the index $\\mathcal{A}^{\\bIpr}_{\\Gamma}$ and use the infix notation; first of all, it is clear how, with such a definition, $\\mathcal{A}^{\\bIpr}_{\\Gamma}=(A^{\\bIpr}_{\\Gamma}, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\uparrow}})$ is indeed a Fidel structure for $\\bI$, presented as a $\\Sigma_{\\bI}^{\\textbf{CPL}}$-multialgebra. We will call $\\mathcal{A}^{\\bIpr}_{\\Gamma}$ the Lindenbaum-Tarski Fidel structure of $\\bIpr$ associated to $\\Gamma$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bIpr$, if $\\Gamma\\Vdash_{\\mathcal{F}}^{\\bIpr}\\varphi$, then $\\Gamma\\vdash_{\\bIpr}\\varphi$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $\\Gamma\\not\\vdash_{\\bIpr}\\varphi$ and take a maximal, with respect to not proving $\\varphi$, closed extension $\\Delta$ of $\\Gamma$ and consider: the Lindenbaum-Tarski Fidel structure $\\mathcal{A}^{\\bIpr}_{\\Delta}$ of $\\bIpr$ associated to $\\Delta$, and the map $\\nu:F(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow A^{\\bIpr}_{\\Delta}$ such that $\\nu(\\alpha)=[\\alpha]$.\n\nClearly $\\nu$ is a $\\Sigma_{\\bI}$-homomorphism, given $\\nu(\\alpha\\#\\beta)=\\nu(\\alpha)\\#\\nu(\\beta)$ for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$ and $\\nu(\\alpha\\uparrow\\beta)\\in[\\alpha]\\uparrow[\\beta]$. It remains for us to show that $\\nu\\in\\mathcal{F}_{\\mathcal{A}}$:\n\\begin{enumerate}\n\\item since $(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha)$ and $(\\beta\\uparrow\\alpha)\\rightarrow(\\alpha\\uparrow\\beta)$ are instances of $\\textbf{Comm}$, we have that $[\\alpha\\uparrow\\beta]=[\\beta\\uparrow\\alpha]$, and therefore $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$;\n\\item given an instance $\\phi$ of $\\textbf{pr}_{\\vee}$, we have that $\\Gamma\\vdash_{\\bIpr}\\phi\\rightarrow\\top_{\\phi}$ and $\\Gamma\\vdash_{\\bIpr}\\top_{\\phi}\\rightarrow\\phi$, meaning $[\\phi]=\\top$ and therefore $\\nu(\\phi)=\\top$;\n\\item for the case of $\\textbf{pr}_{\\wedge}$, the proof follows that of $\\textbf{pr}_{\\vee}$.\n\\end{enumerate}\nWe see that $\\nu(\\alpha)=\\top$ if and only if $\\Delta\\vdash_{\\bIpr}\\alpha$, and so $\\Delta\\not\\Vdash_{\\mathcal{F}}^{\\bIpr}\\varphi$, meaning that $\\Gamma\\not\\Vdash_{\\mathcal{F}}^{\\bIpr}\\varphi$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Decision method}\n\nTake the Fidel structure $\\textbf{2}_{\\bI}$ for $\\bI$: we comment now on how $(\\textbf{2}_{\\bI}, \\{1\\}, \\mathcal{F}_{\\textbf{2}_{\\bIpr}})$, which we shall denote by $\\mathbb{2}_{\\bIpr}$\\label{2bIpr},  is a decision method for $\\bIpr$, where $\\mathcal{F}_{\\textbf{2}_{\\bIpr}}$ is the set of homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\textbf{2}_{\\bI}$ such that:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$, for any formulas $\\alpha$ and $\\beta$;\n\\item for any instance $\\phi$ of $\\textbf{pr}_{\\vee}$, $\\nu(\\phi)=1$;\n\\item for any instance $\\psi$ of $\\textbf{pr}_{\\wedge}$, $\\nu(\\psi)=1$.\n\\end{enumerate}\n\n\\begin{theorem}\n$\\nu:F(\\Sigma_{\\bI},\\mathcal{V})\\rightarrow\\{0,1\\}$ is a bivaluation for $\\bIpr$ if, and only if, it is a $\\Sigma_{\\bI}$-homomorphism from $\\textbf{F}(\\Sigma_{\\bI},\\mathcal{V})$ to $\\textbf{2}_{\\bI}$ that lies in $\\mathcal{F}_{\\textbf{2}_{\\bIpr}}$.\n\\end{theorem}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\bIpr$, $\\Gamma\\vDash_{\\bIpr}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathbb{2}_{\\bIpr}}\\varphi$.\n\\end{theorem}\n\nThe row-branching, row-eliminating truth tables based on $\\mathbb{2}_{\\bIpr}$ are then constructed more or less like the ones for $\\bI$ found in Section \\ref{Decision Method for bI}, the key difference being that rows where $\\alpha\\uparrow\\gamma$ and $\\beta\\uparrow\\gamma$ are both $1$ but $(\\alpha\\vee\\beta)\\uparrow\\gamma$ is $0$, or where $\\alpha\\uparrow\\gamma$ or $\\beta\\uparrow\\gamma$ is $1$ but $(\\alpha\\wedge\\beta)\\uparrow\\gamma$ is $0$, must also be disconsidered.\n\n\n\n\n\n\\section{Collapsing axioms}\\label{Collapsing axioms part 1}\n\n\n\\subsection{$\\textbf{Ex}$}\n\nA question that naturally arises when dealing with incompatibility is the partial equivalence between $\\alpha$ and $\\beta$ being incompatible, and $\\alpha$ and $\\beta$, together, trivializing a logic.\n\nThat is, we know that if $\\alpha$, $\\beta$, and $\\alpha\\uparrow\\beta$ are simultaneously valid, then our logic becomes trivial; but if $\\alpha$ and $\\beta$, just the two of them, can trivialize our logic, does that mean $\\alpha$ and $\\beta$ are incompatible? We believe such a question must reference the philosophical notion hoped for the incompatibility to encompass, and for this end we consider the axiom\\label{Ex}\n\\[\\tag{\\textbf{Ex}}(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta})\\rightarrow(\\alpha\\uparrow\\beta).\\]\nThe logic obtained from $\\bI$ by addition of $\\textbf{Ex}$ will be called $\\bI\\textbf{Ex}$\\label{bIEx}, but that is not really a new logic: we shall prove that, in this case, $\\alpha\\uparrow\\beta$ is equivalent to $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$ or, what is the same, $\\alpha\\Uparrow\\beta={\\sim}(\\alpha\\wedge\\beta)$ when we consider the classical negation definable in $\\bI$, and therefore $\\bI\\textbf{Ex}$ becomes again classical propositional logic with a shorthand notation for $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$, that is, $\\alpha\\uparrow\\beta$.\n\nSo we wish to prove that $\\alpha\\uparrow\\beta$ is equivalent, in this $\\bI\\textbf{Ex}$, to $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$, meaning that $(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta})\\rightarrow(\\alpha\\uparrow\\beta)$ but also that\n\\[(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}).\\]\nOf course, the first direction of the bi-implication follows straightforwardly from $\\textbf{Ex}$, being an instance of the axiom. To prove the second direction, take the instance $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\bot_{\\alpha\\beta}))$ of $\\textbf{Ip}$, and by applying the meta-deduction theorem three times we find that \n\\[\\alpha\\uparrow\\beta, \\alpha, \\beta\\vdash_{\\bI\\textbf{Ex}}\\bot_{\\alpha\\beta}.\\]\nBy applying the deduction meta-theorem to the instances $\\alpha\\wedge\\beta\\rightarrow\\alpha$ and $\\alpha\\wedge\\beta\\rightarrow\\beta$ of, respectively, $\\textbf{Ax\\: 4}$ and $\\textbf{Ax\\: 5}$, we obtain that $\\alpha\\wedge\\beta\\vdash_{\\bI\\textbf{Ex}}\\alpha$ and $\\alpha\\wedge\\beta\\vdash_{\\bI\\textbf{Ex}}\\beta$, and therefore \n\\[\\alpha\\uparrow\\beta, \\alpha\\wedge\\beta\\vdash_{\\bI\\textbf{Ex}}\\bot_{\\alpha\\beta},\\]\nimplying that $\\alpha\\uparrow\\beta\\vdash_{\\bI\\textbf{Ex}}\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$ and, therefore, that $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta})$ is a tautology in $\\bI\\textbf{Ex}$.\n\n\n\n\\subsection{$\\textbf{ciw}^\\uparrow$}\n\nThe axiom $\\textbf{ciw}$, given by\n\\[\\circ\\alpha\\vee(\\alpha\\wedge\\neg \\alpha),\\]\nis a very important one when dealing with paraconsistency, and adding it to $\\textbf{mbC}$ gives us the logic $\\textbf{mbCciw}$. When dealing with incompatibility, we would like to define an equivalent axiom, namely\\label{ciwuparrow}\n\\[\\tag{$\\textbf{ciw}^\\uparrow$}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta).\\]\nThe logic obtained from $\\bI$ by adding $\\textbf{ciw}^\\uparrow$ will be called $\\bI\\textbf{ciw}^\\uparrow$\\label{bIciw}, and we will prove that, as it happens with $\\bI\\textbf{Ex}$, $\\bI\\textbf{ciw}^\\uparrow$ is again equivalent to classical propositional logic, with $\\alpha\\uparrow\\beta$ corresponding to $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$.\n\nThe proof that $\\vdash_{\\bI\\textbf{ciw}^\\uparrow}(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta})$ is the same as the one for the same fact in $\\bI\\textbf{Ex}$, remaining for us to show that $(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta})\\rightarrow(\\alpha\\uparrow\\beta)$ is a tautology in $\\bI\\textbf{ciw}^\\uparrow$.\n\nBy an application of the deduction meta-theorem, the desired result is equivalent to $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\bI\\textbf{ciw}^{\\uparrow}}\\alpha\\uparrow\\beta$. It is clear how $\\alpha\\uparrow\\beta, \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\bI\\textbf{ciw}^{\\uparrow}}\\alpha\\uparrow\\beta$, and by an application of Modus Ponens and the fact $\\bot_{\\alpha\\beta}$ behaves like a bottom element, it is obvious that\n\\[\\alpha\\wedge\\beta, \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\bI\\textbf{ciw}^{\\uparrow}}\\alpha\\uparrow\\beta.\\]\nBy a proof by cases, we get that \n\\[(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta), \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\bI\\textbf{ciw}^{\\uparrow}}\\alpha\\uparrow\\beta,\\]\nand since $(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)$ is an instance of an axiom of $\\bI\\textbf{ciw}^{\\uparrow}$, we get the desired result.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Brandom's notion of incompatibility}\n\nHere, we offer a brief overview of Robert B. Brandom\\index{Brandom} and Alp Aker's work with incompatibility, echoed in Jaroslav Peregrin's\\index{Peregrin} research. The differences between their own methodology and ours are many: they focus on incompatibility between sets of formulas,; they aim to define consequence from incompatibility, while we assume both to coexist; and, perhaps most importantly, although many systems can be retrieved from their methods, those of a paraconsistent behavior are not among them, as the only negations considered by them are classical, or at most intuitionistic. But the connection between both works is there, and quite clear: the attempt to control, in a way or another, explosion by mediating it through incompatibility.\n\n\n\n\\subsection{Brandom and Aker's ``Between saying and doing''}\n\nWe start by pointing out that, although the author of \\cite{Brandom} appears as Robert B. Brandom, he makes it clear that much of the semantics for his logic of incompatibility was developed by his Ph.D. student Alp Aker; given our focus in his book is mostly restricted to the chapters about incompatibility, we will refer to both Brandom and Aker as authors.\n\nIn \\cite{Brandom}, Brandom and Aker propose an approach to logic through incompatibility, instead of consequence; they defend that a modal understanding of incompatibility fits better with the argumentative nature of epistemology, in his neo-pragmatic program. They proceed to redefine consequence, and the usual connectives, starting from incompatibility, or rather incoherence\\index{Incoherence}, as a primitive notion.\n\nInitially, Brandom and Aker start with the notions of commitment\\index{Commitment} and entitlement\\index{Entitlement}, which are very modal in nature. Intuitively, an agent is committed to an statement when stating such statement is necessary for the agent; and the same agent is entitled to a statement when stating it is possible for the agent. Building from this, two statements are incompatible when being committed to one implies not being entitled to the other. A notion of consequence arises from this concept when we define that a statement $p$ implies $q$ whenever every statement incompatible with $q$ is incompatible with $p$; clearly we must reach for second-order logic and quantification over formulas for this procedure to make sense.\n\nOur approach to incompatibility is quite close, in a sense, to this first notion of Brandom and Aker, but they proceed to extend their definition: they sustain that incompatibility must deal with pairs of sets of statements, instead of pairs of statements; their argumentation is that a claim may be incompatible with a set of claims without being incompatible with any one claim of the set. First of all, we believe this problem may be circumvented by a careful distinction between incompatibility in natural discourse and as expressed formally, although this distinction may be blurred given Brandom and Aker want incompatibility to originate logic, what in turn may explain their position; second, we do not aim to encompass every reasoning involving incompatibility as Brandom  and Aker do, but rather prefer to focus on its interplay with paraconsistency and its possible semantics.\n\nHere, a digression seems useful. In the beginning of this chapter we offered interpretations of incompatibility, two of them being related to probability: mutual exclusivity and independence. Now, if $\\{A_{i}\\}_{i\\in I}$ is a collection of pairwise mutually exclusive events, for any $J\\subseteq I$ with at least two elements $j_{1}$ and $j_{2}$ one has that $P(\\bigcap_{j\\in J}A_{j})\\leq P(A_{j_{1}}\\cap A_{j_{2}})=0$, and so $P(\\bigcap_{j\\in J}A_{j})=0$; this means that a set of pairwise mutually exclusive events also has a generalized mutual exclusivity. Meanwhile, if $\\{A_{i}\\}_{i\\in I}$ is instead a collection of pairwise independent events, the fact that, for any two $i_{1}, i_{2}\\in I$, one has $P(A_{i_{1}}\\cap A_{i_{2}})=P(A_{i_{1}})P(A_{i_{2}})$ does not imply that, for any finite $J\\subseteq I$, one also has $P(\\bigcap_{j\\in J}A_{j})=\\prod_{j\\in J}P(A_{j})$; because of that, we define instead that a finite collection of events $\\{A_{i}\\}_{i\\in I}$ is mutually independent, something much stronger than merely pairwise independent, if, for any $J\\subseteq I$, \n\\[P(\\bigcap_{j\\in J}A_{j})=\\prod_{j\\in J}P(A_{j})\\]\nholds. This difference between these two notions carries a lot of similarities with the difference between ours, and Brandom and Aker`s approach: in our understanding of incompatibility, a set of propositions is incompatible if, and only if, any pair of its elements is, itself, incompatible, exactly as in the case of mutually exclusive events; meanwhile, in Brandom and Aker, the more general notion of incompatible set can not be reduced to pairwise incompatibility, as in the problem of mutual independence as opposed to pairwise independence. This suggests interesting connections between distinct incompatibilities and several concepts in probability theory.\n\n\nThen, \\cite{Brandom} demands only two properties of incompatibility, derived from its natural interpretation and intuition:\n\\begin{enumerate}\n\\item if $X$ is incompatible with $Y$, $Y$ is incompatible with $X$ (symmetry)\\index{Symmetry};\n\\item if $X$ is incompatible with $Y$, and $Z$ contains $Y$, then $X$ is incompatible with $Z$ (persistence)\\index{Persistence}.\n\\end{enumerate}\n\nIn our logics of incompatibility, symmetry is analogous to the commutative axiom $\\textbf{Comm}$, where $\\alpha\\uparrow\\beta\\rightarrow\\beta\\uparrow\\alpha$; regarding persistence, ignoring the obvious interpretation in second-order logic, one very natural take would be, in our language, the axiom\n\\[(\\alpha\\uparrow\\beta)\\rightarrow\\big((\\gamma\\rightarrow\\beta)\\rightarrow(\\alpha\\uparrow\\gamma)\\big).\\]\nBut we also may, for sets of formulas $X$ and $Y$, write a generalized incompatibility operator $X\\uparrow Y$ whenever there exist formulas $\\alpha$ and $\\beta$ such that $X\\vdash \\alpha$, $Y\\vdash\\beta$ and $\\alpha\\uparrow\\beta$. Then, if $Z$ contains $Y$ and $X\\uparrow Y$, given formulas $\\alpha$ and $\\beta$ as above, since $Y\\subseteq Z$ one finds $Z\\vdash \\beta$, and therefore $X\\uparrow Z$ and persistence is reobtained on this environment. Of course, this is not to say that our logics of incompatibility naturally model Brandom and Aker's approach, but rather to show they are plastic enough to do so.\n\nBrandom and Aker stress that their incompatibility should not be limited to truth values, as they intend to define the latter starting from the former, but they still mention \\textit{en passant} that the obvious interpretation of, restricting ourselves to single statements once again, $p$ being incompatible to $q$ should be that it is impossible to have both $p$ and $q$ simultaneously true; this, in turn, is of course very distant from our own take on incompatibility, as it limits the concept to a modal version of Sheffer's stroke $\\square{\\sim}(p\\wedge q)$.\n\nTo further distance its incompatibility from ours, \\cite{Brandom} takes as a tacit starting point that a statement should be incompatible with its negation, appealing to classical logic to do so; this may be justified if they intend to replicate merely classical negation from incompatibility, but it seems an inadmissible loss not to consider more modern negations, objects around which our logics of incompatibility are actually built. It seems that, to define a negation from incompatibility, it makes more sense, and is more interesting, to start from the assumption there is no prior connection between incompatibility and negation; then one gains the interesting notion of ``well-behavedness'' among statements, a resource well established amid logics of formal inconsistency. \n\nNevertheless, Brandom and Aker then define the negation of a claim $p$ as the minimum claim incompatible to $p$, meaning a claim ${\\sim}p$ incompatible with $p$ such that, whenever $X$ is a set of statements incompatible with $p$, one has $X\\vdash {\\sim}p$. Notice that, using the classical negation that may be defined in $\\bI$, we find $\\alpha\\uparrow\\beta\\rightarrow(\\alpha\\rightarrow{\\sim}\\beta)$, meaning that if $p$ and $q$ are incompatible, then $p$ implies the negation of $q$; this may seem awfully close to Brandom and Aker's account, but notice ${\\sim}q$ is not, necessarily, incompatible with $q$, and we do not intend to define negation from this relationship, but rather derive the relationship from the definition of the strong negation and the fact $\\alpha\\vee(\\alpha\\rightarrow\\beta)$ is an instance of an axiom. It becomes clear \\cite{Brandom} has no interest in dealing with paraconsistency when it defines an inconsistent set of formulas as any set which derives both a claim and its negation; this, of course, is not interesting to us at all, since paraconsistency intends to precisely avoid this.\n\nBrandom and Aker define the conjunction of statements $p$ and $q$ as the minimal statement incompatible with every set $X$ incompatible with $\\{p,q\\}$: they then go on to show that the logic obtained from both these connectives they have defined is classical propositional logic, what shows one may recover classical accounts of logic from the notion of incompatibility; although not in line with what we hope to accomplish with our own logics of incompatibility, Brandom and Aker's result is no less important, and establishes that incompatibility, at least in classical logic, is a notion no less important than that of deduction.\n\nWe have not studied modalities and their logical counterpart very deeply, but this is one of Brandom and Aker's concern, and they define that something is incompatible with the necessity of a statement $p$, $\\square p$, whenever it is compatible with something that does not imply $p$; meaning, $X$ is incompatible with $\\square p$ if and only if it is compatible with a $Y$ such that $Y\\not\\vdash p$, and one may define necessity as the minimal statement which can replace $\\square p$ in the previous discussion. Very interestingly, this definition makes of the system at hand precisely $S5$. Nevertheless, one may wonder if the definitions \\cite{Brandom} takes are the ideal ones: if we are recovering only the most standard systems, perhaps a little more plasticity should be allowed when redefining connectives from incompatibility.\n\nIn one last stretch, approaching their own account again to ours, Brandom and Aker notice that, in his logic of incompatibility, connectives do not have the semantic sub-formula property, meaning they are not functional: one can not derive the interpretation of a formula by merely looking at the interpretation of its sub-formulas. This non-deterministic behavior is very clear in our logics of incompatibility, and specially clear when one analyses the easiness with which one treats logics such as $\\bI$ and $\\nbI$ (see Chapter \\ref{Chapter8}) with the aid of restricted non-deterministic matrices and, more generally, multialgebras.\n\nIn a more formal treatment of their system, Brandom and Aker wish to analyze when a consequence relation $\\vdash$ may be characterized by incompatibility, in their own terms of ``to be characterized'' as previously described. They reach two necessary and sufficient conditions:\n\\begin{enumerate}\n\\item if $X\\vdash Y$ and $Y\\cup W\\vdash Z$, then $X\\cup W\\vdash Z$ (general transitivity);\n\\item if $X\\not\\vdash Y$, there exist a $W$ and a $Z$ such that, $X\\cup Z\\not\\vdash W$ but, for all $V$, $Y\\cup Z\\vdash V$ (defeasibility)\\index{Defeasibility}.\n\\end{enumerate}\n\nGeneral transitivity is, of course, a reformulation of the cut rule; defeasibility states that, if $Y$ is not a consequence of $X$, then there exists something that when added to $Y$ will collapse the deduction, but not when added to $X$. Although somewhat natural, the way this conditions are used to achieve incompatibility is diametrically opposed to what we look for, as two sets are deemed incompatible whenever their union is inconsistent; disregarding our problem with inconsistency, the axiom schema $\\textbf{Ip}$ invites one to produce instead the concept that, if two sets are incompatible, then their union should be trivializing, and not the other way around. Brandom and Aker go on to give several interesting definitions, but all based on these basic concepts, and therefore distant from our ideal understanding of incompatibility. Concisely, given a fixed set of sets of incoherent (intuitively, inconsistent) formulas, they demand of it only that if $X$ is incoherent and $X\\subseteq Y$, then $Y$ is also incoherent; then, $X$ is incompatible with $Y$ if, and only if, $X\\cup Y$ is incoherent. Then, $X$ derives a statement $p$, regarding their incompatibilities, when every set incompatible to $\\{p\\}$ is incompatible to $X$. Several strong theorems are then provided for those systems, but to showcase how they are not aligned to our notion of incompatibility, one of these theorems is that $p\\wedge{\\sim}p$ derives any set $Y$, meaning their negation is necessarily classically behaved.\n\n\\subsection{Peregrin}\n\n\\subsubsection{``Brandom's incompatibility semantics''}\n\nIn this first article \\cite{Peregrin1}, the author Jaroslav Peregrin studies the notion of incompatibility, and their corresponding semantics, of Brandom and Aker's \\cite{Brandom} from a more philosophical standpoint. More precisely, his main concerns are tied to the problem of whether formal semantics are truly compatible with pragmatic and inferentialist views and of how should such semantics look like; his argument, a very interesting one, may be summarized as stating that, as merely a model for natural processes, formal semantics can indeed be used by the pragmatist, as long as the distinction between model and what is modeled is not ignored.\n\nMore to the point we are interested in, Peregrin asks how should incompatibility come into play in logic: he first stresses how the usual approach is to say that sets of formulas $X$ and $Y$ are incompatible whenever their union can deduce anything; much in line with our own views of the problem, Peregrin continues to state that, however standard this approach may be, reducing incompatibility to inference is, first of all, wasteful, as it disregards the possible intricacies the concept may carry; second, incompatibility, as a byproduct of inference, becomes dependent on how strong is the logic we work over, meaning how expressive it is.\n\n\\cite{Peregrin1} proceeds to point some problematic aspects of Brandom and Aker's interpretation of incompatibility: first of all, when one defines that $X\\vdash p$ whenever everything that is incompatible with $p$ is incompatible with $X$, this may be read as that everything compatible with $X$ is compatible with $p$, implying compatibility is preserved by consequence, and since so is truth, the two concepts may become at moments indistinguishable.  Of course, this is not of much concern for us, since we do not intend to rebuild consequence from incompatibility, but rather have incompatibility to exist in an environment with a predefined consequence operator. A second problem, still not very troublesome for us, is that consequence is usually taken to have a finitary character, while incompatibility, at least in Brandom and Aker's take, often involves quantification over all statements: so, by defining a consequence from incompatibility, one may end with a consequence operator that is more model-theoretic than proof-theoretic, that is, that has a clear non-finitary propensity.\n\nTo us, one of the most important developments found in this article is the connection established between incompatibility and Kripke\\index{Kripke} semantics, \\textit{id est}, semantics of possible worlds. Peregrin defines a possible world, once a concept of coherence is given, as a maximal coherent set of formulas; here, a set is a coherent one once it is not incompatible with itself, a definition that goes back to the work of Brandom and Aker. The truth of a statement on a given world is then taken to be the belonging of this statement to the wold, which is, again, a set of formulas; here, Peregrin notices once again how coherence behaves dangerously alike truth in Brandom and Aker's \\cite{Brandom}. Notice how, given a statement $p$ and a possible world $w$, $w$ proves $p$ in this semantic of possible worlds if and only if $w$ derives $p$, or what is equivalent, when $p$ is compatible with $w$. Reciprocally, he derives a notion of incoherence from a Krikpkean semantics of possible worlds by saying a set of formulas $X$ is incoherent if no formula of this set is validated in any world.\n\nPeregrin also points out how Brandom and Aker's definition of the necessity of a statement $p$ is equivalent to, in his semantic of possible words, validity: so a possible world $w$ verifies $\\square p$ if, and only if, the set $\\{p\\}$ is not incoherent; this of course means that Brandom and Aker's take on modal logic trough incompatibility only derives the most basic interpretations of necessity and possibility. He offers an interesting alternative, in order to add richness to those modal logics: a second level of incompatibility, or rather incoherence, a meta-coherence\\index{Meta-coherence} if you will; Brandom and Aker only take into account a set of sets of formulas, said to be the set of incoherent sets of formulas, while Peregrin goes further and defines a set of sets of sets of formulas, corresponding to a set of incoherent sets of sets of formulas, that offers a second level of control over incompatibility. With this, he is able to characterize modal logics more complex than $S5$, what of course suggests the power such generalizations may carry, although this level of abstractness is not within our scope of research.\n\n\\subsubsection{``Logic as based on incompatibility''}\n\nIn this second of Peregrin's article of great interest to us, he lays clear what he believes are the reasons for Brandom and Aker's incompatibility program: first, to study what the possible minimal foundations of logic may be; and second, to offer a working systems for those philosophers who have based their reasoning on incompatibility alone. While still reasoning about Brandom and Aker's program, the author defends that the most natural logic to arise from defining inference through incompatibility, in the opposite direction of Brandom and Aker's work, is intuitionistic; of course, this is not of great interest to us, since we hope for incompatibility to control \\textit{ex contradictione quolibet} and not the other way around. Peregrin points out that, by defining inference via incompatibility, ``Between saying and doing'' reaches a logic of, instead of intuitionistic, classical character, but goes on to argue that this due not to the nature of incompatibility, but rather to Brandom and Aker's method.\n\n\\cite{Peregrin2} defines then an environment that should deal, simultaneously, with inference and incompatibility: a triple $(S, \\bot, \\vdash)$ is called a generalized inferential structure ($\\textit{gis}$) when $S$ is a set, $\\bot\\subseteq\\mathcal{P}(S)$ and $\\vdash\\subseteq\\mathcal{P}(S)\\times S$; for simplicity, if $X\\in \\bot$ we write $\\bot X$, and if $X\\cup Y\\cup\\{p\\}$ is in $\\bot$, we may simply write $\\bot X,Y,p$. The basic conditions demanded of these objects are the following:\n\\begin{enumerate}\n\\item[$(\\bot)$] if $\\bot X$ and $X\\subseteq Y$, $\\bot Y$;\n\\item[$(\\vdash)$]\\begin{enumerate}\\item $X, p\\vdash p$;\n\\item if $X, p\\vdash q$ and $Y\\vdash p$, then $X,Y\\vdash q$.\n\\end{enumerate}\n\\end{enumerate}\n\nPeregrin suggests a possible interplay between the two concepts:\n\\begin{enumerate}\n\\item[$(\\bot\\vdash)$] if $\\bot X$, then $X\\vdash p$ for every $p$;\n\\item[$(\\vdash\\bot)$] if $X\\vdash p$, then $\\bot Y, p$ implies $\\bot X,Y$ for every $Y$.\n\\end{enumerate}\nOf course, the first condition is very in line with what we expect of incompatibility: that it derives anything and trivializes an argument, or, in other words, a controlled explosion. The second condition has been presented before, in a much more complicated way: it corresponds to Brandom and Aker's defeasibility. The author then points out how assuming $(\\bot\\vdash)$ and its converse amounts to reducing incompatibility to inference, while assuming $(\\vdash\\bot)$ and its converse reduce inference to incompatibility. \\cite{Peregrin2} also has definitions of an incompatibility $\\vartriangle$ defined trough inference and of an inference $\\vartriangleright$ defined trough incompatibility:\n\\begin{enumerate}\n\\item $\\vartriangle X$ if $X\\vdash p$, for every $p$;\n\\item $X\\vartriangleright p$ if, for every $Y$, $\\bot Y, p$ implies $\\bot X,Y$.\n\\end{enumerate}\nPeregrin then offers interesting conditions, related to $(\\bot\\vdash)$ and $(\\vdash\\bot)$, under which $\\vartriangle$ and $\\bot$, and $\\vartriangleright$ and $\\vdash$ are equivalent. But, more importantly, the author states what the minimal requirements for an intuitionistic negation based on incompatibility should be:\n\\begin{enumerate}\n\\item $\\bot p, {\\sim}p$;\n\\item if $\\bot X, p$, then $X\\vdash{\\sim}p$.\n\\end{enumerate}\nOf course, the first condition is not desirable for our systems, while the second is derivable in systems as simple as $\\nbI$ (which we define in Chapter \\ref{Chapter8}); with this, we see that the plasticity that the author is hoping to obtain by modifying Brandom and Aker's stipulations does not encompass paraconsistency, nor is this his objective, as far as we can tell. He seems, instead, more concerned with modal systems, allowing our logics to fill a gap in his approach.\n\nThe additional requirement that $\\bot X, {\\sim}p$ implies $X\\vdash p$ is then equivalent to stating that the negation in question is of classical behavior; Peregrin also shows that the definition Brandom and Aker give of negation is equivalent to his own negation with this last additional requirement, proving that the methodology found ``Between saying and doing'' can only lead to classical negation. Conjunction is defined as in Brandom and Aker, meaning \n\\begin{enumerate}\n\\item if $\\bot X, p\\wedge q$, then $\\bot X, p, q$;\n\\item if $\\bot X, p, q$, then $\\bot X, p\\wedge q$.\n\\end{enumerate}\nAfter comparing Brandom and Aker's, and his own, definitions with those necessary to define classical propositional logic trough inference alone, the author reaches the conclusion that while Brandom and Aker's approach inexorably leads to classical logic, varying the techniques found in his article is significantly more far-reaching. For an example, by removing the very basic condition that\n\\[\\bot X\\quad\\text{and}\\quad X\\subseteq Y\\quad\\text{imply}\\quad \\bot Y,\\]\nwhich is equivalent, given the correct assumptions, to the fact that $X\\vdash p$ implies $X, q\\vdash p$, one finds a system of relevant logic; without going into further details, Peregrin suggests that changing incompatibility to a property between multisets of formulas, instead of sets, leads to linear logic.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{7}\n\\chapter{Adding a negation to logics of incompatibility}\\label{Chapter8}\\label{Chapter 8}\n\nWhen studying paraconsistent logics, our main focus is in the properties of negations, as is the case when working with paracomplete, i.e. intuitionistic, logics. Given such a prominent role negation plays in non-classical logics, is quite natural to shift our focus in the logics of incompatibility from $\\uparrow$ to a non-classical negation, or better yet, to the possible interplay between $\\uparrow$ and such a negation.\n\nOur first step is adding such a negation: we have seen that in any logic extending $\\bI$, is always possible to define a somewhat quite classical negation; for any formulas $\\alpha$ and $\\beta$, the formulas\n\\[\\bot_{\\alpha\\beta}=\\alpha\\wedge(\\beta\\wedge(\\alpha\\uparrow\\beta))\\]\nare all equivalent to each other, and furthermore play the expected role for the bottom, meaning that in the presence of $\\textbf{Ip}$, for any formula $\\gamma$ it is true that $\\bot_{\\alpha\\beta}\\rightarrow\\gamma$. By having a bottom element, we can define a negation of classical behavior ${\\sim} \\alpha$ of a formula $\\alpha$ by $\\alpha\\rightarrow \\bot_{\\alpha\\alpha}$, or any other bottom since all of those are equivalent. \n\nWe, therefore, need a new negation, weaker than that negation intrinsic to $\\bI$. So we need a symbol for it: we define the signature $\\Sigma_{\\nbI}$\\label{SigmanbI} as the signature obtained from $\\Sigma_{\\bI}$ by addition of an unary symbol $\\neg$, that is, $(\\Sigma_{\\nbI})_{1}=\\{\\neg\\}$, $(\\Sigma_{\\nbI})_{2}=\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$ and $(\\Sigma_{\\nbI})_{n}=\\emptyset$ for $n\\notin\\{1,2\\}$.\n\nMost of the research in this chapter was submitted as the preprint \\cite{Frominconsistency}.\n\n\n\n\\section{The logic $\\nbI$}\n\nWe start by adding to the simplest $\\textbf{LIp}$, that is, $\\bI$, a paraconsistent negation: so, to the axiom schemata and rules of inference of $\\bI$, we add\n\\begin{enumerate}\n\\item[$\\textbf{Ax\\: 11}^{*}$] $\\alpha\\vee\\neg \\alpha$.\n\\end{enumerate}\nWe shall call this new logic $\\nbI$\\label{nbI}.\n\n\n\n\\subsection{Bivaluations}\\label{Bivaluations for nbI}\n\nA \\index{Bivaluation for $\\nbI$} bivaluation for $\\nbI$ is a map $\\nu:F(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\{0,1\\}$ such that:\n\\begin{enumerate}\n\\item if $\\nu(\\neg \\alpha)=0$, $\\nu(\\alpha)=1$;\n\\item $\\nu(\\alpha\\vee\\beta)=1$ if and only if $\\nu(\\alpha)=1$ or $\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\wedge\\beta)=1$ if and only if $\\nu(\\alpha)=\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\rightarrow\\beta)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$;\n\\item if $\\nu(\\alpha\\uparrow\\beta)=1$ and $\\nu(\\alpha)=1$, $\\nu(\\beta)=0$;\n\\item $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$.\n\\end{enumerate}\n\nGiven a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\nbI$, we say $\\Gamma$ proves $\\varphi$ according to bivaluations, and write $\\Gamma\\vDash_{\\nbI}\\varphi$\\label{vDashnbI}, if for every bivaluation $\\nu$ for $\\nbI$ we have that, if $\\nu(\\Gamma)\\subseteq\\{1\\}$, then $\\nu(\\varphi)=1$.\n\nOnce most conditions demanded of a bivaluation for $\\nbI$ are essentially the same as those demanded of a bivaluation for $\\bI$ in Section \\ref{bivaluations for bI}, we can easily find that \"$\\vDash_{\\nbI}$\" models those axioms and rules of inference of $\\bI$: to see that it also models $\\textbf{Ax\\: 11}^{*}$, take a formula $\\alpha$ and a bivaluation $\\nu$ for $\\nbI$\n\\begin{enumerate}\n\\item If $\\nu(\\neg \\alpha)=1$, $\\nu(\\alpha\\vee\\neg \\alpha)=1$ from the behavior of bivaluations for $\\nbI$ regarding disjunctions.\n\\item If $\\nu(\\neg \\alpha)=0$, we find $\\nu(\\alpha)=1$ and again $\\nu(\\alpha\\vee\\neg \\alpha)=1$.\n\\end{enumerate}\nEither way, one sees that \"$\\vDash_{\\nbI}$\" also models $\\textbf{Ax\\: 11}^{*}$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\nbI$, if $\\Gamma\\vdash_{\\nbI}\\varphi$ then $\\Gamma\\vDash_{\\nbI}\\varphi$.\n\\end{theorem}\n\nThe reciprocal result goes as it would be expected: for a closed, non-trivial set of formulas $\\Gamma$ of $\\nbI$ maximal with respect to not proving $\\varphi$, we define $\\nu:F(\\Sigma_{\\nbI}, \\mathcal{V})\\rightarrow\\{0,1\\}$ by means of $\\nu(\\gamma)=1$ if and only if $\\gamma\\in\\Gamma$; $\\nu$ is then a bivaluation for $\\nbI$.\n\nThat it is a bivaluation for $\\bI$ is quite obvious, remaining for us to show that, if $\\nu(\\neg \\alpha)=0$, then $\\nu(\\alpha)=1$. So, suppose $\\nu(\\neg \\alpha)=0$, meaning $\\neg \\alpha\\not\\in\\Gamma$: since $\\alpha\\vee\\neg \\alpha$ is an instance of $\\textbf{Ax\\: 11}^{*}$ and $\\Gamma$ is closed, $\\alpha\\vee\\neg \\alpha\\in\\Gamma$. \n\nIf we had $\\nu(\\alpha)=0$, then we would have $\\nu(\\alpha\\vee\\neg \\alpha)=0$, implying $\\alpha\\vee\\neg \\alpha\\not\\in\\Gamma$, contradicting our previous remark. We must have then that $\\nu(\\alpha)=1$, what ends the proof that $\\nu$ is a bivaluation for $\\nbI$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\nbI$, if $\\Gamma\\vDash_{\\nbI}\\varphi$ then $\\Gamma\\vdash_{\\nbI}\\varphi$.\n\\end{theorem}\n\n\n\n\n\\subsection{Fidel structures}\n\nOnce again we find ourselves in need of bottom and top elements, and a classical negation, so that we must define the signature\\label{SigmanbICPL} $\\Sigma_{\\nbI}^{\\textbf{CPL}}$ with $(\\Sigma_{\\nbI}^{\\textbf{CPL}})_{0}=\\{\\bot, \\top\\}$, $(\\Sigma_{\\nbI}^{\\textbf{CPL}})_{1}=\\{{\\sim},\\neg\\}$, $(\\Sigma_{\\nbI}^{\\textbf{CPL}})_{2}=\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$ and $(\\Sigma_{\\nbI}^{\\textbf{CPL}})_{n}=\\emptyset$ for $n>2$.\n\nSo, it becomes easy to define Fidel structures for $\\nbI$: \\index{Fidel structure for $\\nbI$}a Fidel structure for $\\nbI$ is any $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\nbI}^{\\textbf{CPL}}})$ such that:\n\\begin{enumerate}\n\\item $(A,\\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra;\n\\item for all $a\\in A$ and $b\\in \\neg _{\\mathcal{A}}(a)$, $a\\vee b=\\top$;\n\\item for all $a,b\\in A$ and $c\\in\\uparrow_{\\mathcal{A}}(a,b)$, $a\\wedge(b\\wedge c)=\\bot$;\n\\item for all $a, b\\in A$, $a\\uparrow b=b\\uparrow a$.\n\\end{enumerate}\n\nAs we did before, we shall drop the index $\\mathcal{A}$ and use the infix notation.\n \nGiven a Fidel structure $\\mathcal{A}$, presented as a $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra, for $\\nbI$, a valuation for $\\mathcal{A}$ is any $\\Sigma_{\\nbI}$-homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\nbI}, \\mathcal{V})\\rightarrow \\mathcal{A}$; and, for every Fidel structure $\\mathcal{A}$ for $\\nbI$, we will consider the restricted Nmatrix $(\\mathcal{A}, \\{\\top\\}, \\mathcal{F}_{\\mathcal{A}})$, where $\\mathcal{F}_{\\mathcal{A}}$ is the set of valuations $\\nu:\\textbf{F}(\\Sigma_{\\nbI}, \\mathcal{V})\\rightarrow\\mathcal{A}$ such that \n\\[\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha),\\]\nfor any two formulas $\\alpha$ and $\\beta$ in $F(\\Sigma_{\\nbI}, \\mathcal{V})$; it is clear how these RNmatrices are structural. If $\\Gamma$ proves $\\varphi$ according to such restricted Nmatrices, we will write\\label{VdashFnbI} $\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbI}\\varphi$.\n\nIt is easy to see that if $\\phi$ is an instance of an axiom of $\\bI$, $\\Vdash_{\\mathcal{F}}^{\\nbI}\\phi$, and if $\\Vdash_{\\mathcal{F}}^{\\nbI}\\alpha$ and $\\Vdash_{\\mathcal{F}}^{\\nbI}\\alpha\\rightarrow\\beta$ then $\\Vdash_{\\mathcal{F}}^{\\nbI}\\beta$: we would like to prove that, for any formula $\\alpha$ of $\\nbI$, we also have \n\\[\\Vdash_{\\mathcal{F}}^{\\nbI}\\alpha\\vee\\neg\\alpha,\\]\nimplying that \"$\\Vdash_{\\mathcal{F}}^{\\nbI}$\" models the axiom schemata and rules of inference of $\\nbI$.\n\nSo, for any Fidel structure $\\mathcal{A}$ for $\\nbI$ and valuation $\\nu$ over $\\mathcal{A}$, we have that $\\nu(\\alpha\\vee\\neg \\alpha)=\\nu(\\alpha)\\vee\\nu(\\neg \\alpha)$; now, $\\nu(\\neg \\alpha)\\in\\neg\\nu(\\alpha)$, and therefore, from the conditions demanded of a Fidel structure for $\\nbI$, $\\nu(\\alpha)\\vee\\nu(\\neg \\alpha)=\\top$, what finishes the desired proof.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\nbI$, if $\\Gamma\\vdash_{\\nbI}\\varphi$ then $\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbI}\\varphi$.\n\\end{theorem}\n\nTo prove the reciprocal result, we define the equivalence relation between formulas of $\\nbI$ such that, for a fixed set of formulas $\\Gamma$, \\label{equivnbI}$\\alpha\\equiv_{\\Gamma}^{\\nbI}\\beta$ if and only if $\\Gamma\\vdash_{\\nbI}\\alpha\\rightarrow\\beta$ and $\\Gamma\\vdash_{\\nbI}\\beta\\rightarrow\\alpha$; this relation is also a congruence with respect to the connectives in $\\{\\vee, \\wedge,\\rightarrow\\}$, allowing us to make $A^{\\nbI}_{\\Gamma}=F(\\Sigma_{\\nbI},\\mathcal{V})\/\\equiv_{\\Gamma}^{\\nbI}$ into the universe of a Boolean algebra. By defining, for classes of formulas $[\\alpha]$ and $[\\beta]$,\n\\[\\neg[\\alpha]=\\{[\\neg \\varphi]\\ :\\  \\varphi\\in[\\alpha]\\}\\]\nand\n\\[[\\alpha]\\uparrow[\\beta]=\\{[\\varphi\\uparrow\\psi]\\ :\\  \\varphi\\in[\\alpha], \\psi\\in[\\beta]\\},\\]\nwe can prove \\label{AnbIGamma}$\\mathcal{A}^{\\nbI}_{\\Gamma}=(A^{\\nbI}_{\\Gamma},\\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\nbI}^{\\textbf{CPL}}})$ is a Fidel structure, presented as a $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra, for $\\nbI$, which we shall call the Lindenbaum-Tarski Fidel structure of $\\nbI$ associated to $\\Gamma$.\n\nThe proof that, for $[\\gamma]\\in [\\alpha]\\uparrow[\\beta]$, we have $[\\alpha]\\wedge([\\beta]\\wedge[\\gamma])=\\bot$, and that $[\\alpha]\\uparrow[\\beta]=[\\beta]\\uparrow[\\alpha]$ are exactly the same as those for the same facts in $\\bI$ (found in Section \\ref{Completeness fo Fidel structures for bI}), so we will rather focus on proving that, for any class of formulas $[\\alpha]$ and any $[\\beta]\\in \\neg[\\alpha]$, $[\\alpha]\\vee[\\beta]=\\top$: by definition, there exists $\\varphi\\in\\alpha$ such that $[\\beta]=[\\neg\\varphi]$.\n\nThen $[\\alpha]\\vee[\\beta]=[\\alpha\\vee\\neg \\varphi]$: very clearly $\\Gamma\\vdash_{\\nbI}\\alpha\\vee\\neg \\varphi\\rightarrow\\top_{\\alpha}$, remaining for us to show that $\\Gamma\\vdash_{\\nbI}\\top_{\\alpha}\\rightarrow\\alpha\\vee\\neg \\varphi$ or, better yet, that $\\Gamma\\vdash_{\\nbI}\\alpha\\vee\\neg \\varphi$, by one application of the deduction meta-theorem and the fact that a top element is always a tautology.\n\nWe know $\\varphi\\vee\\neg \\varphi$ is an instance of $\\textbf{Ax\\: 11}^{*}$ and $\\Gamma\\vdash_{\\nbI}\\varphi\\rightarrow\\alpha$:\n\\begin{enumerate}\n\\item $\\Gamma,\\varphi\\vdash_{\\nbI}\\alpha$ by the deduction meta-theorem, and from the instance $\\alpha\\rightarrow\\alpha\\vee\\neg \\varphi$ of $\\textbf{Ax\\: 6}$ and one application of Modus Ponens we get that $\\Gamma,\\varphi\\vdash_{\\nbI}\\alpha\\vee\\neg \\varphi$;\n\\item $\\Gamma,\\neg \\varphi\\vdash_{\\nbI}\\neg \\varphi$, and from the instance $\\neg \\varphi\\rightarrow\\alpha\\vee\\neg \\varphi$ of $\\textbf{Ax\\: 7}$ and the deduction meta-theorem we get that $\\Gamma,\\neg \\varphi\\vdash_{\\nbI}\\alpha\\vee\\neg \\varphi$;\n\\end{enumerate}\nfrom this, we get that $\\Gamma, \\varphi\\vee\\neg \\varphi\\vdash_{\\nbI}\\alpha\\vee\\neg \\varphi$ by a proof by cases, and since $\\varphi\\vee\\neg \\varphi$ is an instance of an axiom, $\\Gamma\\vdash_{\\nbI}\\alpha\\vee\\neg \\varphi$, as we wanted to prove.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\nbI$, if $\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbI}\\varphi$ then $\\Gamma\\vdash_{\\nbI}\\varphi$.\n\\end{theorem}\n\n\n\n\n\n\\subsection{Decision method}\n\nWe take the Boolean algebra $\\textbf{2}$ once again and define the unary multioperation $\\neg:\\{0,1\\}\\rightarrow\\mathcal{P}(\\{0,1\\})\\setminus\\{\\emptyset\\}$ trough $\\neg 0=\\{1\\}$ and $\\neg 1=\\{0,1\\}$,\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n& $\\neg $\\\\ \\hline\n$0$ & $\\{1\\}$\\\\ \\hline\n$1$ & $\\{0,1\\}$\\\\\\hline\n\\end{tabular}\n\\caption*{Table for our Paraconsistent Negation}\n\\end{figure}\n\nand, as we did in Section \\ref{Decision Method for bI}, we define $1\\uparrow 1=\\{0\\}$ and $x\\uparrow y=\\{0,1\\}$ for any other values of $x$ and $y$. By adding the previous operations to $\\textbf{2}$, we transform it into a $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra\\label{2nbI} $\\textbf{2}_{\\nbI}$ with a finite universe $\\{0,1\\}$. It is easy to prove that:\n\\begin{enumerate}\n\\item $(\\{0,1\\}, \\{\\sigma_{\\textbf{2}_{\\nbI}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra, since it equals $\\textbf{2}$;\n\\item for any $x,y\\in\\{0,1\\}$ and $z\\in x\\uparrow y$, $x\\wedge(y\\wedge z)=0$;\n\\item for any $x, y\\in\\{0,1\\}$, $x\\uparrow y=y\\uparrow x$.\n\\end{enumerate}\n\nIt remains to be shown, to prove that $\\textbf{2}_{\\nbI}$ is a Fidel structure presented as a $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra for $\\nbI$, that for any $x\\in \\{0,1\\}$ and $y\\in \\neg x$, $x\\vee y=1$. If $x=0$, we must have $y=1$, and then $x\\vee y=0\\vee 1=1$; if $x=1$, we have that, either $y=0$, when $x\\vee y=1\\vee 0=1$, or $y=1$, when $x\\vee y=1\\vee1=1$, so we are done\n\n\\begin{theorem}\n$\\nu:F(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\{0,1\\}$ is a bivaluation for $\\nbI$ if, and only if, it is a $\\Sigma_{\\nbI}$-homomorphism from $\\textbf{F}(\\Sigma_{\\nbI}, \\mathcal{V})$ to $\\textbf{2}_{\\nbI}$ which lies in $\\mathcal{F}_{\\textbf{2}_{\\nbI}}$.\n\\end{theorem}\n\nAs expected, we will denote the restricted Nmatrix $(\\textbf{2}_{\\nbI}, \\{1\\}, \\mathcal{F}_{\\textbf{2}_{\\nbI}})$ simply by $\\mathbb{2}_{\\nbI}$.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\nbI$, $\\Gamma\\vDash_{\\nbI}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathbb{2}_{\\nbI}}\\varphi$.\n\\end{theorem}\n\n\nSo, take the Nmatrix $(\\textbf{2}_{\\nbI}, \\{1\\})$ subjacent to the RNmatrix $(\\textbf{2}_{\\nbI}, \\{1\\}, \\mathcal{F}_{\\textbf{2}_{\\nbI}})$: if one writes the row-branching truth table for the said Nmatrix to test a formula $\\varphi$ of $\\nbI$ and simply erases the rows where $\\alpha\\uparrow\\beta$ and $\\beta\\uparrow\\alpha$ are given different values, we get a row-branching, row-eliminating truth table that decides the validity, in a finite number of steps, of $\\varphi$ in $\\nbI$. Essentially, this is the same decision method as the one presented for $\\bI$ in Section \\ref{Decision Method for bI}, with an extra multi-operation standing for negation.\n\n\n\\subsection{Another decision method}\n\nGiven the similitude between $\\bI$ and $\\nbI$, one should hope that a tableau calculus for $\\nbI$, that we will name $\\mathbb{T}_{\\nbI}$, could be easily obtained from $\\mathbb{T}_{\\bI}$ of Section \\ref{Tableaux for bI}: this is, in fact, the case. It is sufficient to add one new rule governing the case in which $\\neg\\varphi$ is false, \n\\[\\frac{\\textsf{0}(\\neg\\varphi)}{\\textsf{1}(\\varphi)}\\]\nand change the conditions for a branch to be complete: a branch $\\theta$ is complete, now in $\\mathbb{T}_{\\nbI}$, when it contains, for every $\\textsf{L}(\\gamma)$ in $\\theta$ neither of the form $\\textsf{0}(\\varphi\\uparrow\\psi)$ or $\\textsf{1}(\\neg\\varphi)$ and with $\\gamma$ not a variable, all the labeled formulas of one of the branches in the rule headed by $\\textsf{L}(\\gamma)$.\n\n\n\\section{Collapsing axioms}\n\n\n\\subsection{$\\textbf{ci}^{\\uparrow}$}\\label{ciuparrow}\n\nNow that we have in our signature a paraconsistent negation, we are capable of doing to the axioms schemata $\\textbf{ci}$ and $\\textbf{cl}$ much the same we did to $\\textbf{ciw}$ when we transformed it into $\\textbf{ciw}^{\\uparrow}$: consider\\label{ciuparrow}\n\\[\\tag{$\\textbf{ci}^{\\uparrow}$} \\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\]\nand the logic $\\nbI\\textbf{ci}^{\\uparrow}$ obtained from $\\nbI$ by addition of $\\textbf{ci}^{\\uparrow}$; we shall prove, once again, that $\\nbI\\textbf{ci}^{\\uparrow}$ collapses back to $\\textbf{CPL}$, and $\\alpha\\uparrow\\beta$ is equivalent to $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$.\n\nIt is clear how $\\alpha\\uparrow\\beta\\vdash_{\\nbI\\textbf{ci}^{\\uparrow}}\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$, since such a deduction could be made in $\\bI$ itself, so let us concentrate in proving that $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{ci}^{\\uparrow}}\\alpha\\uparrow\\beta$.\n\nIt is also clear that \n\\[\\alpha\\uparrow\\beta, \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{ci}^{\\uparrow}}\\alpha\\uparrow\\beta,\\]\nbut from the instance $\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)$ of $\\textbf{ci}^{\\uparrow}$ plus Modus Ponens we can similarly get \n\\[\\neg(\\alpha\\uparrow\\beta), \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{ci}^{\\uparrow}}\\alpha\\uparrow\\beta,\\]\nand using a proof by cases we get\n\\[(\\alpha\\uparrow\\beta)\\vee\\neg(\\alpha\\uparrow\\beta), \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{ci}^{\\uparrow}}\\alpha\\uparrow\\beta.\\]\nSince $(\\alpha\\uparrow\\beta)\\vee\\neg(\\alpha\\uparrow\\beta)$ is an instance of $\\textbf{Ax\\: 11}^{*}$, we get the desired result.\n\nBut we can prove an even nicer result, that implies the previous: an instance of $\\textbf{ci}^{\\uparrow}$ actually implies its corresponding instance of $\\textbf{ciw}^{\\uparrow}$ in $\\nbI$, that is,\n\\[\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta).\\]\nWe have that\n\\[\\neg(\\alpha\\uparrow\\beta),\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)\\]\nby an application of Modus Ponens and remembering $(\\alpha\\wedge\\beta)\\rightarrow[(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)]$ is an instance of $\\textbf{Ax\\: 7}$; furthermore, we trivially find\n\\[\\alpha\\uparrow\\beta,\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)\\]\nby remembering $(\\alpha\\uparrow\\beta)\\rightarrow[(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)]$ is an instance of $\\textbf{Ax\\: 6}$. By a proof by cases, \n\\[(\\alpha\\uparrow\\beta)\\vee\\neg(\\alpha\\uparrow\\beta),\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta),\\]\nand since $(\\alpha\\uparrow\\beta)\\vee\\neg(\\alpha\\uparrow\\beta)$ is an instance of $\\textbf{Ax\\: 11}^{*}$, we derive the aforementioned result.\n\n\n\\subsection{$\\textbf{cl}^{\\uparrow}$}\\label{cluparrow}\n\nAs one should expect, the axiom schema\\label{cluparrow}\n\\[\\tag{$\\textbf{cl}^{\\uparrow}$} \\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\]\nwill collapse its corresponding logic $\\nbI\\textbf{cl}^{\\uparrow}$ back to $\\textbf{CPL}$, with $\\alpha\\uparrow\\beta$ standing for $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}$.\n\nTo see that $\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{cl}^{\\uparrow}}\\alpha\\uparrow\\beta$, we begin by noticing\n\\[\\alpha\\wedge\\beta, \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{cl}^{\\uparrow}}\\alpha\\uparrow\\beta,\\]\nby use of Modus Ponens and the fact $\\bot_{\\alpha\\beta}$ behaves like a bottom element; at the same time,\n\\[\\neg(\\alpha\\wedge\\beta), \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{cl}^{\\uparrow}}\\alpha\\uparrow\\beta,\\]\ngiven the instance $ \\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)$ of $\\textbf{cl}^{\\uparrow}$. With a proof by cases,\n\\[(\\alpha\\wedge\\beta)\\vee\\neg(\\alpha\\wedge\\beta), \\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta}\\vdash_{\\nbI\\textbf{cl}^{\\uparrow}}\\alpha\\uparrow\\beta,\\]\nand since $(\\alpha\\wedge\\beta)\\vee\\neg(\\alpha\\wedge\\beta)$ is an instance of $\\textbf{Ax\\: 11}^{*}$ we obtain the desired result.\n\nHowever, we can again prove a stronger result: an instance of $\\textbf{cl}^{\\uparrow}$ can prove, in $\\nbI$, the corresponding instance of $\\textbf{ciw}^{\\uparrow}$, what proves, as a special case, the previous collapse. To summarize, we wish to prove\n\\[\\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta).\\]\nThis is pretty simple:\n\\[\\alpha\\wedge\\beta, \\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta),\\]\nby the instance $(\\alpha\\wedge\\beta)\\rightarrow[(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)]$ of $\\textbf{Ax\\: 7}$, and\n\\[\\neg(\\alpha\\wedge\\beta), \\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)\\]\nby Modus Ponens and the instance $(\\alpha\\uparrow\\beta)\\rightarrow[(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)]$ of $\\textbf{Ax\\: 6}$; with a proof by cases, \n\\[(\\alpha\\wedge\\beta)\\vee\\neg(\\alpha\\wedge\\beta), \\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta),\\]\nand since $(\\alpha\\wedge\\beta)\\vee\\neg(\\alpha\\wedge\\beta)$ is an instance of $\\textbf{Ax\\: 11}^{*}$, the proof is done.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The logics $\\nbIciw$, $\\nbIci$ and $\\nbIcl$}\n\nOne notices that one of the basic axiom schema, $\\textbf{Ip}$, of $\\bI$ is\n\\[(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\gamma)),\\]\nwhich can be seem as the basic axiom of $\\textbf{mbC}$, $\\textbf{bc1}$,\n\\[\\circ\\alpha\\rightarrow(\\alpha\\rightarrow(\\neg\\alpha\\rightarrow\\beta)),\\]\nwith $\\neg\\alpha$ replaced by $\\beta$ and $\\circ\\alpha$, that is, that $\\alpha$ is consistent, replaced by $\\alpha\\uparrow\\beta$, that is, that $\\alpha$ is incompatible with $\\beta$.\n\nAn attempt to make something similar to the axiom $\\textbf{ciw}$,\n\\[\\circ\\alpha\\vee(\\alpha\\wedge\\neg\\alpha),\\]\nis not successful, as the axiom $\\textbf{ciw}^{\\uparrow}$\n\\[(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)\\]\ncollapses the incompatibility operator back to its classical interpretation. However, in $\\nbI$, where we find at our disposal a non-classical negation, we can take an adaptation of the axiom $\\textbf{ciw}$ to the language $\\Sigma_{\\nbI}$, instead of the generalization $\\textbf{ciw}^{\\uparrow}$, and study the resulting logic. We will do the same with the axioms $\\textbf{ci}$ and $\\textbf{cl}$, producing three logics that mix paraconsistency and incompatibility.\n\nSo, over the signature $\\Sigma_{\\nbI}$, we consider the systems:\n\\begin{enumerate}\n\\item $\\nbIciw$\\label{nbIciw}, obtained from $\\nbI$ by addition of the axiom schema\\label{ciw*}\n\\[\\tag{$\\textbf{ciw}^{*}$} (\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha);\\]\n\\item $\\nbIci$\\label{nbIci}, obtained from $\\nbI$ by adding the axiom schema\\label{ci*}\n\\[\\tag{$\\textbf{ci}^{*}$}\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha);\\]\n\\item $\\nbIcl$\\label{nbIcl}, obtained from $\\nbI$ by adding\\label{cl*}\n\\[\\tag{$\\textbf{cl}^{*}$}\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha).\\]\n\\end{enumerate}\n\n\n\\subsection{Bivaluations}\n\n\\begin{definition}\\label{definition of Bivaluation for nbIciw, nbIci, ...}\nA \\index{Bivaluation for $\\nbIciw$}\\index{Bivaluation for $\\nbIci$}\\index{Bivaluation for $\\nbIcl$}bivaluation for $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$ is a valuation for $\\nbI$ satisfying in addition that:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\neg\\alpha)=0$;\n\\item \\begin{enumerate}\n\\item if $\\mathcal{L}=\\nbIci$ and $\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))=1$, then $\\nu(\\alpha)=\\nu(\\neg\\alpha)=1$;\n\\item if $\\mathcal{L}=\\nbIcl$ and $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=1$, then $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{definition}\n\nAs usual, given a set of formulas $\\Gamma\\cup\\{\\varphi\\}$ on the signature $\\Sigma_{\\nbI}$, we say $\\Gamma$ proves $\\varphi$ according to bivaluations for $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, and write \\label{vDashnbIciw}\\label{vDashnbIci}\\label{vDashnbIcl}$\\Gamma\\vDash_{\\mathcal{L}}\\varphi$, if, for every bivaluation $\\nu$ for $\\mathcal{L}$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$, one has $\\nu(\\varphi)=1$.\n\nSince bivaluations for $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$ are bivaluations for $\\nbI$ satisfying some additional property, it is easy to see that \"$\\vDash_{\\mathcal{L}}$\" models all those axiom schemata of $\\nbI$, plus Modus Ponens, its only inference rule; we wish now to prove that \"$\\vDash_{\\nbIciw}$\" also models $\\textbf{ciw}^{*}$, and an analogous result holds for the other logics.\n\n\\begin{enumerate}\n\\item Take an instance $(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)$ of $\\textbf{ciw}^{*}$ and a bivaluation $\\nu$ for $\\nbIciw$, and we see that \n\\[\\nu((\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha))=0\\]\nif and only if $\\nu(\\alpha\\uparrow\\neg\\alpha)=0$ and $\\nu(\\alpha\\wedge\\neg\\alpha)=0$; the first of these equalities holds if and only $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=1$, what would imply that $\\nu(\\alpha\\wedge\\neg\\alpha)=1$, reaching a contradiction.\n\n\\item Take a bivaluation $\\nu$ for $\\nbIci$ and suppose \n\\[\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha))=0,\\]\nwhat happens if and only if $\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))=1$ but $\\nu(\\alpha\\wedge\\neg\\alpha)=0$; from the first equality, $\\nu(\\alpha)=\\nu(\\neg\\alpha)=1$, what again contradicts $\\nu(\\alpha\\wedge\\neg\\alpha)=0$.\n\n\\item Finally, we take a bivaluation $\\nu$ for $\\nbIcl$ and assume it is possible to have\n\\[\\nu(\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha))=0.\\]\nThis is equivalent to having $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=1$ and $\\nu(\\alpha\\uparrow\\neg\\alpha)=0$, but the first equality implies $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$, which is evidently contradictory.\n\\end{enumerate}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\nbI}$ and $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, if $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ then $\\Gamma\\vDash_{\\mathcal{L}}\\varphi$.\n\\end{theorem}\n\nTo prove the reciprocal, take a set of formulas $\\Gamma$ maximal with respect to not proving $\\varphi$ in $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$ which is also closed (again, with respect to $\\mathcal{L}$) and non-trivial. We define $\\nu:F(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\{0,1\\}$ such that $\\nu(\\gamma)=1$ if and only if $\\gamma\\in\\Gamma$, and wish to prove that $\\nu$ is a bivaluation for $\\mathcal{L}$.\n\nIt is clear $\\Gamma$ does not prove $\\varphi$ in $\\nbI$, since $\\mathcal{L}$ extends $\\nbI$, and also that $\\Gamma$ is closed in $\\nbI$, given it is closed according to $\\mathcal{L}$ and this logic is stronger than the previous; this implies $\\nu$ is at least a bivaluation for $\\nbI$, according to the definition found in Section \\ref{Bivaluations for nbI}.\n\n\\begin{enumerate}\n\\item For any of the logics $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, if $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$, suppose $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=1$: by definition of $\\nu$, this means $\\alpha\\uparrow\\neg\\alpha, \\alpha, \\neg\\alpha\\in\\Gamma$, and since $\\Gamma$ is closed and \n\\[(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\rightarrow(\\neg\\alpha\\rightarrow\\varphi))\\]\nis an instance of $\\textbf{Ip}$, we get that $\\varphi\\in\\Gamma$, what is a contradiction, We must then have either $\\nu(\\alpha)=0$ or $\\nu(\\neg\\alpha)=0$.\n\nSuppose, reciprocally, that $\\nu(\\alpha)=0$ or $\\nu(\\neg\\alpha)=0$, and for a proof by contradiction, let us take $\\nu(\\alpha\\uparrow\\neg\\alpha)=0$, meaning $\\alpha\\uparrow\\neg\\alpha\\notin\\Gamma$ and either $\\alpha\\notin\\Gamma$ or $\\neg\\alpha\\notin\\Gamma$, implying $\\alpha\\wedge\\neg\\alpha\\notin\\Gamma$. By the maximality of $\\Gamma$, one has $\\Gamma, \\alpha\\uparrow\\neg\\alpha\\vdash_{\\mathcal{L}}\\varphi$ and $\\Gamma,\\alpha\\wedge\\neg\\alpha\\vdash_{\\mathcal{L}}\\varphi$, and by a proof by cases\n\\[\\Gamma, (\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)\\vdash_{\\mathcal{L}}\\varphi.\\]\nSince $(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)$ is an instance of $\\textbf{ciw}^{*}$, if $\\mathcal{L}=\\nbIciw$ we get $\\Gamma\\vdash_{\\nbIciw}\\varphi$, a contradiction; for the other two logics, given that, from Sections \\ref{ciuparrow} and \\ref{cluparrow},\n\\[\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)\\]\nand\n\\[\\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta),\\]\nby replacing $\\beta$ with $\\neg\\alpha$ we obtain that we still have $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$, so still a contradiction. This means we must have $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$.\n\n\\item If $\\mathcal{L}=\\nbIci$ and $\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))=1$, this means $\\neg(\\alpha\\uparrow\\neg\\alpha)\\in\\Gamma$; since $\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha)$ is an instance of $\\textbf{ci}^{*}$ and $\\Gamma$ is closed, $\\alpha\\wedge\\neg\\alpha\\in\\Gamma$, implying that $\\alpha, \\neg\\alpha\\in\\Gamma$ and therefore $\\nu(\\alpha)=1$ and $\\nu(\\neg\\alpha)=1$.\n\n\\item If $\\mathcal{L}=\\nbIcl$ and $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=1$, we have $\\neg(\\alpha\\wedge\\neg\\alpha)\\in\\Gamma$; since $\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha)$ is an instance of $\\textbf{cl}^{*}$ and $\\Gamma$ is closed, $\\alpha\\uparrow\\neg\\alpha\\in\\Gamma$, and therefore $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$.\n\\end{enumerate}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\nbI}$ and $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, if $\\Gamma\\vDash_{\\mathcal{L}}\\varphi$ then $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$.\n\\end{theorem}\n\nIn the spirit of the arguments just used, since\n\\[\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)\\]\nand\n\\[\\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)\\vdash_{\\nbI}(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta),\\]\nby replacing $\\beta$ with $\\neg\\alpha$ we discover both $\\nbIci$ and $\\nbIcl$ are extensions of $\\nbIciw$, and with the aid of bivaluations, we can prove even more.\n\n\\begin{proposition}\n\\begin{enumerate}\n\\item $\\nbIciw$ can not prove $\\textbf{ci}^{*}$, and therefore $\\nbIci$ is strictly stronger than $\\nbIciw$;\n\\item $\\nbIciw$ can not prove $\\textbf{cl}^{*}$, and therefore $\\nbIcl$ is strictly stronger than $\\nbIciw$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Take a bivaluation $\\nu$ for $\\nbIciw$ such that $\\nu(\\alpha)=0$, $\\nu(\\neg\\alpha)=0$ (and therefore $\\nu(\\alpha\\wedge\\neg\\alpha)=0$), $\\nu(\\alpha\\uparrow\\neg\\alpha)=1$ and $\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))=1$: then\n\\[\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha))=0.\\]\n\\item Take a bivaluation $\\nu$ for $\\nbIciw$ such that $\\nu(\\alpha)=1$, $\\nu(\\neg\\alpha)=1$ (and therefore $\\nu(\\alpha\\wedge\\neg\\alpha)=1$), $\\nu(\\alpha\\uparrow\\neg\\alpha)=0$ and $\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))=1$: then \n\\[\\nu(\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha))=0.\\]\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theorem}\n\\begin{enumerate}\n\\item The formula $(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)$ is a tautology of $\\nbIci$;\n\\item $\\nbIci$ is obtained from $\\nbIciw$ by adding the axiom schema\\label{cc*}\n\\[\\tag{$\\textbf{cc}^{*}$} (\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha).\\]\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item It is clear that\n\\[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIci}(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha),\\]\nbut from the instance\n\\[\\neg[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]\\rightarrow[(\\alpha\\uparrow\\neg\\alpha)\\wedge\\neg(\\alpha\\uparrow\\neg\\alpha)]\\]\nof $\\textbf{ci}^{*}$ we see that\n\\[\\neg[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]\\vdash_{\\nbIci}(\\alpha\\uparrow\\neg\\alpha)\\wedge\\neg(\\alpha\\uparrow\\neg\\alpha).\\]\n$(\\alpha\\uparrow\\neg\\alpha)\\wedge\\neg(\\alpha\\uparrow\\neg\\alpha)$ easily deduces both $\\alpha\\uparrow\\neg\\alpha$ and $\\neg(\\alpha\\uparrow\\neg\\alpha)$, and from this last formula and the instance $\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha)$ of $\\textbf{ci}^{*}$, we get $\\neg[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]$ deduces $\\alpha\\uparrow\\neg\\alpha$, $\\alpha$ and $\\neg\\alpha$: from the instance of $\\textbf{Ip}$\n\\[(\\alpha\\uparrow\\neg\\alpha)\\rightarrow\\big[\\alpha\\rightarrow\\big[\\neg\\alpha\\rightarrow\\big((\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)\\big)\\big]\\big],\\]\nthis means \n\\[\\neg[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]\\vdash_{\\nbIci}(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha),\\]\nand by a proof by cases \n\\[[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]\\vee\\neg[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]\\vdash_{\\nbIci}(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha).\\]\nSince the antecedent in this argument is an instance of $\\textbf{Ax\\: 11}^{*}$, we obtain that $(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)$ is a tautology.\n\n\\item So we have that $\\nbIci$ proves any instance of $\\textbf{ciw}^{*}$ and $\\textbf{cc}^{*}$, being therefore stronger than the logic obtained from $\\nbIciw$ by addition of $\\textbf{cc}^{*}$, which we will briefly call $\\nbIciw+$. To prove that $\\nbIciw+$ is as strong as $\\nbIci$, and therefore that both are equal, we only need to prove that any instance of $\\textbf{ci}^{*}$ is a tautology in $\\nbIciw+$, or what is equivalent, that\n\\[\\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIciw+}\\alpha\\wedge\\neg\\alpha.\\]\nObviously\n\\[\\alpha\\wedge\\neg\\alpha, \\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIciw+}\\alpha\\wedge\\neg\\alpha;\\]\nnow, from the instance\n\\[[(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)]\\rightarrow\\big[(\\alpha\\uparrow\\neg\\alpha)\\rightarrow\\big[\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha)\\big]\\big]\\]\nof $\\textbf{Ip}$ and the fact $(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)$ is an instance of $\\textbf{cc}^{*}$, we can see by applying the deduction meta-theorem as needed that\n\\[\\alpha\\uparrow\\neg\\alpha, \\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIciw+}\\alpha\\wedge\\neg\\alpha.\\]\n\nBy a proof by cases,\n\\[(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha), \\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIciw+}\\alpha\\wedge\\neg\\alpha,\\]\nand since $(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)$ is an instance of $\\textbf{ciw}^{*}$, we discover that $\\nbIci$ is $\\nbIciw+$, as we wanted to show.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Fidel structures}\n\nA \\index{Fidel structure for $\\nbIciw$}\\index{Fidel structure for $\\nbIci$}\\index{Fidel structure for $\\nbIcl$}Fidel structure, presented as a $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra, for $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$ is any $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra $\\mathcal{A}=(A,\\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\nbI}^{\\textbf{CPL}}})$ that is a Fidel structure for $\\nbI$ and satisfies, additionally:\n\\begin{enumerate}\n\\item if $b\\in\\neg a$, ${\\sim}(a\\wedge b)\\in a\\uparrow b$;\n\\item\\begin{enumerate}\n\\item if $\\mathcal{L}=\\nbIci$ and $b\\in\\neg a$, $a\\wedge b\\in\\neg{\\sim}(a\\wedge b)$;\n\\item if $\\mathcal{L}=\\nbIcl$ and $b\\in\\neg a$, ${\\sim}(a\\wedge b)\\in\\neg(a\\wedge b)$.\n\\end{enumerate}\n\\end{enumerate}\n\n\nA valuation for a Fidel structure $\\mathcal{A}$, presented as a $\\Sigma_{\\nbI}^{\\textbf{CPL}}$-multialgebra, for $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, will still be any $\\Sigma_{\\nbI}$-homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\mathcal{A}$; and now we will consider the restricted Nmatrices $(\\mathcal{A},\\{\\top\\},\\mathcal{F}_{\\mathcal{A}}^{\\mathcal{L}})$, where $\\mathcal{A}$ is a Fidel structure for $\\mathcal{L}$ and $\\mathcal{F}_{\\mathcal{A}}^{\\mathcal{L}}$ is the set of valuations $\\nu:\\textbf{F}(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\mathcal{A}$ such that, for any formulas $\\alpha$ and $\\beta$ over the signature $\\Sigma_{\\nbI}$:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$;\n\\item $\\nu(\\alpha\\uparrow\\neg\\alpha)={\\sim}(\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha))$;\n\\item \\begin{enumerate}\n\\item if $\\mathcal{L}=\\nbIci$, $\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))=\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha)$;\n\\item if $\\mathcal{L}=\\nbIcl$, $\\nu(\\alpha\\uparrow\\neg\\alpha)=\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))$.\n\\end{enumerate}\n\\end{enumerate}\n\n\\begin{proposition}\nThe RNmatrices $(\\mathcal{A}, \\{\\top\\}, \\mathcal{F}_{\\mathcal{A}}^{\\mathcal{L}})$ as described above, for $\\mathcal{L}\\in$\\\\$\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, are structural.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\sigma:\\textbf{F}(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\textbf{F}(\\Sigma_{\\nbI},\\mathcal{V})$ be a $\\Sigma_{\\nbI}$-homomorphism and $\\nu\\in\\mathcal{F}_{\\mathcal{A}}^{\\mathcal{L}}$: it is clear that $\\nu\\circ\\sigma: \\textbf{F}(\\Sigma_{\\nbI},\\mathcal{V})\\rightarrow\\mathcal{A}$ is always a $\\Sigma_{\\nbI}$-homomorphism, satisfying additionally that for all formulas $\\alpha$ and $\\beta$ on the signature $\\Sigma_{\\nbI}$, $\\nu\\circ\\sigma(\\alpha\\uparrow\\beta)=\\nu\\circ\\sigma(\\beta\\uparrow\\alpha)$.\n\n\\begin{enumerate}\n\\item We must prove that $\\nu\\circ\\sigma(\\alpha\\uparrow\\neg\\alpha)={\\sim}(\\nu\\circ\\sigma(\\alpha)\\wedge\\nu\\circ\\sigma(\\neg\\alpha))$, what is easy since\n\\[\\nu\\circ\\sigma(\\alpha\\uparrow\\neg\\alpha)=\\nu(\\sigma(\\alpha\\uparrow\\neg\\alpha))=\\nu(\\sigma(\\alpha)\\uparrow\\sigma(\\neg\\alpha))=\\nu(\\sigma(\\alpha)\\uparrow\\neg\\sigma(\\alpha))=\\]\n\\[{\\sim}\\big(\\nu(\\sigma(\\alpha))\\wedge\\neg\\nu(\\sigma(\\alpha))\\big)={\\sim}(\\nu\\circ\\sigma(\\alpha)\\wedge\\nu\\circ\\sigma(\\neg\\alpha)).\\]\n\n\\item\\begin{enumerate}\n\\item If $\\mathcal{L}=\\nbIci$, it remains for us to show that $\\nu\\circ\\sigma(\\neg(\\alpha\\uparrow\\neg\\alpha))=\\nu\\circ\\sigma(\\alpha)\\wedge\\nu\\circ\\sigma(\\neg\\alpha)$, and we see that\n\\[\\nu\\circ\\sigma(\\neg(\\alpha\\uparrow\\neg\\alpha))=\\nu\\big(\\neg(\\sigma(\\alpha)\\uparrow\\neg\\sigma(\\alpha))\\big)=\\nu(\\sigma(\\alpha))\\wedge\\nu(\\neg\\sigma(\\alpha))=\\nu\\circ\\sigma(\\alpha)\\wedge\\nu\\circ\\sigma(\\neg\\alpha).\\]\n\n\\item If $\\mathcal{L}=\\nbIcl$, we only need to prove that $\\nu\\circ\\sigma(\\alpha\\uparrow\\neg\\alpha)=\\nu\\circ\\sigma(\\neg(\\alpha\\wedge\\neg\\alpha))$:\n\\[\\nu\\circ\\sigma(\\alpha\\uparrow\\neg\\alpha)=\\nu(\\sigma(\\alpha)\\uparrow\\neg\\sigma(\\alpha))=\\nu\\big(\\neg(\\sigma(\\alpha)\\wedge\\neg\\sigma(\\alpha))\\big)=\\nu\\circ\\sigma(\\neg(\\alpha\\wedge\\neg\\alpha)).\\]\n\\end{enumerate}\n\\end{enumerate}\n\\end{proof}\n\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\nbI}$, if $\\Gamma$ proves $\\varphi$ according to such restricted Nmatrices we will write \\label{VdashFnbIciw}\\label{VdashFnbIci}\\label{VdashFnbIcl}$\\Gamma\\Vdash_{\\mathcal{F}}^{\\mathcal{L}}\\varphi$, and say that $\\Gamma$ proves $\\varphi$ according to Fidel structures for $\\mathcal{L}$.\n\nIt is easy to see how \"$\\Vdash_{\\mathcal{F}}^{\\mathcal{L}}$\" models the axiom schemata of $\\nbI$ and its rules of inference, but we can also prove \"$\\Vdash_{\\mathcal{F}}^{\\nbIciw}$\" proves any instance of $\\textbf{ciw}^{*}$, \"$\\Vdash_{\\mathcal{F}}^{\\nbIci}$\" proves any instance of $\\textbf{ci}^{*}$ and \"$\\Vdash_{\\mathcal{F}}^{\\nbIcl}$\" proves any instance of $\\textbf{cl}^{*}$, so take a formula $\\alpha$ over the signature $\\Sigma_{\\nbI}$.\n\n\\begin{enumerate}\n\\item Given an instance $(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)$ of $\\textbf{ciw}^{*}$, we have for a Fidel structure $\\mathcal{A}$ for $\\nbIciw$ and a $\\nu\\in \\mathcal{F}_{\\mathcal{A}}^{\\nbIciw}$ that\n\\[\\nu((\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha))=\\nu(\\alpha\\uparrow\\neg\\alpha)\\vee\\nu(\\alpha\\wedge\\neg\\alpha)=\\]\n\\[{\\sim}(\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha))\\vee(\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha))=({\\sim}\\nu(\\alpha)\\vee{\\sim}\\nu(\\neg\\alpha))\\vee(\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha))=\\]\n\\[\\big[({\\sim}\\nu(\\alpha)\\vee{\\sim}\\nu(\\neg\\alpha))\\vee\\nu(\\alpha)\\big]\\wedge\\big[({\\sim}\\nu(\\alpha)\\vee{\\sim}\\nu(\\neg\\alpha))\\vee\\nu(\\neg\\alpha)\\big]=\\]\n\\[[{\\sim}\\nu(\\neg\\alpha)\\vee\\top]\\wedge[{\\sim}\\nu(\\alpha)\\vee\\top]=\\top\\wedge\\top=\\top.\\]\n\n\\item Take an instance $\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha)$ of $\\textbf{ci}^{*}$, a Fidel structure $\\mathcal{A}$ for $\\nbIci$ and a $\\nu\\in\\mathcal{F}_{\\mathcal{A}}^{\\nbIci}$:\n\\[\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha))=\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))\\rightarrow\\nu(\\alpha\\wedge\\neg\\alpha)=\\]\n\\[\\big[\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha)\\big]\\rightarrow\\big[\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha)\\big]=\\top.\\]\n\n\\item For $\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha)$ an instance of $\\textbf{cl}^{*}$, a Fidel structure $\\mathcal{A}$ for $\\nbIcl$ and a $\\nu\\in\\mathcal{F}_{\\mathcal{A}}^{\\nbIcl}$,\n\\[\\nu(\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha))=\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))\\rightarrow\\nu(\\alpha\\uparrow\\neg\\alpha)=\\nu(\\alpha\\uparrow\\neg\\alpha)\\rightarrow\\nu(\\alpha\\uparrow\\neg\\alpha)=\\top.\\]\n\\end{enumerate}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\nbI}$, for any $\\mathcal{L}\\in$\\\\$\\{\\nbIciw, \\nbIci, \\nbIcl\\}$ we have that, if $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$, then $\\Gamma\\vDash_{\\mathcal{F}}^{\\mathcal{L}}\\varphi$.\n\\end{theorem}\n\nReciprocally, we define, for a logic $\\mathcal{L}\\in\\{\\nbIciw,\\nbIci,\\nbIcl\\}$ and a set of formulas $\\Gamma$ over the signature $\\Sigma_{\\nbI}$, the equivalence relation such that, for formulas $\\alpha$ and $\\beta$ still over the signature $\\Sigma_{\\nbI}$, $\\alpha\\equiv^{\\mathcal{L}}_{\\Gamma}\\beta$ if and only if $\\Gamma\\vdash_{\\mathcal{L}}\\alpha\\rightarrow\\beta$ and $\\Gamma\\vdash_{\\mathcal{L}}\\beta\\rightarrow\\alpha$.\n\nAs we did several times before, the well-defined quotient $A^{\\mathcal{L}}_{\\Gamma}=F(\\Sigma_{\\nbI}, \\mathcal{V})\/\\equiv^{\\mathcal{L}}_{\\Gamma}$ is made into a Boolean algebra, and by defining, for classes of formulas $[\\alpha]$ and $[\\beta]$, $\\neg\\alpha=$\\\\$\\{[\\neg\\varphi]\\ :\\  \\varphi\\in[\\alpha]\\}$ and\n\\begin{enumerate}\n\\item if $[\\beta]\\in \\neg[\\alpha]$,\n\\[[\\alpha]\\uparrow[\\beta]=[{\\sim}(\\alpha\\wedge\\beta)];\\]\n\\item otherwise,\n\\[[\\alpha]\\uparrow[\\beta]=\\{[\\varphi\\uparrow\\psi]\\ :\\  \\varphi\\in[\\alpha], \\psi\\in[\\beta]\\};\\]\n\\end{enumerate}\nwe shall prove $\\mathcal{A}^{\\mathcal{L}}_{\\Gamma}=(A^{\\mathcal{L}}_{\\Gamma},\\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in \\Sigma_{\\nbI}^{\\textbf{CPL}}})$ is a Fidel structure for $\\mathcal{L}$, which we shall call the Lindenbaum-Tarski Fidel structure of $\\mathcal{L}$ associated to $\\Gamma$.\n\nQuite clearly $(A^{\\mathcal{L}}_{\\Gamma},\\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra; concerning the paraconsistent negation, for every $[\\beta]\\in\\neg[\\alpha]$, there exists $\\varphi\\equiv^{\\mathcal{L}}_{\\Gamma}\\alpha$ such that $\\beta\\equiv^{\\mathcal{L}}_{\\Gamma}\\neg\\varphi$ and then $[\\alpha]\\vee[\\beta]=[\\alpha\\vee\\beta]=[\\varphi\\vee\\neg\\varphi]=\\top$; regarding the incompatibility connective, we begin by proving the operation is well-defined in the case that $[\\beta]\\in\\neg[\\alpha]$.\n\nIf $\\varphi\\in[\\alpha]$ and $\\psi\\in[\\beta]$, we have that $\\alpha\\equiv^{\\mathcal{L}}_{\\Gamma}\\varphi$ and $\\beta\\equiv^{\\mathcal{L}}_{\\Gamma}\\psi$, implying that ${\\sim}(\\alpha\\wedge\\beta)\\equiv^{\\mathcal{L}}_{\\Gamma}{\\sim}(\\varphi\\wedge\\psi)$ given \"$\\equiv^{\\mathcal{L}}_{\\Gamma}$\" is a congruence for the connectives in $\\{\\vee, \\wedge, \\rightarrow, {\\sim}\\}$, and therefore $[\\alpha]\\uparrow[\\beta]=[\\varphi]\\uparrow[\\psi]$. In the case of $[\\alpha]\\uparrow[\\beta]$ with $[\\beta]\\not\\in\\neg[\\alpha$, we use the same reasoning that worked for $\\bI$ in Section \\ref{Completeness fo Fidel structures for bI}.\n\nTo prove $\\mathcal{A}^{\\mathcal{L}}_{\\Gamma}$ is a Fidel structure for $\\nbI$, we are yet to prove that, if $[\\beta]\\in\\neg[\\alpha]$, for every value $[\\gamma]$ in $[\\alpha]\\uparrow[\\beta]$ (in this case, there is only one), we have $[\\alpha]\\wedge([\\beta]\\wedge[\\gamma])=\\bot$, the result being clear in the case that $[\\beta]\\notin\\neg[\\alpha]$. Since $\\gamma\\equiv^{\\mathcal{L}}_{\\Gamma}{\\sim}(\\alpha\\wedge\\beta)$,\n\\[\\alpha\\wedge(\\beta\\wedge\\gamma)\\equiv^{\\mathcal{L}}_{\\Gamma}\\alpha\\wedge(\\beta\\wedge{\\sim}(\\alpha\\wedge\\beta))\\equiv^{\\mathcal{L}}_{\\Gamma}\\bot,\\]\nand the result holds; the case in which $[\\alpha]\\in\\neg[\\beta]$ is analogous.\n\nIt only remains to be shown that $\\mathcal{A}^{\\mathcal{L}}_{\\Gamma}$ is a Fidel structure for $\\mathcal{L}$.\n\\begin{lemma}\\label{some equivalences in extensions of nbI}\n\\begin{enumerate}\n\\item For any logic $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, $\\alpha\\uparrow\\neg\\alpha$ and ${\\sim}(\\alpha\\wedge\\neg\\alpha)$ are equivalent.\n\\item $\\neg(\\alpha\\uparrow\\neg\\alpha)$ and $\\alpha\\wedge\\neg\\alpha$ are equivalent in $\\nbIci$.\n\\item $\\neg(\\alpha\\wedge\\neg\\alpha)$ and $\\alpha\\uparrow\\neg\\alpha$ are equivalent in $\\nbIcl$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item For simplicity, we denote $\\bot_{\\alpha\\wedge\\neg\\alpha,\\alpha\\wedge\\neg\\alpha}$ simply by $\\bot$. Applying the deduction meta-theorem, we must prove $\\alpha\\uparrow\\neg\\alpha, \\alpha\\wedge\\neg\\alpha\\vdash_{\\mathcal{L}}\\bot$ and $(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\bot\\vdash_{\\mathcal{L}}\\alpha\\uparrow\\neg\\alpha$.\n\nThe first implication is obvious, since $\\alpha\\uparrow\\neg\\alpha$ together with $\\alpha\\wedge\\neg\\alpha$ is exactly $\\bot_{\\alpha,\\neg\\alpha}$, and all bottom elements are equivalent to each other. Reciprocally, quite obviously\n\\[(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\bot, \\alpha\\uparrow\\neg\\alpha\\vdash_{\\mathcal{L}}\\alpha\\uparrow\\neg\\alpha,\\]\nand\n\\[(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\bot, \\alpha\\wedge\\neg\\alpha\\vdash_{\\mathcal{L}}\\alpha\\uparrow\\neg\\alpha\\]\nby Modus Ponens and the fact a bottom element implies any formula. With a proof by cases and the fact that $(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)$ is an instance of $\\textbf{ciw}^{*}$, which is a tautology in any of the logics $\\mathcal{L}$, we get the desired result.\n\n\\item We must prove $\\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIci}\\alpha\\wedge\\neg\\alpha$ and vice-versa, being the first direction a clear application of $\\textbf{ci}^{*}$. Reciprocally, we have \n\\[\\alpha\\wedge\\neg\\alpha, \\neg(\\alpha\\uparrow\\neg\\alpha)\\vdash_{\\nbIci}\\neg(\\alpha\\uparrow\\neg\\alpha)\\]\nwith easy, while \n\\[\\alpha\\wedge\\neg\\alpha, \\alpha\\uparrow\\neg\\alpha\\vdash_{\\nbIci}\\neg(\\alpha\\uparrow\\neg\\alpha)\\]\nfollows from the fact $\\alpha\\wedge\\neg\\alpha$ implies both $\\alpha$ and $\\neg\\alpha$, followed by an application of $\\textbf{Ip}$. With a proof by cases and the fact $(\\alpha\\uparrow\\neg\\alpha)\\vee\\neg(\\alpha\\uparrow\\neg\\alpha)$ is an instance of $\\textbf{Ax\\: 11}^{*}$, the result follows.\n\n\\item Now, we must prove $\\neg(\\alpha\\wedge\\neg\\alpha)\\vdash_{\\nbIcl}\\alpha\\uparrow\\neg\\alpha$ and its reciprocal, being the first implication a direct application of $\\textbf{cl}^{*}$. Reciprocally, \n\\[\\alpha\\uparrow\\neg\\alpha, \\neg(\\alpha\\wedge\\neg\\alpha)\\vdash_{\\nbIcl}\\neg(\\alpha\\wedge\\neg\\alpha)\\]\nand, from the fact $\\alpha\\wedge\\neg\\alpha$ implies both $\\alpha$ and $\\neg\\alpha$, and by applying $\\textbf{Ip}$,\n\\[\\alpha\\uparrow\\neg\\alpha, \\alpha\\wedge\\neg\\alpha\\vdash_{\\nbIcl}\\neg(\\alpha\\wedge\\neg\\alpha);\\]\nthrough a proof by cases and the fact $(\\alpha\\wedge\\neg\\alpha)\\vee\\neg(\\alpha\\wedge\\neg\\alpha)$ is an instance of $\\textbf{Ax\\: 11}^{*}$, we conclude the proof.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{enumerate}\n\\item By definition, if $[\\beta]\\in\\neg[\\alpha]$, then $[\\alpha]\\uparrow[\\beta]$ is single-valued and equal to ${\\sim}([\\alpha]\\wedge[\\beta])$.\n\\item \\begin{enumerate}\n\\item If $\\mathcal{L}=\\nbIci$ and $[\\beta]\\in\\neg[\\alpha]$, implying that there exists $\\phi\\equiv^{\\nbIci}_{\\Gamma}\\alpha$ such that $\\beta\\equiv^{\\nbIci}_{\\Gamma}\\neg\\phi$, by use of Lemma \\ref{some equivalences in extensions of nbI} we find\n\\[[{\\sim}(\\alpha\\wedge\\beta)]=[{\\sim}(\\phi\\wedge\\neg\\phi)]=[\\phi\\uparrow\\neg\\phi],\\]\nand therefore $\\neg[{\\sim}(\\alpha\\wedge\\beta)]$ contains $[\\neg(\\phi\\uparrow\\neg\\phi)]=[\\phi\\wedge\\neg\\phi]$, which equals $[\\alpha]\\wedge[\\beta]$, as we wanted to show.\n\\item If $\\mathcal{L}=\\nbIcl$ and $[\\beta]\\in\\neg[\\alpha]$, let $\\phi\\equiv^{\\nbIcl}_{\\Gamma}\\alpha$ be an element of $[\\alpha]$ such that $\\beta\\equiv^{\\nbIcl}_{\\Gamma}\\neg\\phi$, and then $[\\alpha\\wedge\\beta]=[\\phi\\wedge\\neg\\phi]$, while $[\\phi\\uparrow\\neg\\phi]=[\\neg(\\phi\\wedge\\neg\\phi)]\\in \\neg[\\phi\\wedge\\neg\\phi]$. \n\nSince $[\\phi\\uparrow\\neg\\phi]=[{\\sim}(\\phi\\wedge\\neg\\phi)]$ and ${\\sim}([\\phi]\\wedge[\\neg\\phi])={\\sim}([\\alpha]\\wedge[\\beta])$, the result is done: ${\\sim}([\\alpha]\\wedge[\\beta])\\in \\neg([\\alpha]\\wedge[\\beta])$.\n\\end{enumerate}\n\\end{enumerate}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{\\nbI}$, for any logic $\\mathcal{L}\\in$\\\\$\\{\\nbIciw, \\nbIci, \\nbIcl\\}$ we have that, if $\\Gamma\\Vdash_{\\mathcal{F}}^{\\mathcal{L}}\\varphi$, then $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\\subsection{Decision method}\\label{Decision method for nbIciw, ...}\n\nWe take the Boolean algebra $\\textbf{2}$ and extend it to the $\\Sigma^{\\textbf{CPL}}_{\\textbf{nbI}}$-multialgebra $\\textbf{2}_{\\textbf{nbIciw}}$ with operations given by the tables below.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|r|}\n\\hline\n& $\\neg$\\\\ \\hline\n$0$ & $\\{1\\}$\\\\ \\hline\n$1$ & $\\{0,1\\}$\\\\\\hline\n\\end{tabular}\n\\caption*{Negation}\n\\end{minipage}\n\\hspace{3cm}\n\\centering\n\\begin{minipage}[t]{4cm}\n\\centering\n\\begin{tabular}{|l|c|r|}\n\\hline\n$\\uparrow$ & $0$ & $1$\\\\ \\hline\n$0$ & $\\{0,1\\}$ & $\\{0, 1\\}$\\\\ \\hline\n$1$ & $\\{0, 1\\}$ & $\\{0\\}$\\\\\\hline\n\\end{tabular}\n\\caption*{Incompatibility}\n\\end{minipage}\n\\end{figure}\n\nIt is easy to see that:\n\\begin{enumerate}\n\\item $(\\{0,1\\}, \\{\\sigma_{\\textbf{2}_{\\textbf{nbIciw}}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra (that is, $\\textbf{2}$);\n\\item for any $x,y\\in\\{0,1\\}$ and $z\\in x\\uparrow y$, $x\\wedge(y\\wedge z)=0$;\n\\item for any $x\\in \\{0,1\\}$ and $y\\in\\neg x$, $x\\vee y=1$, since in the case that $x=0$ we must have $y=1$.\n\\end{enumerate}\n\nSo $\\textbf{2}_{\\textbf{nbIciw}}$ is a Fidel structure for $\\textbf{nbI}$. But we can prove even more, that it is a Fidel structure for $\\mathcal{L}\\in\\{\\textbf{nbIciw}, \\textbf{nbIci}, \\textbf{nbIcl}\\}$.\n\n\\begin{enumerate}\n\\item We see that, for any values of $x$ and $y$ in $\\{0,1\\}$, ${\\sim}(x\\wedge y)\\in x\\uparrow y$ even if $y\\not\\in\\neg x$;\n\\item\n\\begin{enumerate}\n\\item suppose that, for some values $a$ and $b$ in $\\{0,1\\}$ such that $b\\in\\neg a$, $a\\wedge b$ is not in $\\neg{\\sim}(a\\wedge b)$, and since $\\neg 1=\\{0,1\\}$, we must then have ${\\sim}(a\\wedge b)=0$, that is, $a\\wedge b=1$ and therefore $a=b=1$; but then $\\neg{\\sim}(a\\wedge b)=\\{1\\}$, which contains exactly the value of $a\\wedge b=1$, leading to a contradiction; so, to summarize, for any values $a, b\\in\\{0,1\\}$ we have $a\\wedge b\\in\\neg{\\sim}(a\\wedge b)$;\n\\item suppose that, for some values $a$ and $b$ in $\\{0,1\\}$ such that $b\\in\\neg a$, ${\\sim}(a\\wedge b)$ is not in $\\neg(a\\wedge b)$, which means $a\\wedge b=0$ (since $\\neg 1=\\{0,1\\}$); but then ${\\sim}(a\\wedge b)=1$ and $\\neg(a\\wedge b)=\\{1\\}$, what is a contradiction; so, for any values $a,b\\in\\{0,1\\}$, ${\\sim}(a\\wedge b)\\in\\neg(a\\wedge b)$.\n\\end{enumerate}\n\\end{enumerate}\n\nFinally, we can define, for $\\mathcal{L}\\in\\{\\textbf{nbIciw}, \\textbf{nbIci}, \\textbf{nbIcl}\\}$, the restricted Nmatrix \n\\[\\mathbb{2}_{\\mathcal{L}}=(\\textbf{2}_{\\textbf{nbIciw}}, \\{1\\}, \\mathcal{F}_{\\textbf{2}_{\\textbf{nbIciw}}}^{\\mathcal{L}})\\]\nsuch that $\\mathcal{F}_{\\textbf{2}_{\\textbf{nbIciw}}}^{\\mathcal{L}}$ is the set of homomorphisms $\\nu:\\textbf{F}(\\Sigma_{\\textbf{nbI}}, \\mathcal{V})\\rightarrow \\textbf{2}_{\\textbf{nbIciw}}$ satisfying that, for any formulas $\\alpha$ and $\\beta$:\n\\begin{enumerate}\n\\item $\\nu(\\alpha\\uparrow\\beta)=\\nu(\\beta\\uparrow\\alpha)$;\n\\item $\\nu(\\alpha\\uparrow\\neg\\alpha)={\\sim}(\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha))$;\n\\item\\begin{enumerate}\n\\item if $\\mathcal{L}=\\textbf{nbIci}$, $\\nu(\\neg(\\alpha\\uparrow\\neg\\alpha))=\\nu(\\alpha)\\wedge\\nu(\\neg\\alpha)$;\n\\item if $\\mathcal{L}=\\textbf{nbIcl}$, $\\nu(\\alpha\\uparrow\\neg\\alpha)=\\nu(\\neg(\\alpha\\wedge\\neg\\alpha))$.\n\\end{enumerate}\n\\end{enumerate}\nClearly such RNmatrices are structural.\n\n\\begin{theorem}\n$\\nu: F(\\Sigma_{\\textbf{nbI}},\\mathcal{V})\\rightarrow\\{0,1\\}$ is a bivaluation for $\\mathcal{L}\\in\\{\\textbf{nbIciw}, \\textbf{nbIci}, \\textbf{nbIcl}\\}$ if, and only if, it is a $\\Sigma_{\\textbf{nbI}}$-homomorphism from $\\textbf{F}(\\Sigma_{\\textbf{nbI}}, \\mathcal{V})$ to $\\textbf{2}_{\\textbf{nbIciw}}$ which lies in $\\mathcal{F}_{\\textbf{2}_{\\textbf{nbIciw}}}^{\\mathcal{L}}$.\n\\end{theorem}\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\mathcal{L}\\in\\{\\textbf{nbIciw}, \\textbf{nbIci}, \\textbf{nbIcl}\\}$, $\\Gamma\\vDash_{\\mathcal{L}}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathbb{2}_{\\mathcal{L}}}\\varphi$.\n\\end{theorem}\n\n\nOur finite RNmatrices again lead to decision methods through row-branching, row-eliminating truth tables, and for all three of the logics $\\nbIciw$, $\\nbIci$ and $\\nbIcl$. We believe the subjacent Nmatrices are explicit enough, so let us focus on the conditions for a row to be erased.\n\\begin{enumerate}\n\\item If $\\alpha\\uparrow\\beta$ and $\\beta\\uparrow\\alpha$ have different values.\n\\item If $\\alpha\\uparrow\\neg\\alpha$ or $\\neg\\alpha\\uparrow\\alpha$ is $0$, and either $\\alpha$ or $\\neg\\alpha$ is also $0$.\n\\item \\begin{enumerate}\n\\item If the logic is $\\nbIci$: if $\\neg(\\alpha\\uparrow\\neg\\alpha)$ is $1$ but either $\\alpha$ or $\\neg\\alpha$ is $0$.\n\\item If the logic is $\\nbIcl$, $\\neg(\\alpha\\wedge\\neg\\alpha)$ is $1$ and: either $\\alpha\\uparrow\\neg\\alpha$ or $\\neg\\alpha\\uparrow\\alpha$ is $0$, or $\\alpha$ and $\\neg\\alpha$ are both $1$.\n\\end{enumerate}\n\\end{enumerate}\n\n\\subsection{Another decision method}\n\nWe can add rules to the tableau calculus $\\mathbb{T}_{\\nbI}$ inspired by the tables found in Section \\ref{Decision method for nbIciw, ...} to obtain tableau calculi $\\mathbb{T}_{\\nbIciw}$, $\\mathbb{T}_{\\nbIci}$ and $\\mathbb{T}_{\\nbIcl}$ capable of characterizing their respective logics. So, consider the following tableau rules:\n\n\n$$\n\\begin{array}{cp{1cm}cp{1cm}cp{1cm}c}\n\\displaystyle \\frac{\\textsf{0}(\\varphi\\uparrow\\neg\\varphi)}{\\begin{array}{c}\\textsf{1}(\\varphi) \\\\ \\textsf{1}(\\neg\\varphi)\\end{array}} & &\n\\displaystyle \\frac{\\textsf{0}(\\neg\\varphi\\uparrow\\varphi)}{\\begin{array}{c}\\textsf{1}(\\varphi) \\\\ \\textsf{1}(\\neg\\varphi)\\end{array}} & & \n\\displaystyle \\frac{\\textsf{1}(\\neg(\\varphi\\uparrow\\neg\\varphi))}{\\begin{array}{c}\\textsf{1}(\\varphi) \\\\ \\textsf{1}(\\neg\\varphi)\\end{array}} & & \n\\displaystyle \\frac{\\textsf{1}(\\neg(\\varphi\\wedge\\neg\\varphi))}{\\textsf{0}(\\varphi)\\mid\\textsf{0}(\\neg\\varphi)}\\\\[2mm]\n\\end{array}\n$$\n\nThen, by addition to $\\mathbb{T}_{\\nbI}$ of:\n\\begin{enumerate}\n\\item the two first rules, one obtains $\\mathbb{T}_{\\nbIciw}$;\n\\item the three first rules, one obtains $\\mathbb{T}_{\\nbIci}$;\n\\item the two first rules plus the fourth, one obtains $\\mathbb{T}_{\\nbIcl}$.\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Our logics are not algebraizable by Blok and Pigozzi}\n\n\\subsection{$\\bI$ is not algebraizable by Blok and Pigozzi}\\label{bI is not algebraizable}\n\nWe wish to prove $\\bI$ is not algebraizable even by some quite expressive standards: specifically, we wish to show it is not \\index{Algebraizable}algebraizable according to Blok and Pigozzi \\cite{BlokPigozzi}. To accomplish that we will use \\index{Lewin}Lewin, Mikenberg and Schwarze's construction found in\\cite{Lewin}, and prove as they do that there is a model for $\\bI$ for which the Leibniz operator $\\Omega_{\\bI}$, that sends a $\\bI$-filter to its largest compatible congruence, is not a bijection; from Theorem $5.1$ of \\cite{BlokPigozzi}, this proves $\\bI$ is not algebraizable.\n\n\\begin{definition}\nGiven a signature $\\Sigma$ and a $\\Sigma$-algebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$, a \\index{Congruence}congruence in $\\mathcal{A}$ is a relation $\\theta$ on $A\\times A$ such that, if $\\sigma\\in\\Sigma_{n}$ and $a_{1}, b_{1}, \\dotsc  , a_{n}, b_{n}\\in A$ are elements such that $a_{1}\\theta b_{1}, \\dotsc , a_{n}\\theta b_{n}$, then\n\\[\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\theta\\sigma_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{n}).\\]\n\\end{definition}\n\n\\begin{definition}\nGiven a signature $\\Sigma$, a logic $\\mathcal{L}$ and a $\\Sigma$-algebra $\\mathcal{A}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma})$, an \\index{Filter, $\\mathcal{L}$-}$\\mathcal{L}$-filter in $\\mathcal{A}$ is a subset $F\\subseteq A$ such that \n\\[\\Gamma\\vdash_{\\mathcal{L}}\\varphi\\quad\\text{implies}\\quad \\Gamma\\vDash_{(\\mathcal{A}, F)}\\varphi,\\]\nfor every set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over $\\Sigma$.\n\\end{definition}\n\nThe \\index{Congruence, Largest compatible}largest compatible congruence $\\theta$ to a filter $F$ is the largest congruence such that, if $a\\theta b$ and $a\\in F$, then $b\\in F$.\n\nSo, over the signature $\\Sigma_{\\bI}$, for which $(\\Sigma_{\\bI})_{2}=\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$ and $(\\Sigma_{\\bI})_{n}=\\emptyset$ for every $n\\neq2$, we consider the $\\Sigma_{\\bI}$-algebra $\\mathfrak{L}$ with universe $L=\\{u, 1, a, b, 0\\}$ and operations given by the tables bellow (all, with the exception of incompatibility, extracted from \\cite{Lewin}).\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{l|ccccr}\n$\\vee$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $u$ & $u$ & $u$ & $u$ & $u$\\\\\n$1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n$a$ & $u$ & $1$ & $a$ & $1$ & $a$\\\\\n$b$ & $u$ & $1$ & $1$ & $b$ & $b$\\\\\n$0$ & $u$ & $1$ & $a$ & $b$ & $0$\n\\end{tabular}\n\\caption*{Disjunction}\n\\end{minipage}\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{l|ccccr}\n$\\wedge$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $u$ & $1$ & $a$ & $b$ & $0$\\\\\n$1$ & $1$ & $1$ & $a$ & $b$ & $0$\\\\\n$a$ & $a$ & $a$ & $a$ & $0$ & $0$\\\\\n$b$ & $b$ & $b$ & $0$ & $b$ & $0$\\\\\n$0$ & $0$ & $0$ & $0$ & $0$ & $0$\n\\end{tabular}\n\\caption*{Conjunction}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{l|ccccr}\n$\\rightarrow$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $u$ & $u$ & $a$ & $b$ & $0$\\\\\n$1$ & $u$ & $1$ & $a$ & $b$ & $0$\\\\\n$a$ & $u$ & $1$ & $1$ & $b$ & $b$\\\\\n$b$ & $u$ & $1$ & $a$ & $1$ & $a$\\\\\n$0$ & $u$ & $1$ & $1$ & $1$ & $1$\n\\end{tabular}\n\\caption*{Implication}\n\\end{minipage}\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{l|ccccr}\n$\\uparrow$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $0$ & $0$ & $0$ & $0$ & $1$\\\\\n$1$ & $0$ & $0$ & $b$ & $a$ & $1$\\\\\n$a$ & $0$ & $b$ & $b$ & $1$ & $1$\\\\\n$b$ & $0$ & $a$ & $1$ & $a$ & $1$\\\\\n$0$ & $1$ & $1$ & $1$ & $1$ & $1$\n\\end{tabular}\n\\caption*{Incompatibility}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzcd}\n & u & \\\\\n & 1 \\arrow[u] & \\\\\n a \\arrow[ru] & & b \\arrow[lu]\\\\\n & 0 \\arrow[ru]\\arrow[lu] & \\\\\n  \\end{tikzcd}\n\\caption*{The lattice $(L, \\vee, \\wedge)$}\n\\end{figure}\n\nWe then consider the logical matrix $\\mathfrak{M}=(\\mathfrak{L}, D)$, with $D=\\{u,1\\}$. \n\n\\subsubsection{$\\mathfrak{M}$ is a model of $\\bI$}\n\nNotice that $\\textbf{Ax\\: 1}$, $\\textbf{Ax\\: 3}$, $\\textbf{Ax\\: 4}$, $\\textbf{Ax\\: 5}$, $\\textbf{Ax\\: 6}$, $\\textbf{Ax\\: 7}$, $\\textbf{Ax\\: 8}$ and Modus Ponens of the definition of $\\bI$ ate the beginning of Section \\ref{Defining bI} correspond, respectively, to the axiom schemata and rules $1$, $6$, $4$, $5$, $7$, $8$, $9$ and $10$ of $\\textbf{C}_{1}$ as defined in \\cite{Lewin}; since the operations $\\vee$, $\\wedge$ and $\\rightarrow$ in $\\mathfrak{L}$ are exactly the same as the ones in the algebra of Lewin, Mikenberg and Schwarze's article, and since the logical matrix formed by that algebra and $D$ models $\\textbf{C}_{1}$, we obtain that $\\mathfrak{M}$ models at least these axiom schemata and rules of inference. It remains to be shown it also models $\\textbf{Ax\\: 2}$, $\\textbf{Ax\\: 9}^{*}$, $\\textbf{Ip}$ and $\\textbf{Comm}$.\n\n\\begin{enumerate}\n\\item Concerning $\\textbf{Ax\\: 2}$ of $\\bI$, $(\\alpha\\rightarrow(\\beta\\rightarrow \\gamma)\\big)\\rightarrow\\big((\\alpha\\rightarrow\\beta)\\rightarrow(\\alpha\\rightarrow\\gamma))$, notice that by two applications of the deduction meta-theorem we have that the validity of the axiom schema $2$ of \\cite{Lewin} in $\\bI$ is equivalent to stating that $\\alpha\\rightarrow \\beta, \\alpha\\rightarrow(\\beta\\rightarrow \\gamma)\\vdash_{\\bI}\\alpha\\rightarrow\\gamma$, which by two new applications of the deduction meta-theorem is now equivalent to the validity of $\\textbf{Ax\\: 2}$. Since the matrix of Lewin, Mikenberg and Schwarze's algebra validates the axiom schema $2$, we have $\\mathfrak{M}$ validates $\\textbf{Ax\\: 2}$.\n\n\\item For $\\textbf{Ax\\: 9}^{*}$, $(\\alpha\\rightarrow\\beta)\\vee\\alpha$, we see from the tables below that the image of $\\textbf{Ax\\: 9}^{*}$ under any homomorphism is in $D$, and therefore this axiom schema is validated by $\\mathfrak{M}$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|ccccc|ccccr}\n& \\multicolumn{5}{c|}{$x\\rightarrow y$} & \\multicolumn{5}{c}{$(x\\rightarrow y)\\vee x$}\\\\ \\hline\n\\diag{.1em}{0.2cm}{$x$}{$y$} & $u$ & $1$ & $a$ & $b$ & $0$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $u$ & $u$ & $a$ & $b$ & $0$ & $u$ & $u$ & $u$ & $u$ & $u$\\\\\n$1$ & $u$ & $1$ & $a$ & $b$ & $0$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n$a$ & $u$ & $1$ & $1$ & $b$ & $b$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n$b$ & $u$ & $1$ & $a$ & $1$ & $a$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n$0$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\n\\end{tabular}\n\\caption*{Table for $\\textbf{Ax\\: 9}^{*}$}\n\\end{figure}\n\n\n\n\\item To prove $\\mathfrak{M}$ models $\\textbf{Ip}$, given by $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\gamma))$, we begin with the two tables below.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{l|ccccr}\n\\diag{.1em}{0.2cm}{$y$}{$z$} & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $u$ & $u$ & $a$ & $b$ & $0$\\\\\n$1$ & $u$ & $1$ & $a$ & $b$ & $0$\\\\\n$a$ & $u$ & $1$ & $1$ & $b$ & $b$\\\\\n$b$ & $u$ & $1$ & $a$ & $1$ & $a$\\\\\n$0$ & $u$ & $1$ & $1$ & $1$ & $1$\n\\end{tabular}\n\\caption*{Table for $y\\rightarrow z$}\n\\end{minipage}\n\\centering\n\\begin{minipage}[t]{5cm}\n\\centering\n\\begin{tabular}{l|ccccr}\n\\diag{.1em}{0.2cm}{$x$}{$y$} & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $0$ & $0$ & $0$ & $0$ & $1$\\\\\n$1$ & $0$ & $0$ & $b$ & $a$ & $1$\\\\\n$a$ & $0$ & $b$ & $b$ & $1$ & $1$\\\\\n$b$ & $0$ & $a$ & $1$ & $a$ & $1$\\\\\n$0$ & $1$ & $1$ & $1$ & $1$ & $1$\n\\end{tabular}\n\\caption*{Table for $x\\uparrow y$}\n\\end{minipage}\n\\end{figure}\n\nBy looking at the table for $\\textbf{Ip}$ below, we see that, since the image of $\\textbf{Ip}$ under any homomorphism is always in $D$, we have $\\mathfrak{M}$ validates this axiom schema.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|c|c|ccccc|C{1.2em}C{1.2em}C{1.2em}C{1.2em}C{1.2em}}\n& & $x\\uparrow y$ & \\multicolumn{5}{c|}{$x\\rightarrow(y\\rightarrow z)$} & \\multicolumn{5}{c}{$(x\\uparrow y)\\rightarrow(x\\rightarrow(y\\rightarrow z))$}\\\\ \\hline\n$x$ & \\diag{.1em}{0.2cm}{$y$}{$z$} & & $u$ & $1$ & $a$ & $b$ & $0$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\ \\hline\n\\multirow{5}{*}{$u$} & $u$ & $0$ & $u$ & $u$ & $a$ & $b$ & $0$ & $u$ & $u$ & $1$ & $1$ & $1$\\\\\n& $1$ & $0$ & $u$ & $u$ & $a$ & $b$ & $0$ & $u$ & $u$ & $1$ & $1$ & $1$\\\\\n& $a$ & $0$ & $u$ & $u$ & $u$ & $b$ & $b$ & $u$ & $u$ & $u$ & $1$ & $1$\\\\\n& $b$ & $0$ & $u$ & $u$ & $a$ & $u$ & $a$ & $u$ & $u$ & $1$ & $u$ & $1$\\\\\n& $0$ & $1$ & $u$ & $u$ & $u$ & $u$ & $u$ & $u$ & $u$ & $u$ & $u$ & $u$\\\\ \\hline\n\\multirow{5}{*}{$1$} & $u$ & $0$ & $u$ & $u$ & $a$ & $b$ & $0$ & $u$ & $u$ & $1$ & $1$ & $1$\\\\\n& $1$ & $0$ & $u$ & $1$ & $a$ & $b$ & $0$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $a$ & $b$ & $u$ & $1$ & $1$ & $b$ & $b$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $b$ & $a$ & $u$ & $1$ & $a$ & $1$ & $a$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $0$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\ \\hline\n\\multirow{5}{*}{$a$} & $u$ & $0$ & $u$ & $u$ & $1$ & $b$ & $b$ & $u$ & $u$ & $1$ & $1$ & $1$\\\\\n& $1$ & $b$ & $u$ & $1$ & $1$ & $b$ & $b$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $a$ & $b$ & $u$ & $1$ & $1$ & $b$ & $b$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $b$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $0$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\ \\hline\n\\multirow{5}{*}{$b$} & $u$ & $0$ & $u$ & $u$ & $a$ & $1$ & $a$ & $u$ & $u$ & $1$ & $1$ & $1$\\\\\n& $1$ & $a$ & $u$ & $1$ & $a$ & $1$ & $a$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $a$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $b$ & $a$ & $u$ & $1$ & $a$ & $1$ & $a$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $0$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\ \\hline\n\\multirow{5}{*}{$0$} & $u$ & $1$ & $u$ & $u$ & $1$ & $1$ & $1$ & $u$ & $u$ & $1$ & $1$ & $1$\\\\\n& $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $a$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $b$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n& $0$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$ & $u$ & $1$ & $1$ & $1$ & $1$\\\\\n\\end{tabular}\n\\caption*{Table for $\\textbf{Ip}$}\n\\end{figure}\n\n\\item It only remains to show $\\mathfrak{M}$ validates $\\textbf{Comm}$, an axiom schema given by $(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha)$, what is done on the following table; notice that the tables for $x\\uparrow y$ and $y\\uparrow x$ are the same since incompatibility is commutative in $\\mathfrak{M}$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|ccccc|ccccr}\n& \\multicolumn{5}{c|}{$x\\uparrow y=y\\uparrow x$} & \\multicolumn{5}{c}{$(x\\uparrow y)\\rightarrow(y\\uparrow x)$}\\\\\\hline\n\\diag{.1em}{0.2cm}{$x$}{$y$} & $u$ & $1$ & $a$ & $b$ & $0$ & $u$ & $1$ & $a$ & $b$ & $0$ \\\\\\hline\n$u$ & $0$ & $0$ & $0$ & $0$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$\\\\\n$1$ & $0$ & $0$ & $b$ & $a$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$\\\\\n$a$ & $0$ & $b$ & $b$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$\\\\\n$b$ & $0$ & $a$ & $1$ & $a$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$\\\\\n$0$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$\n\\end{tabular}\n\\caption*{Table for $\\textbf{Comm}$}\n\\end{figure}\n\n\\end{enumerate}\n\nWith this, we have proved that $\\mathfrak{M}$ is a model for $\\bI$, meaning that, for any formulas $\\Gamma\\cup\\{\\varphi\\}$ on the signature $\\Sigma_{\\bI}$, $\\Gamma\\vdash_{\\bI}\\varphi$ implies $\\Gamma\\vDash_{\\mathfrak{M}}\\varphi$.\n\n\\subsubsection{There are no non-trivial congruences on $\\mathfrak{L}$}\n\nNow, we wish to prove that the $\\Sigma_{\\bI}$-algebra $\\mathfrak{L}$ has only two congruences, the ones we call trivial: \\label{nabla}$\\nabla$, equal to $L\\times L$, and \\label{Delta}$\\Delta$, given by $\\{(x,x)\\ :\\  x\\in L\\}$. Here, we will perpetrate an abuse of notation: take a signature $\\Sigma$, a $\\Sigma$-algebra $\\mathcal{A}$ with universe $A$, a congruence $\\theta$ on $\\mathcal{A}$, $\\sigma\\in\\Sigma_{n}$ and $\\omega\\in\\Sigma_{m}$, and $a_{1}, \\dotsc  , a_{n}, b_{1}, \\dotsc  , b_{m}\\in A$: then, if $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})=a$ and $\\omega_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{m})=b$, and $\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\theta\\omega_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{m})$, we may simply write $a=\\sigma_{\\mathcal{A}}(a_{1}, \\dotsc  , a_{n})\\theta\\omega_{\\mathcal{A}}(b_{1}, \\dotsc  , b_{m})=b$ to make it clear that $a\\theta b$.\n\n\\begin{enumerate}\n\\item \n\\begin{enumerate}\n\\item If $u\\theta 1$, $0=(u\\uparrow a)\\theta(1\\uparrow a)=b$, $0=(u\\uparrow b)\\theta(1\\uparrow b)=a$ and $1=(a\\rightarrow a)\\theta (b\\rightarrow a)=a$, and therefore $\\theta=\\nabla$.\n\\item If $u\\theta a$, then $u=(u\\vee 1)\\theta (a\\vee 1)=1$ (meaning $a\\theta 1$), $b=(u\\wedge b)\\theta (a\\wedge b)=0$ and $0=(u\\uparrow b)\\theta(1\\uparrow b)=a$, implying $\\theta=\\nabla$.\n\\item If $u\\theta b$, then $u=(u\\vee 1)\\theta(b\\vee 1)=1$, $a=(u\\wedge a)\\theta(b\\wedge a)=0$ and $0=(u\\uparrow a)\\theta(1\\uparrow a)=b$, and again $\\theta=\\nabla$.\n\\item If $u\\theta 0$, then for every $x$ we have $u=(u\\vee x)\\theta (0\\vee x)=x$, meaning $\\theta=\\nabla$.\n\\end{enumerate}\nWith this, we have proved that, if any pair $(u,x)$, with $x\\in L\\setminus\\{u\\}$, is in $\\theta$, then $\\theta=\\nabla$.\n\\item We will ignore the case $1\\theta u$, since it is equivalent to $u\\theta 1$, a case considered before.\n\\begin{enumerate}\n\\item If $1\\theta a$, $b=(1\\wedge b)\\theta (a\\wedge b)=0$, $u=(u\\rightarrow 1)\\theta(u\\rightarrow a)=a$ (and therefore $u\\theta 1$) and $0=(u\\uparrow b)\\theta(1\\uparrow b)=a$, meaning $\\theta=\\nabla$.\n\\item If $1\\theta b$, $a=(1\\wedge a)\\theta(b\\wedge a)=0$, $u=(u\\rightarrow1)\\theta(u\\rightarrow b)=b$ (and therefore $u\\theta 1$) and $0=(u\\uparrow a)\\theta(1\\uparrow a)=b$, and so $\\theta=\\nabla$.\n\\item If $1\\theta 0$, $a=(1\\rightarrow a)\\theta(0\\rightarrow a)=1$, $b=(1\\rightarrow b)\\theta(0\\rightarrow b)=1$ and $u=(u\\rightarrow 1)\\theta (u\\rightarrow 0)=0$, and therefore $\\theta=\\nabla$.\n\\end{enumerate}\nOnce again, we have that if any pair $(1,x)$, with $x\\in L\\setminus\\{1\\}$, is in $\\theta$, then $\\theta=\\nabla$.\n\\item The cases $a\\theta u$ and $a\\theta 1$ correspond to the cases $u\\theta a$ and $1\\theta a$ above.\n\\begin{enumerate}\n\\item If $a\\theta 0$, $1=(a\\vee b)\\theta(0\\vee b)=b$, $u=(u\\rightarrow1)\\theta(u\\rightarrow b)=b$ (and therefore $u\\theta 1$) and $0=(u\\uparrow a)\\theta(1\\uparrow a)=b$, what implies $\\theta=\\nabla$.\n\\item If $a\\theta b$, $b=(a\\rightarrow b)\\theta (b\\rightarrow b)=1$ (implying $a\\theta1$), $0=(a\\wedge b)\\theta(b\\wedge b)=b$ and $u=(u\\rightarrow 1)\\theta(u\\rightarrow a)=a$, what means $\\theta=\\nabla$.\n\\end{enumerate}\nSo, if $(a,x)\\in\\theta$, for $x\\in L\\setminus\\{a\\}$, $\\theta=\\nabla$.\n\\item The cases $b\\theta u$, $b\\theta 1$ and $b\\theta a$ equal the previous cases, respectively, $u\\theta b$, $1\\theta b$ and $a\\theta b$. \n\\begin{enumerate}\n\\item If $b\\theta 0$, $1=(a\\vee b)\\theta(a\\vee 0)=a$, $u=(u\\rightarrow 1)\\theta(u\\rightarrow a)=a$ (and therefore $u\\theta 1$) and $0=(u\\uparrow b)\\theta(1\\uparrow b)=a$, and therefore $\\theta=\\nabla$.\n\\end{enumerate}\nAgain, if $(b,x)\\in \\theta$, for $b\\in L\\setminus\\{b\\}$, then $\\theta=\\nabla$.\n\\item Since the cases $0\\theta u$, $0\\theta1$, $0\\theta a$ and $0\\theta b$ correspond, respectively, to $u\\theta 0$, $1\\theta0$, $a\\theta0$ and $b\\theta0$, we have that, if $(0,x)\\in\\theta$, for $x\\in L\\setminus\\{0\\}$, then $\\theta=\\nabla$.\n\\end{enumerate}\n\nWith all of this, we discover that for any congruence $\\theta$ in $\\mathfrak{L}$, if $(x,y)\\in \\theta$, for $x\\neq y$, then $\\theta=\\nabla$, implying as we had mentioned that the only two congruences in $\\mathfrak{L}$ are $\\nabla$ and $\\Delta$.\n\n\\subsubsection{Two filters whose largest compatible congruence is $\\nabla$}\n\n\\begin{lemma}\\label{Classifying filters}\nGiven a signature $\\Sigma$, a $\\Sigma$-algebra $\\mathcal{A}$ and a logic $\\mathcal{L}$ over $\\Sigma$, $F$ is a $\\mathcal{L}$-filter in $\\mathcal{A}$ if, and only if, the following are satisfied:\n\\begin{enumerate}\n\\item for every $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ and instance of axiom $\\psi$ of $\\mathcal{L}$, $\\sigma(\\psi)\\in F$;\n\\item for every $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ and instance of a rule of inference\\\\ $\\psi_{1}, \\dotsc  , \\psi_{n}|\\psi$ of $\\mathcal{L}$, if $\\sigma(\\psi_{1}), \\dotsc  , \\sigma(\\psi_{n})\\in F$ then $\\sigma(\\psi)\\in F$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose $F$ is a $\\mathcal{L}$-filter in $\\mathcal{A}$ and $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ is a $\\Sigma$-homomorphism.\n\\begin{enumerate}\n\\item If $\\psi$ is an instance of an axiom, $\\vdash_{\\mathcal{L}}\\psi$, meaning $\\vDash_{(\\mathcal{A}, F)}\\psi$ and, therefore, $\\sigma(\\psi)\\in F$.\n\\item If $\\psi_{1}, \\dotsc  , \\psi_{n}|\\psi$ is an instance of an inference rule, $\\psi_{1}, \\dotsc  , \\psi_{n}\\vdash_{\\mathcal{L}}\\psi$, meaning $\\psi_{1}, \\dotsc  , \\psi_{n}$\\\\$\\vDash_{(\\mathcal{A}, F)}\\psi$ and, therefore, if $\\sigma(\\psi_{1}), \\dotsc  , \\sigma(\\psi_{n})\\in F$, then $\\sigma(\\psi)\\in F$.\n\\end{enumerate}\n\nReciprocally, suppose $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ and that $\\alpha_{1}, \\dotsc  , \\alpha_{n}$ is a proof of $\\varphi$ from $\\Gamma$ with $\\alpha_{n}=\\varphi$; we shall prove that, for any $\\Sigma$-homomorphism $\\sigma:\\textbf{F}(\\Sigma, \\mathcal{V})\\rightarrow\\mathcal{A}$ such that $\\sigma(\\Gamma)\\subseteq F$, $\\sigma(\\alpha_{1}), \\dotsc  , \\sigma(\\alpha_{n})\\in F$, and therefore $\\sigma(\\varphi)\\in F$ and $\\Gamma\\vDash_{(\\mathcal{A}, F)}\\varphi$.\n\nFor induction hypothesis, being $\\alpha_{1}$ either an element of $\\Gamma$ or an instance of an axiom, assume $\\alpha_{1}$ through $\\alpha_{i-1}$ are mapped, by $\\sigma$, into $F$.\n\\begin{enumerate}\n\\item If $\\alpha_{i}$ is in $\\Gamma$, by hypothesis on $\\sigma$ we have $\\sigma(\\alpha_{i})\\in F$.\n\\item If $\\alpha_{i}$ is an instance of an axiom, by our hypothesis on $F$ we have $\\sigma(\\alpha_{i})\\in F$.\n\\item Finally, if there are $\\alpha_{i_{1}}, \\dotsc  , \\alpha_{i_{n}}$, with $i_{1}, \\dotsc  , i_{n}\\in\\{1, \\dotsc  , i-1\\}$, such that $\\alpha_{i_{1}}, \\dotsc  , \\alpha_{i_{n}}|\\alpha_{i}$ is an instance of a rule of inference, by induction hypothesis $\\sigma(\\alpha_{i_{1}}), \\dotsc  , \\sigma(\\alpha_{i_{n}})\\in F$, and again by our hypothesis on $F$ we have $\\sigma(\\alpha_{i})\\in F$.\n\\end{enumerate}\n\\end{proof}\n\nSo, we take the subsets $F_{a}=\\{u, 1, a\\}$ and $F_{b}=\\{u, 1, b\\}$ of $L$, and state that both are $\\bI$-filters. First of all, for any $\\Sigma_{\\bI}$-homomorphism $\\sigma:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\mathfrak{L}$ and any instance of an axiom schema $\\psi$ of $\\bI$, since $\\mathfrak{M}=(\\mathfrak{L}, D)$ models $\\bI$, $\\sigma(\\psi)\\in D=\\{u,1\\}\\subseteq F_{a}$ and in much the same way $\\sigma(\\psi)\\in F_{b}$, implying that both $F_{a}$ and $F_{b}$ satisfy the first condition of Lemma \\ref{Classifying filters} for being an $\\bI$-filter.\n\nFurthermore, there is only one rule of inference to analyze, that of Modus Ponens. From the table for implication, we see that if $x\\rightarrow y$ is in $F_{a}$, then either $y\\in F_{a}$ or $x\\in L\\setminus F_{a}$; so, if both $x$ and $x\\rightarrow y$ are in $F_{a}$, then $y\\in F_{a}$. Similarly, if both $x$ and $x\\rightarrow y$ are in $F_{b}$, $y$ is also in $F_{b}$, what implies that both $F_{a}$ and $F_{b}$ are $\\bI$-filters by Lemma \\ref{Classifying filters}.\n\nBut we state that the largest compatible congruence to both $F_{a}$ and $F_{b}$ is $\\Delta$, meaning the Leibniz operator $\\Omega_{\\bI}$ of $\\bI$ is not injective and therefore $\\bI$ is not algebraizable by Blok and Pigozzi. This is easy to prove: $\\nabla$ is not compatible to neither $F_{a}$ nor $F_{b}$, since $u\\Delta 0$ and $u\\in F_{a}\\cap F_{b}$, but $0$ is not in $F_{a}$ nor in $F_{b}$; clearly $\\Delta$ is compatible to both $F_{a}$ and $F_{b}$, and since there are no congruences larger than $\\Delta$ different from $\\nabla$, we obtain the aforementioned result.\n\n\\subsection{$\\nbI$ is not algebraizable by Blok and Pigozzi}\n\nNow, we will follow the reasoning found in Section \\ref{bI is not algebraizable} to show that $\\nbI$ is also not algebraizable by Blok And Pigozzi, showing that its Leibniz operator is not bijective, the only difference being that we must add a negation to $\\mathfrak{L}$. So, consider the $\\Sigma_{\\nbI}$-algebra $\\mathfrak{L}_{\\nbI}$ with universe $L=\\{u,1,a,b,0\\}$, $\\sigma_{\\mathfrak{L}_{\\nbI}}$ equal to $\\sigma_{\\mathfrak{L}}$ for any $\\sigma\\in\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$ and negation defined by the table below; we drop $\\mathfrak{L}_{\\nbI}$ from indexing the operations for simplicity.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|ccccr}\n$x$ & $u$ & $1$ & $a$ & $b$ & $0$\\\\ \\hline\n$\\neg x$ & $1$ & $0$ & $b$ & $a$ & $1$\n\\end{tabular}\n\\caption*{Table for Negation}\n\\end{figure}\n\nWe then define the logical matrix $\\mathfrak{M}_{\\nbI}=(\\mathfrak{L}_{\\nbI}, D)$, with $D=\\{u,1\\}$. Since the operations associated to the symbols in $\\{\\vee, \\wedge, \\rightarrow, \\uparrow\\}$ are the same as those of $\\mathfrak{L}$, we see that $\\mathfrak{M}_{\\nbI}$ models the axiom schemata $\\textbf{Ax\\: 1}$ through $\\textbf{Ax\\: 8}$ of $\\bI$, plus $\\textbf{Ax\\: 9}^{*}$, $\\textbf{Ip}$ and $\\textbf{Comm}$, since those axiom schemata involve only the connectives on $\\Sigma_{\\bI}$.\n\nThe proof that $\\mathfrak{M}_{\\nbI}$ models $\\textbf{Ax\\: 11}^{*}$ is not necessary since it corresponds to the axiom scheme number $3$ for $\\textbf{C}_{1}$ in the axiomatization found in \\cite{Lewin}, and we use the same negation.\n\nThe proof that there are only two congruences on $\\mathfrak{L}_{\\nbI}$ goes as expected, beginning with the supposition that a pair $(x,y)$, with $x\\neq y$, is in $\\theta$, what then implies $\\theta=\\nabla$.\n\nFinally, we trivially find both $F_{a}=\\{u,1,a\\}$ and $F_{b}=\\{u,1,b\\}$ are $\\nbI$-filters whose largest compatible congruence is $\\Delta$, what proves $\\nbI$ is not-algebraizable according to Blok and Pigozzi.\n\n\\subsection{$\\nbIciw$, $\\nbIci$ and $\\nbIcl$ are not algebraizable by Blok and Pigozzi}\n\nWe state that $\\mathfrak{M}_{\\nbI}$ also models $\\nbIciw$, $\\nbIci$ and $\\nbIcl$, and since $F_{a}$ and $F_{b}$ are still, respectively, $\\mathcal{L}$-filters, for $\\mathcal{L}\\in\\{\\nbIciw, \\nbIci, \\nbIcl\\}$, we prove that none of these three is algebraizable by Blok and Pigozzi. The proof that $\\mathfrak{M}$ models $\\textbf{ciw}^{*}$, $\\textbf{ci}^{*}$ and $\\textbf{cl}^{*}$ is in the following tables.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|cccc}\n$x$ & $\\neg x$ & $x\\wedge\\neg x$ & $x\\uparrow\\neg x$ & $(x\\uparrow\\neg x)\\vee(x\\wedge\\neg x)$ \\\\\\hline\n$u$ &   $1$ &              $1$ &                     $0$ &                                     $1$ \\\\\n$1$ &   $0$ &              $0$ &                     $1$ &                                     $1$ \\\\\n$a$ &   $b$ &              $0$ &                     $1$ &                                     $1$ \\\\\n$b$ &   $a$ &              $0$ &                     $1$ &                                     $1$ \\\\\n$0$ &   $1$ &              $0$ &                     $1$ &                                     $1$  \n\\end{tabular}\n\\caption*{Table for $\\textbf{ciw}^{*}$}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|ccccc}\n$x$ & $\\neg x$ & $x\\wedge\\neg x$ & $x\\uparrow\\neg x$ & $\\neg(x\\uparrow\\neg x)$ & $\\neg(x\\uparrow\\neg x)\\rightarrow(x\\wedge\\neg x)$ \\\\\\hline\n$u$ &   $1$ &              $1$ &                     $0$ &                               $1$ &                                                      $1$ \\\\\n$1$ &   $0$ &              $0$ &                     $1$ &                               $0$ &                                                      $1$ \\\\\n$a$ &   $b$ &              $0$ &                     $1$ &                               $0$ &                                                      $1$\\\\\n$b$ &   $a$ &              $0$ &                     $1$ &                               $0$ &                                                      $1$\\\\\n$0$ &   $1$ &              $0$ &                     $1$ &                               $0$  &                                                      $1$\n\\end{tabular}\n\\caption*{Table for $\\textbf{ci}^{*}$}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{l|ccccc}\n$x$ & $\\neg x$ & $x\\wedge\\neg x$ & $x\\uparrow\\neg x$ & $\\neg(x\\wedge\\neg x)$ & $\\neg(x\\wedge\\neg x)\\rightarrow(x\\uparrow\\neg x)$ \\\\\\hline\n$u$ &   $1$ &              $1$ &                     $0$ &                               $0$ &                                                      $1$ \\\\\n$1$ &   $0$ &              $0$ &                     $1$ &                               $1$ &                                                      $1$ \\\\\n$a$ &   $b$ &              $0$ &                     $1$ &                               $1$ &                                                      $1$\\\\\n$b$ &   $a$ &              $0$ &                     $1$ &                               $1$ &                                                      $1$\\\\\n$0$ &   $1$ &              $0$ &                     $1$ &                               $1$  &                                                      $1$\n\\end{tabular}\n\\caption*{Table for $\\textbf{cl}^{*}$}\n\\end{figure}\n\n\n\n\n\n\\section{$\\bI$ and $\\nbI$ are not characterizable by finite Nmatrices}\\label{bI is not characterized by Nmatrices section}\n\nIt is well known that da Costa's hierarchy is both not algebraizable according to Blok and Pigozzi (\\cite{Lewin}, \\cite{Mortensen80}) and not characterizable by a finite Nmatrix (\\cite{Avron}). We have already shown that $\\bI$, and the other systems of incompatibility we have here defined, are not algebraizable according to Blok and Pigozzi, giving them a difficulty to be approached close to that of the logics $C_{n}$; but here, we show furthermore that $\\bI$ and $\\nbI$ are not characterizable by finite Nmatrices, and are, therefore, specially hard systems from a semantical standpoint. Interestingly, most proofs found so far in this chapter, such as those for the facts that bivaluations and Fidel structures both characterize our logics, or that these systems are not algebraizable according to Blok and Pigozzi, are incredibly similar to the proofs of these results for logics of formal inconsistency, the only difference being that some amount of care must be taking while dealing with incompatibility; however, the proofs in this section require formulas $\\phi_{ij}$ specially composed to prove the non-characterizability (by either finite Nmatrices or finite Rmatrices) of systems of incompatibility, and are therefore intrinsically different from any demonstrations produced on the field of $\\textbf{LFI}$'s.\n\n\nWe start by supposing there exists an Nmatrix $\\mathcal{M}=(\\mathcal{A}, D)$ that characterizes $\\bI$, with $A$ the universe of $\\mathcal{A}$ and $U=A\\setminus D$ the set of undesignated elements; we also remember we have defined $\\bot_{\\alpha\\beta}$ as $\\alpha\\wedge(\\beta\\wedge(\\alpha\\uparrow\\beta))$ and ${\\sim}\\alpha$ as $\\alpha\\rightarrow\\bot_{\\alpha\\alpha}$, since both of these will play an important role in the proof to come. For simplicity, we will drop the indexes from the operations on $\\mathcal{A}$ and use the infix notation.\n\n\\begin{lemma}\\label{basic facts about Nmatrix semantics}\nSuppose $d_{1}, d_{2}\\in D$ and $u_{1}, u_{2}\\in U$:\n\\begin{enumerate}\n\\item $d_{1}\\wedge d_{2}\\subseteq D$, while $d_{1}\\wedge u_{1}$, $u_{1}\\wedge d_{1}$ and $u_{1}\\wedge u_{2}$ are subsets of $U$;\n\\item$d_{1}\\rightarrow d_{2}$, $u_{1}\\rightarrow d_{1}$ and $u_{1}\\rightarrow u_{2}$ are subsets of $D$, while $d_{1}\\rightarrow u_{1}\\subseteq U$;\n\\item $d_{1}\\vee d_{2}$, $d_{1}\\vee u_{1}$ and $u_{1}\\vee d_{1}$ are subsets of $D$, while $u_{1}\\vee u_{2}\\subseteq U$;\n\\item for any formulas $\\alpha$ and $\\beta$ and valuation $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow \\mathcal{A}$ for $\\mathcal{A}$, $\\nu(\\bot_{\\alpha\\beta})\\in U$;\n\\item for any formula $\\alpha$ and valuation $\\nu$, if $\\nu(\\alpha)\\in D$, then $\\nu({\\sim}\\alpha)\\in U$; and if $\\nu(\\alpha)\\in U$, then $\\nu({\\sim}\\alpha)\\in D$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}\n\\item Given variables $p$ and $q$, from $\\textbf{Ax\\: 4}$ and $\\textbf{Ax\\: 5}$ one gets $p\\wedge q\\vdash_{\\bI}p$ and $p\\wedge q\\vdash_{\\bI}q$; given a valuation $\\nu$ for $\\mathcal{A}$, if $\\nu(p)\\in U$ or $\\nu(q)\\in U$, then $\\nu(p\\wedge q)\\in U$, and therefore $\\nu(p)\\wedge\\nu(q)\\subseteq U$: this is the case since, otherwise, one could define a valuation $\\nu^{*}: \\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow \\mathcal{A}$ such that $\\nu^{*}(p)=\\nu(p)$, $\\nu^{*}(q)=\\nu(q)$ but $\\nu^{*}(p\\wedge q)\\in D$, meaning that either $p\\wedge q\\vdash_{\\bI}p$ or $p\\wedge q\\vdash_{\\bI}q$ is not validated by $\\mathcal{M}$.\n\nFrom $\\textbf{Ax\\: 3}$ and the deduction meta-theorem, we obtain $p, q\\vdash_{\\bI}p\\wedge q$, meaning that if $\\nu(p), \\nu(q)\\in D$, then $\\nu(p\\wedge q)\\in D$, and so $\\nu(p)\\wedge\\nu(q)\\subseteq D$.\n\n\\item We have that, for variables $p$ and $q$, $q\\rightarrow (p\\rightarrow q)$ is an instance of an axiom schema of $\\bI$, and therefore $q\\vdash_{\\bI}p\\rightarrow q$; taking a homomorphism $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow \\mathcal{A}$ such that $\\nu(q)\\in D$, we must have $\\nu(p\\rightarrow q)\\in D$ and therefore $\\nu(p)\\rightarrow \\nu(q)\\subseteq D$. This, of course, corresponds to both $d_{1}\\rightarrow d_{2}$ and $u_{1}\\rightarrow d_{1}$ being subsets of $D$.\n\nNow, $p, p\\rightarrow q\\vdash_{\\bI} q$, and therefore if $\\nu(q)\\in U$, we must have either $\\nu(p)\\in U$ or $\\nu(p\\rightarrow q)\\in U$; so, if $\\nu(q)\\in U$ and $\\nu(p)\\in D$, one must necessarily have $\\nu(p\\rightarrow q)\\in U$, meaning $\\nu(p)\\rightarrow\\nu(q)\\subseteq U$, corresponding to $d_{1}\\rightarrow u_{1}\\subseteq U$. \n\nFinally, suppose $\\nu(p), \\nu(q)\\in U$ and, without loss of generality, $\\nu(r)\\in D$; from $\\textbf{Ax\\: 2}$ and the deduction meta-theorem, $p\\rightarrow (q\\rightarrow r)\\vdash_{\\bI}(p\\rightarrow q)\\rightarrow(p\\rightarrow r)$ and, from the previous investigations, one finds that $\\nu(q\\rightarrow r)\\in D$ and $\\nu(p\\rightarrow(q\\rightarrow r))\\in D$, meaning $\\nu((p\\rightarrow q)\\rightarrow(p\\rightarrow r))\\in D$. Since $\\nu(p\\rightarrow r)\\in D$, one necessarily obtains that $\\nu(p\\rightarrow q)\\in D$, forcibly implying that $\\nu(p)\\rightarrow\\nu(q)\\subseteq D$, what corresponds to $u_{1}\\rightarrow u_{2}\\subseteq D$.\n\n\\item Given variables $p$ and $q$, we have that $p\\vdash_{\\bI}p\\vee q$ and $q\\vdash_{\\bI}p\\vee q$ from axiom schemata $\\textbf{Ax\\: 6}$ and $\\textbf{Ax\\: 7}$ and the deduction meta-theorem. So, if $\\nu(p)\\in D$ or $\\nu(q)\\in D$, one necessarily finds $\\nu(p\\vee q)\\in D$, and therefore $\\nu(p)\\rightarrow\\nu(q)\\subseteq D$.\n\nNow, from axiom schema $\\textbf{Ax\\: 8}$ and the deduction meta-theorem, $p\\rightarrow r, q\\rightarrow r\\vdash_{\\bI}(p\\vee q)\\rightarrow r$; if $\\nu(p)\\in U$ and $\\nu(q)\\in U$, suppose, without loss of generality, that $\\nu(r)\\in U$; this means $\\nu(p\\rightarrow r)$ and $\\nu(q\\rightarrow r)$ are both in $D$, and so must be $\\nu((p\\vee q)\\rightarrow r)$, what only happens if $\\nu(p\\vee q)\\in U$. This, of course, implies that $\\nu(p)\\vee\\nu(q)\\subseteq U$.\n\n\\item Let $p$, $q$ and $r$ be propositional variables: from $\\textbf{Ip}$ and the deduction meta-theorem one finds that $p, q, p\\uparrow q\\vdash_{\\bI}r$, yet $p, q\\not\\vdash_{\\bI}r$.\\footnote{To see that, take a bivaluation $\\nu$ for $\\bI$ with $\\nu(p)=\\nu(q)=1$ and $\\nu(r)=0$} If we suppose $\\nu$ is a valuation for $\\mathcal{A}$ such that $\\nu(p), \\nu(q)\\in D$ and $\\nu(r)\\in U$, it becomes clear that one must have $\\nu(p\\uparrow q)\\in U$, since otherwise one would forcibly have $\\nu(r)\\in D$. This means we can never have all three $\\nu(\\alpha)$, $\\nu(\\beta)$ and $\\nu(\\alpha\\uparrow\\beta)$ in $D$, and therefore \n\\[\\nu(\\bot_{pq})=(\\nu(p)\\wedge\\nu(q))\\wedge\\nu(p\\uparrow q)\\in U.\\]\nOf course, for arbitrary formulas $\\alpha$ and $\\beta$, $\\bot_{\\alpha\\beta}\\vdash_{\\bI}\\bot_{pq}$ and $\\bot_{pq}\\vdash_{\\bI}\\bot_{\\alpha\\beta}$, meaning that, for any $\\nu$, $\\nu(\\bot_{\\alpha\\beta})\\in U$.\n\n\\item If $\\nu(\\alpha)\\in D$, since $\\nu(\\bot_{\\alpha\\alpha})\\in U$ we obtain $\\nu({\\sim}\\alpha)=\\nu(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\in U$. Reciprocally, if $\\nu(\\alpha)\\in U$, $\\nu({\\sim}\\alpha)=\\nu(\\alpha\\rightarrow\\bot_{\\alpha\\alpha})\\in D$.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\\begin{lemma}\n\\begin{enumerate}\n\\item For any two elements $a, b\\in A$, either $a\\uparrow b\\subseteq D$ or $a\\uparrow b\\subseteq U$;\n\\item for any two elements $a, b\\in A$, either both $a\\uparrow b$ and $b\\uparrow a$ are subsets of $D$, or both are subsets of $U$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose that there are values $d, u\\in a\\uparrow b$ such that $d\\in D$ and $u\\in U$, and let $p$ and $q$ be propositional variables throughout the proof. Since $b\\uparrow a$ is necessarily not empty, it must contain either an element $d^{*}\\in D$, or an $u^{*}\\in U$.\n\\begin{enumerate}\n\\item In the first case, take a valuation $\\nu:\\textbf{F}(\\Sigma_{\\bI}, \\mathcal{V})\\rightarrow\\mathcal{A}$ satisfying $\\nu(p)=a$, $\\nu(q)=b$, $\\nu(p\\uparrow q)=u$ and $\\nu(q\\uparrow p)=d^{*}$, and then we have that $\\nu((q\\uparrow p)\\rightarrow(p\\uparrow q))\\in d^{*}\\rightarrow u$, which by Lemma \\ref{basic facts about Nmatrix semantics} is contained in $U$. \n\nThis shows $\\nu$ does not validate $\\textbf{Comm}$, what is absurd given that $\\mathcal{M}$ characterizes $\\bI$.\n\\item In the second case, take now a valuation $\\nu$ with $\\nu(p)=a$, $\\nu(q)=b$, $\\nu(p\\uparrow q)=d$ and $\\nu(q\\uparrow p)=u^{*}$. Then $\\nu((p\\uparrow q)\\rightarrow(q\\uparrow p))\\in d\\rightarrow u^{*}$, again contained in $U$ according to Lemma \\ref{basic facts about Nmatrix semantics}.\n\\end{enumerate}\n\nEither way we reach a contradiction, an the conclusion must be that either $a\\uparrow b$ is contained in $D$, or it is contained in $U$.\n\nFinally, suppose that there are values $a, b\\in A$ with $a\\uparrow b\\subseteq D$ and $b\\uparrow a\\subseteq U$, and then, for any valuation $\\nu$ with $\\nu(p)=a$ and $\\nu(q)=b$ (and there are many of them), one necessarily finds that $\\nu(p\\uparrow q)\\in D$ and $\\nu(q\\uparrow p)\\in U$, and from Lemma \\ref{basic facts about Nmatrix semantics} we have $\\nu((p\\uparrow q)\\rightarrow(q\\uparrow p))\\in U$; this again contradicts the fact that $\\mathcal{M}$ characterizes $\\bI$, since it implies that $\\nu$ does not model $\\textbf{Comm}$.\n\\end{proof}\n\nThe following fact is trivial, but important in the following discussion so we make a point of proving it: suppose $\\Gamma$ is a set of formulas of $\\bI$ such that there exists $\\varphi$ with the property that $\\Gamma\\not\\vdash_{\\bI}\\varphi$; then we state that there exists a valuation $\\nu$ for $\\mathcal{A}$ with $\\nu(\\Gamma)\\subseteq D$ but $\\nu(\\varphi)\\in U$. This is true since, otherwise, if we had that for every valuation $\\nu$ for $\\mathcal{A}$ satisfying $\\nu(\\Gamma)\\subseteq D$ one also had $\\nu(\\varphi)\\in D$, this would imply $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$. Since $\\mathcal{M}$ characterizes $\\bI$, this would mean $\\Gamma\\vdash_{\\bI}\\varphi$, against our suppositions.\n\nConsider then two disjoint sets of distinct variables $\\{p_{n}\\ :\\ n\\in\\mathbb{N}\\}$ and $\\{q_{n}\\ :\\ n\\in\\mathbb{N}\\}$ and the formulas, for $i, j\\in\\mathbb{N}$,\n\\[\\phi_{ij}=\\begin{cases*}\n      \\quad p_{i}\\uparrow q_{j} & if $i<j$ \\\\\n      {\\sim}(p_{i}\\uparrow q_{j})   & otherwise\n    \\end{cases*};\\]\nwe also define, for any $n\\in\\mathbb{N}$, \n\\[\\Gamma_{n}=\\{\\phi_{ij}\\ :\\  0\\leq i, j\\leq n\\}.\\]\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{cccc}\n${\\sim}(p_{0}\\uparrow q_{0})$ & $p_{0}\\uparrow q_{1}$ & $\\cdots$ & $p_{0}\\uparrow q_{n}$\\\\\n${\\sim}(p_{1}\\uparrow q_{0})$ & ${\\sim}(p_{1}\\uparrow q_{1})$ & $\\cdots$ & $p_{1}\\uparrow q_{n}$\\\\\n$\\vdots$ & $\\vdots$ & $\\ddots$ & $\\vdots$\\\\\n${\\sim}(p_{n}\\uparrow q_{0})$ & ${\\sim}(p_{n}\\uparrow q_{1})$ & $\\cdots$ & ${\\sim}(p_{n}\\uparrow q_{n})$\n\\end{tabular}\n\\caption*{The formulas in $\\Gamma_{n}$}\n\\end{figure}\n\nWe state that, for all $n$, $\\Gamma_{n}\\not\\vdash_{\\bI}p_{0}$: to see that, we consider the bivaluation $\\nu$ for which $\\nu(p_{i})=\\nu(p_{j})=0$, for all $i, j\\in\\mathbb{N}$, and \n\\[\\nu(p_{i}\\uparrow q_{j})=\\begin{cases*}\n      1 & if $i<j$ \\\\\n      0   & otherwise\n    \\end{cases*}.\\]\nSuch a bivaluation is possible since: first of all, given all $p_{i}$ and $q_{j}$ are mapped into $0$, $p_{i}\\uparrow q_{j}$ may assume any value; second, assigning a value to $p_{i}\\uparrow q_{j}$ does not interfere with the value assigned to $p_{k}\\uparrow q_{l}$, for $(i,j)\\neq(k,l)$, given that the only restriction imposed by the fact $\\nu(p_{i}\\uparrow q_{j})$ has a certain value is that $\\nu(q_{j}\\uparrow p_{i})$ has the same value. Furthermore, Lemma \\ref{basic facts about Nmatrix semantics} tells us that $\\nu({\\sim}\\alpha)=1$ if, and only if, $\\nu(\\alpha)=0$, giving us $\\nu(\\phi_{ij})=1$ for all $0\\leq i, j\\leq n$; so $\\nu(\\Gamma_{n})=1$ but $\\nu(p_{0})=0$, and since bivaluations characterize $\\bI$, we have proved $\\Gamma_{n}\\not\\vdash_{\\bI}p_{0}$.\n\nThis means that, for any $n\\in\\mathbb{N}$, there exists a valuation $\\nu$ for $\\mathcal{A}$ such that $\\nu(\\Gamma_{n})\\subseteq D$ but $\\nu(p_{0})\\in U$; however, we will prove that, assuming $\\mathcal{A}$ has a finite universe with $n$ elements, there can not exist a valuation $\\nu$ satisfying $\\nu(\\Gamma_{n})\\subseteq D$. This is rather simple: suppose there exists such a valuation $\\nu$; given there are $n+1$ elements among $p_{0}, p_{1}, \\dotsc  , p_{n}$ and $\\mathcal{A}$ has only $n$ elements, by the pigeonhole principle one finds there exist $0\\leq i<j\\leq n$ such that $\\nu(p_{i})=\\nu(p_{j})$.\n\nSince $\\nu(\\Gamma_{n})\\subseteq D$, $\\nu(\\phi_{ij})=\\nu(p_{i}\\uparrow q_{j})\\in D$, implying $\\nu(p_{i})\\uparrow\\nu(q_{j})\\subseteq D$. Nevertheless, $\\nu(p_{i})=\\nu(p_{j})$ gives us that \n\\[\\nu(p_{j}\\uparrow q_{j})\\in \\nu(p_{j})\\uparrow\\nu(q_{j})=\\nu(p_{i})\\uparrow\\nu(q_{j})\\subseteq D,\\]\nand therefore $\\nu(\\phi_{jj})=\\nu({\\sim}(p_{j}\\uparrow q_{j}))\\in U$, contradicting the supposition that $\\nu(\\Gamma_{n})\\subseteq D$.\n\n\\begin{theorem}\nThere exists no finite Nmatrix which characterizes $\\bI$.\n\\end{theorem}\n\nOf course, this also implies $\\nbI$ is not characterizable by finite Nmatrices: if it were characterizable by some $\\mathcal{M}=(\\mathcal{A}, D)$, for $\\mathcal{A}$ a $\\Sigma_{\\nbI}$-multialgebra with universe $A$, by ignoring the paraconsistent negation and defining $\\mathcal{A}_{-}=(A, \\{\\sigma_{\\mathcal{A}}\\}_{\\sigma\\in\\Sigma_{\\bI}})$ one would find that $\\mathcal{M}_{-}=(\\mathcal{A}_{-}, D)$ characterizes $\\bI$, contradicting our previous theorem.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\bI$ is not characterizable by a finite Rmatrix}\n\nAlthough Rmatrices haven't been as well studied as logical matrices, or even Nmatrices, one could ask themselves whether Rmatrix semantics wouldn't offer easier decision methods for $\\bI$ and other logics of incompatibility than RNmatrices; so we stop for a moment and show that, already for $\\bI$, it is not possible to find a characterizing finite Rmatrix.\n\nAssume $\\mathcal{M}=(\\mathcal{A}, D, \\mathcal{F})$ is a finite Rmatrix (meaning the universe $A$ of $\\mathcal{A}$ is finite) which characterizes $\\bI$, and consider again: two disjoint sets of distinct variables $\\{p_{n}\\ :\\ n\\in\\mathbb{N}\\}$ and $\\{q_{n}\\ :\\ n\\in\\mathbb{N}\\}$; the formulas, for $i, j\\in\\mathbb{N}$,\n\\[\\phi_{ij}=\\begin{cases*}\n      \\quad p_{i}\\uparrow q_{j} & if $i<j$ \\\\\n      {\\sim}(p_{i}\\uparrow q_{j})   & otherwise\n    \\end{cases*};\\]\nand the sets of formulas, for any $n\\in\\mathbb{N}$, $\\Gamma_{n}=\\{\\phi_{ij}\\ :\\  0\\leq i, j\\leq n\\}$.\n\n\\begin{lemma}\\label{basic facts about Rmatrix semantics}\nIf the Rmatrix $\\mathcal{M}$ characterizes $\\bI$ and the image of $\\nu\\in\\mathcal{F}$ is not contained in $D$, then $\\nu(\\alpha)$ and $\\nu({\\sim}\\alpha)$ can not both belong to $D$.\n\\end{lemma}\n\n\\begin{proof}\nWe know that, since \"$\\sim$\" behaves classically, it satisfies \n\\[\\alpha\\rightarrow({\\sim}\\alpha\\rightarrow\\beta)\\]\nfor any formula $\\beta$, implying by the deduction meta-theorem that $\\alpha, {\\sim}\\alpha\\vdash_{\\bI}\\beta$. Let $\\gamma$ be a formula such that $\\nu(\\gamma)\\notin D$: if $\\nu(\\alpha), \\nu({\\sim}\\alpha)\\in D$, then $\\nu$ would not validate the deduction $\\alpha, {\\sim}\\alpha\\vdash_{\\bI}\\gamma$, and therefore at most one among $\\nu(\\alpha)$ and $\\nu({\\sim}\\alpha)$ belongs to $D$.\n\\end{proof}\n\nAs we know, $\\Gamma_{n}\\not\\vdash_{\\bI}p_{0}$ for any $n\\in\\mathbb{N}$, which means there must exist a valuation $\\nu\\in\\mathcal{F}$ with $\\nu(\\Gamma_{n})\\subseteq D$ but $\\nu(\\varphi)\\not\\in D$, since otherwise one would have, for all $\\nu\\in\\mathcal{F}$, satisfying that $\\nu(\\Gamma_{n})\\subseteq D$, that $\\nu(\\varphi)\\in D$; this, of course, would show that $\\Gamma_{n}\\vDash_{\\mathcal{M}}p_{0}$ and, since $\\mathcal{M}$ characterizes $\\bI$, that $\\Gamma_{n}\\vdash_{\\bI}p_{0}$, which we know not to be true. So there exists a $\\nu\\in\\mathcal{F}$ with undesignated elements in its image and $\\nu(\\Gamma_{n})\\subseteq D$. If $A$ has cardinality $n$, by the pigeonhole principle there must exist two elements $p_{i}$ and $p_{j}$ among $\\{p_{n}\\ :\\  n\\in\\mathbb{N}\\}$ such that $\\nu(p_{i})=\\nu(p_{j})$, what leads us to the following problem: assume, without loss of generality, $i<j$; then \n\\[\\nu(p_{i}\\uparrow q_{j})=\\nu(p_{i})\\uparrow\\nu(q_{j})=\\nu(p_{j})\\uparrow\\nu(q_{j})=\\nu(p_{j}\\uparrow q_{j}).\\]\nSince $\\nu(\\Gamma_{n})\\subseteq D$, $\\nu(p_{i}\\uparrow q_{j}), \\nu({\\sim}(p_{j}\\uparrow q_{j}))\\in D$, which means $\\nu$ is a valuation in $\\mathcal{F}$ (with image not contained in $D$) satisfying that $\\nu(p_{j}\\uparrow q_{j})$ and $\\nu({\\sim}(p_{j}\\uparrow q_{j}))$ are both in $D$, a contradiction given Lemma \\ref{basic facts about Rmatrix semantics}.\n\nThe necessary conclusion is that no finite Rmatrix can characterize $\\bI$. A similar argument applies to $\\nbI$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Comparing matrix semantics}\n\n\nSo, once we introduced the notion of an RNmatrix, or restricted Nmatrix, extending the notions of Rmatrix and Nmatrix, it becomes clear that we are considering generalizations of a logical matrix in two distinct, and independent, directions: we extend matrices by allowing their underlying algebras to become multialgebras, hence exchanging deterministic operations for multioperations, non-deterministic operations; and in a second direction, we restrict ourselves to a subset of all valuations, selecting which homomorphisms to take into consideration.\n\nThis gives the semantics at hand, if one interprets independent directions as it would be done in linear algebra, two dimensions of generality. Many of such dimensions are already quite standard in the literature, if we take as starting point that a logical matrix should merely be a pair $\\mathcal{M}=(\\mathcal{A}, D)$, with $\\mathcal{A}$ a finite algebra; we list a few ways below to extend such a matrix.\n\\begin{enumerate}\n\\item Allowing $\\mathcal{A}$ to become an infinite algebra, that is, an algebra with an infinite universe (and, reciprocally, if the underlying algebraic structure of the matrix is actually a multialgebra, allowing it to have an infinite universe). This generalization turns a logical matrix from a rather strict notion of a decision method to a broader semantical approach having universes of any desired cardinality.\n\n\\item Considering, instead of an unique matrix $\\mathcal{M}$, a class $\\mathbb{M}$ of matrices over which we define a semantical deduction operator by $\\Gamma\\vDash_{\\mathbb{M}}\\varphi$ if and only if $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$ for all $\\mathcal{M}$ in $\\mathbb{M}$; these matrices may be of any desired type, but it was already established by W\\'ojcicki \\cite{Woj, Woj2}, who envisioned this generalization, that every tarskian logic is characterizable by a class of potentially infinite logical matrices.\n\n\\item Permitting that multiple different sets of distinguished elements coexist: meaning a pair $\\mathcal{M}=(\\mathcal{A}, \\{D_{\\lambda}\\}_{\\lambda\\in\\Lambda})$, with $\\bigcup_{\\lambda\\in\\Lambda}D_{\\lambda}$ contained in $\\mathcal{A}$'s universe, when we define $\\Gamma\\vDash_{\\mathcal{M}}\\varphi$ if, and only if, for every valuation $\\nu$ and index $\\lambda$ one has that $\\nu(\\Gamma)\\subseteq D_{\\lambda}$ implies $\\nu(\\varphi)\\in D_{\\lambda}$. Again a development by W\\'ojcicki, usually referred to as generalized matrices\\index{Matrix, Generalized} or Gmatrices\\index{Gmatrix}, he proved in \\cite{Woj, Woj2} that every tarskian logic can also be characterized by a potentially infinite matrix with multiple sets of distinguished elements.\n\\end{enumerate}\n\nOf course, the ideal scenario remains that of a simple, old-fashioned finite logical matrix, and the topic of how we are able to navigate the space of possible generalizations is a rather interesting one. Look at the example of the logic of incompatibility $\\bI$: we know that a restricted Nmatrix $\\textbf{2}(\\bI)$ (Theorem \\ref{2-valued RNmatrix}) with two elements may characterize it, as well as the previously defined $\\mathbb{2}_{\\bI}$ (Section \\ref{Decision Method for bI}) based on bivaluations, also with only two elements. But the differences between the two are rather obvious: $\\mathbb{2}_{\\bI}$ is clearly more desirable as a semantical tool and proves to be a very efficient decision method; additionally, $\\mathbb{2}_{\\bI}$ has far more restrictive operations, and in exchange is far more lenient on the subject of which homomorphisms to consider.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[axis lines = left, \nxlabel={Restrictions on the Operations}, \nylabel={Restrictions on the homomorphisms}, \naxis equal, \nxmin=0, \nxmax=5, \nymin=0, \nymax=5, \nxtick={4},\nxticklabels={Deterministic},\nytick={1},\nyticklabels={All},\nscatter\/@pre marker code\/.code={},\nscatter\/@post marker code\/.code={\\node [above right] {\\pgfplotspointmeta};}]\n\\draw (1,1) rectangle (4,4);\n\\draw [line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth] (0,1) -- (5,1);\n\\draw [line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth] (4,0) -- (4,5);\n\\addplot[scatter,only marks]\n    plot[scatter src=explicit symbolic]\n    coordinates {\n(1,1) [Nmatrices]\n(1,4) [RNmatrices]\n(4,1) [logical matrices]\n(4,4) [Rmatrices]\n};\n\\draw [->, line width=1pt, line cap=round, dash pattern=on 0pt off 2\\pgflinewidth] (1.25,3.75) -- (3.75, 1.25) node[midway,above right] {Ideal};\n\\end{axis}\n\n\\end{tikzpicture}\n\\end{figure}\n\nUsing the previous diagram, we can see that, in $\\bI$'s case, we are able to start at some point in the far top-left, where $\\textbf{2}(\\bI)$ lies, and navigate to the bottom-right without ever leaving the, at least on the diagram, loosely-defined environment of restricted non-deterministic matrices, arriving at $\\mathbb{2}_{\\bI}$; notice this is very much in line with what we previously observed, that going from the first of these RNmatrices to the second involves restricting the underlying operations while expanding the accepted homomorphisms. Worthy of notice is that, while working in $\\bI$, we are never able to leave RNmatrices since $\\bI$ can not be characterized by either (finite) logical matrices, Nmatrices or Rmatrices.\n\nIn a different case, consider $\\bI^{-}$: we know that it admits a finite RNmatrix, namely $\\textbf{2}(\\bI^{-})$, located somewhere on the top-left region of the schematic diagram; but, in this case, we also have a finite Nmatrix which characterizes $\\bI^{-}$, although no (finite) logical matrices are available as far as we know. So, at least in this logic's case, navigating ever bottom-right indeed leads outside the environment of RNmatrices and into the methodology of Nmatrices.\n\nNow, as we have assigned numbers to the other generalizations of a finite logical matrix, let us take $(4)$ as exchanging the underlying algebra of a matrix for a multialgebra (making a logical matrix into an Nmatrix, an Rmatrix into an RNmatrix and so on), and $(5)$ as restricting the valuations to be taken into consideration (turning a logical matrix into an Rmatrix and so on).\n\n\\begin{problem}\nWhich combinations of generalizations of a (finite) logical matrix have the expressive power to characterize every tarskian logic?\n\\end{problem}\n\nW\\'ojciki proved, in \\cite{Woj} and \\cite{Woj2}, that the combinations of $(1)$ and $(2)$, and $(1)$ and $(3)$, corresponding to classes of potentially infinite logical matrices and potentially infinite matrices with multiple sets of distinguished elements, are enough to characterize all tarskian logics; Piochi, in \\cite{Piochi}, proved that $(1)$ plus $(5)$ is also enough, and as we have proved that every tarskian logic is characterizable by a two-elements RNmatrix (again Theorem \\ref{2-valued RNmatrix}), $(4)$ plus $(5)$ is also enough. From this, it is clear that combining any four of these conditions characterizes all such logics, although it seems the same can not yet be said about combinations of three of them, as, for example, we were not able to find a reference concerning $(2)$ plus $(3)$ and $(5)$ or any subset thereof. So, we have the following table to summarize these observations.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{cc|ccccc}\n& & \\multicolumn{5}{c}{Generalizations} \\\\[5pt]\n     & & (1)    &  (2)   & (3)    & (4)    & (5)   \\\\[5pt] \\cline{2-7}\n\\multirow{5}{*}{\\rotatebox[origin=c]{90}{Generalizations}}\n&  (1)  &  &  \\checkmark &  \\checkmark &  \\checkmark &  ? \\\\[5pt]\n & (2)  & \\checkmark &   &  ? &  ? &  ? \\\\[5pt]\n  & (3)  & \\checkmark &  ? &  &  ? &  ? \\\\[5pt]\n&   (4) & \\checkmark &  ? &  ? &   &  \\checkmark \\\\[5pt]\n  & (5) & ? &  ? &  ? &  \\checkmark &  \\\\[5pt]\n\\end{tabular}\n\\end{figure}\n\nWe have left the diagonal of the table empty, as it does not deal with combinations of generalizations \\textit{per se}, but rather the generalizations themselves. Of course, there is nothing stopping us from considering which, if any, of the generalizations can characterize all tarskian logics, leading to our second problem.\n\n\\begin{problem}\nAre there any generalizations which, alone, can characterize all tarskian logics?\n\\end{problem}\n\nAs we have shown, $\\bI$ is not characterizable by either finite Nmatrices or finite Rmatrices, so we can already place some limitatitive results in our table.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tabular}{cc|ccccc}\n& & \\multicolumn{5}{c}{Generalizations} \\\\[5pt]\n     & & (1)    &  (2)   & (3)    & (4)    & (5)   \\\\[5pt] \\cline{2-7}\n\\multirow{5}{*}{\\rotatebox[origin=c]{90}{Generalizations}}\n&  (1)  & ? &  \\checkmark &  \\checkmark &  \\checkmark &  ? \\\\[5pt]\n & (2)  & \\checkmark & ?  &  ? &  ? &  ? \\\\[5pt]\n  & (3)  & \\checkmark &  ? & ? &  ? &  ? \\\\[5pt]\n&   (4) & \\checkmark &  ? &  ? & \\ding{55}   &  \\checkmark \\\\[5pt]\n  & (5) & ? &  ? &  ? &  \\checkmark & \\ding{55} \\\\[5pt]\n\\end{tabular}\n\\end{figure}\n\nOf course, the first course of action one probably thinks of, when finding the problems mentioned above and looking at the tables, is attempting to fill in the missing results; this does not seem impossible, but it does not seem trivial either. Proving that certain semantics can not characterize all tarskian logics involves most probably presenting counter-examples, preferably examples among already known and studied systems.\n\nBut one can always increase the rows and columns in our little illustrative tables, by considering other generalizations of logical matrices. Here, for one, we have not included making the operations of a matrix partial, instead of non-deterministic: when applied to Nmatrices, this procedure returns the semantical objects known as PNmatrices. These play a unique role indeed, as they are weaker than RNmatrices, yet we do not know if strictly so or if both semantics characterize the same logics. We are then tempted to consider a hierarchy of strength among combinations of generalizations of logical matrices, instead of the simply binary ``do they characterize all tarskian logics?'', and things start to become involved... \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\\setcounter{chapter}{8}\n\\chapter{Translating paraconsistent logics}\\label{Chapter9}\\label{Chapter 9}\n\n\n\n\nTake the signature \\label{SigmaLFICPL}$\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}}$ such that $(\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}})_{0}=\\{\\bot, \\top\\}$, $(\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}})_{1}=\\{\\neg, {\\sim}, \\circ\\}$, $(\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}})_{2}=\\{\\vee, \\wedge, \\rightarrow\\}$ and $(\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}})_{n}=\\emptyset$ for $n>2$.\n\nWe classically define a Fidel structure to be a $\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}}-$multialgebra $\\mathcal{E}=(A, \\{\\sigma_{\\mathcal{E}}\\}_{\\sigma \\in\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}}})$ such that:\n\\begin{enumerate}\n\\item $(A, \\{\\sigma_{\\mathcal{E}}\\}_{\\sigma\\in\\Sigma^{\\textbf{CPL}}})$ is a Boolean algebra;\n\\item for every $a\\in A$ and $b\\in \\neg a$, $a\\vee b=\\top$;\n\\item for every $a\\in A$ and $b\\in \\neg a$, there exists a non-empty subset $O_{ab}$ of $\\circ a$, defined case by case, designed to capture the logical structure intended to be emulated by the Fidel structure.\n\\end{enumerate}\n\nIntuitively, one looks at $\\neg a$ as all possible negations of $a$, and at $O_{ab}$ as all possible consistencies for $a$ given that $b$ is its negation. To give one example, in $\\textbf{mbC}$, where we usually denote $O_{ab}$ by $BC_{ab}$, we require that\n\\[a\\wedge (b\\wedge c)=\\bot,\\quad \\forall b\\in \\neg a,\\quad \\forall c\\in BC_{ab}.\\]\n\nWhen looking at previous instances in this text of a \"Fidel structure\", maybe the most important distinction was the intuitive replacement of consistency for incompatibility: so, for example, instead of the axiom schema $\\textbf{bc1}$ of $\\textbf{mbC}$ given by $\\circ\\alpha\\rightarrow(\\alpha\\rightarrow(\\neg\\alpha\\rightarrow\\beta))$, we used a similar, but distinct, schema $\\textbf{Ip}$, given by $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow\\gamma))$.\n\nOne clearly notices how our binary incompatibility operator, the generalized Sheffer's stroke, as studied above seems inherently distinct from the unary consistency operator \"$\\circ$\". In fact, one must translate accordingly the axiomatization of $\\textbf{LFI}$'s to that of our $\\textbf{LIp}$'s to show that the latter generalize the former, and logics such as $\\textbf{mbC}$, $\\textbf{mbCciw}$, $\\textbf{mbCci}$ and $\\textbf{mbCcl}$ become sublogics of, respectively, $\\nbI$, $\\nbIciw$, $\\nbIci$ and $\\nbIcl$. \n\nBut, and this is the important finding of this chapter, our intuition can still be validated: inconsistency, at least as found in the simpler paraconsistent logics here exhibited, can be obtained from incompatibility, well-behaved formulas being precisely those formulas incompatible with their negations. Since there are systems on which incompatibility clearly does not reduce back to inconsistency, such as $\\textbf{bI}$ (which does not even have a paraconsistent negation), we must reach the conclusion that incompatibility appears to non-trivially generalize the notion of inconsistency, giving some extra validation to the work we performed so far. \n\nThe developments found here have been submitted, as a preprint, in \\cite{Frominconsistency}.\n\n\\section{Preliminaries}\n\nSo, let us define a translation from the usual signature $\\Sigma_{\\textbf{LFI}}$ for $\\textbf{LFI}'s$, with $(\\Sigma_{\\textbf{LFI}})_{1}=\\{\\neg , \\circ\\}$, $(\\Sigma_{\\textbf{LFI}})_{2}=\\{\\vee, \\wedge, \\rightarrow\\}$ and $(\\Sigma_{\\textbf{LFI}})_{n}=\\emptyset$ for $n\\notin\\{1,2\\}$, to the signature $\\Sigma_{\\nbI}$. For $\\mathcal{V}$ a countable set of propositional variables, consider the function\\label{T} \n\\[T:F(\\Sigma_{\\textbf{LFI}}, \\mathcal{V})\\rightarrow F(\\Sigma_{\\nbI}, \\mathcal{V})\\]\nsuch that:\n\\begin{enumerate}\n\\item $T(p)=p$ for every $p\\in \\mathcal{V}$;\n\\item $T(\\neg\\alpha)=\\neg T(\\alpha)$;\n\\item $T(\\alpha\\#\\beta)=T(\\alpha)\\# T(\\beta)$ for every $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$;\n\\item $T(\\circ\\alpha)=T(\\alpha)\\uparrow\\neg T(\\alpha)$.\n\\end{enumerate}\nEssentially, $T$ changes all occurrences of the form $\\circ\\alpha$ to $\\alpha\\uparrow\\neg\\alpha$. What one does is then to consider for a set of axiom schemata $\\Gamma$ of an $\\textbf{LFI}$ its translation $T(\\Gamma)=\\{T(\\psi)\\ :\\  \\psi\\in \\Gamma\\}$: to give one example, many instances of $\\textbf{Ip}$ are translations of instances of $\\textbf{bc1}$.\n\n\\begin{proposition}\n$T$ is an injective function.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $T(\\alpha)=T(\\beta)$: we proceed by double induction on the orders of $\\alpha$ and $\\beta$, to show that in this case $\\alpha=\\beta$. \n\nIf $\\alpha$ is of order $0$, then $\\alpha=p$ for some $p\\in \\mathcal{V}$ and therefore $T(\\alpha)=\\alpha$: if $\\beta$ is of order $0$, then it is a propositional variable $q$, and then again $T(\\beta)=\\beta$, implying we have $\\alpha=T(\\alpha)=T(\\beta)=\\beta$.\n\nSo, assume $\\beta$ is of order at least $1$, and we show that we can not actually have, in this case, $T(\\alpha)=T(\\beta)$:\n\\begin{enumerate}\n\\item if $\\beta=\\neg\\beta_{0}$, $T(\\beta)=\\neg T(\\beta_{0})=T(\\alpha)$, which is absurd since $T(\\alpha)$ is a propositional variable and can not contain an unary connective;\n\\item if $\\beta=\\beta_{0}\\#\\beta_{1}$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $T(\\beta)=T(\\beta_{0})\\# T(\\beta_{1})=T(\\alpha)$, which is again absurd since $T(\\alpha)$ is a propositional variable;\n\\item if $\\beta=\\circ\\beta_{0}$, $T(\\beta)=T(\\beta_{0})\\uparrow\\neg T(\\beta_{0})$, which clearly contains even more than one connective and therefore can not equal $T(\\alpha)$.\n\\end{enumerate}\n\nNow, suppose that for every $\\alpha$ of order at most $m$ we have that, if $T(\\alpha)=T(\\beta)$, then $\\alpha=\\beta$, and take a formula $\\alpha$ of order $m+1$: we have that either $\\alpha=\\alpha_{0}\\#\\alpha_{1}$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\alpha=\\neg\\alpha_{0}$ or $\\alpha=\\circ\\alpha_{0}$, with the orders of $\\alpha_{0}$ and $\\alpha_{1}$ being at most $m$.\n\nIf $\\beta$ is of order $0$, then $\\beta=q$ for some $q\\in \\mathcal{V}$, and in that case $T(\\beta)=\\beta$ is a formula of order $0$: we then can not really have $T(\\alpha)=T(\\beta)$, since $T(\\alpha)$ is either $\\neg T(\\alpha_{0})$, $T(\\alpha_{0})\\# T(\\alpha_{1})$ or $T(\\alpha_{0})\\uparrow\\neg T(\\alpha_{0})$, none of which is of order $0$; by vacuity, the lemma holds. \n\nInductively, suppose that for all $\\beta$ of order at most $n$, if $T(\\alpha)=T(\\beta)$ then $\\alpha=\\beta$, and take $\\beta$ of order $n+1$: we have that either $\\beta=\\beta_{0}\\ast\\beta_{1}$, for $\\ast\\in\\{\\vee, \\wedge, \\rightarrow\\}$, $\\beta=\\neg\\beta_{0}$ or $\\beta=\\circ\\beta_{0}$, with the orders of $\\beta_{0}$ and $\\beta_{1}$ being at most $n$.\n\nIf $\\alpha=\\alpha_{0}\\#\\alpha_{1}$, $T(\\alpha)=T(\\alpha_{0})\\# T(\\alpha_{1})$: in this case, $\\beta$ can not equal $\\neg\\beta_{0}$, for in this case we would have $T(\\beta)=\\neg T(\\beta_{0})$ which has a leading connective of arity different from that of $T(\\alpha)$; for similar reasons we can not have $\\beta=\\circ\\beta_{0}$ or $\\beta=\\beta_{0}\\ast\\beta_{1}$ for $\\ast$ different of $\\#$. The same can be done when $\\alpha=\\neg\\alpha_{0}$, in which case $\\beta$ must also be of the form $\\neg\\beta_{0}$, and when $\\alpha=\\circ\\alpha_{0}$, when we must have $\\beta=\\circ\\beta_{0}$.\n\nSo there remains three cases to check:\n\\begin{enumerate}\n\\item if $\\alpha=\\alpha_{0}\\#\\alpha_{1}$ and $\\beta=\\beta_{0}\\#\\beta_{1}$, $T(\\alpha)=T(\\beta)$ implies that $T(\\alpha_{0})\\# T(\\alpha_{1})=T(\\beta_{0})\\# T(\\beta_{1})$, and so $T(\\alpha_{0})=T(\\beta_{0})$ and $T(\\alpha_{1})=T(\\beta_{1})$; by our induction hypothesis, $\\alpha_{0}=\\beta_{0}$ and $\\alpha_{1}=\\beta_{1}$, so that $\\alpha=\\beta$;\n\\item if $\\alpha=\\neg\\alpha_{0}$ and $\\beta=\\neg\\beta_{0}$, $T(\\alpha)=T(\\beta)$ implies that $\\neg T(\\alpha_{0})=\\neg T(\\beta_{0})$ and so $T(\\alpha_{0})=T(\\beta_{0})$; by our induction hypothesis, $\\alpha_{0}=\\beta_{0}$ and therefore $\\alpha=\\beta$;\n\\item if $\\alpha=\\circ\\alpha_{0}$ and $\\beta=\\circ\\beta_{0}$, $T(\\alpha)=T(\\beta)$ implies that \n\\[T(\\alpha_{0})\\uparrow\\neg  T(\\alpha_{0})=T(\\beta_{0})\\uparrow\\neg  T(\\beta_{0})\\]\nand so $T(\\alpha_{0})=T(\\beta_{0})$; by our induction hypothesis, $\\alpha_{0}=\\beta_{0}$ and therefore $\\alpha=\\beta$.\n\\end{enumerate}\nThis, of course, finishes the proof.\n\\end{proof}\n\nOne important thing to notice is that a formula $\\alpha$ in $F(\\Sigma_{\\nbI},\\mathcal{V})$ is not in $T(F(\\Sigma_{\\textbf{LFI}},\\mathcal{V}))$ if and only if it contains a subformula $\\beta_{1}\\uparrow\\beta_{2}$ such that $\\beta_{2}\\neq\\neg\\beta_{1}$.\n\nOne direction is clear: if $\\alpha$ contains a subformula $\\beta_{1}\\uparrow\\beta_{2}$ with $\\beta_{2}\\neq\\neg \\beta_{1}$ then it is not in $T(F(\\Sigma_{\\textbf{LFI}},\\mathcal{V}))$, since such a formula is not a translation of anything over the signature $\\Sigma_{\\textbf{LFI}}$.\n\nReciprocally, we proceed by induction on the order of $\\alpha$: if it is $0$, $\\alpha$ is a propositional variable, and it is always the case that $\\alpha$ is a translation; so assume that $\\alpha$ is of order $n>1$ and that the result holds for formulas of order smaller than $n$. Then we have three cases to consider:\n\\begin{enumerate}\n\\item if $\\alpha=\\alpha_{0}\\#\\alpha_{1}$, for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, and $\\alpha_{0}$ and $\\alpha_{1}$ are translations, so is $\\alpha$; therefore, if $\\alpha\\notin T(F(\\Sigma_{\\textbf{LFI}},\\mathcal{V}))$, then one of $\\alpha_{0}$ or $\\alpha_{1}$ has, by induction hypothesis, a subformula $\\beta_{1}\\uparrow\\beta_{2}$ with $\\beta_{2}\\neq\\neg \\beta_{1}$, and the result holds;\n\\item if $\\alpha=\\neg\\alpha_{0}$ and $\\alpha_{0}$ is a translation, so is $\\alpha$; therefore, if $\\alpha\\notin T(F(\\Sigma_{\\textbf{LFI}},\\mathcal{V}))$, then $\\alpha_{0}\\notin T(F(\\Sigma_{\\textbf{LFI}},\\mathcal{V}))$ and thus has, by induction hypothesis, a subformula of the desired form, making the result hold once again;\n\\item finally, if $\\alpha=\\alpha_{0}\\uparrow\\alpha_{1}$ but $\\alpha\\notin T(F(\\Sigma_{\\textbf{LFI}},\\mathcal{V}))$, either $\\alpha_{0}$ and $\\alpha_{1}$ are translations but $\\alpha_{1}\\neq\\neg\\alpha_{0}$, and the result holds; or one of $\\alpha_{0}$ and $\\alpha_{1}$ is not a translation, and therefore contains a subformula $\\beta_{1}\\uparrow\\beta_{2}$ with $\\beta_{2}\\neq\\neg\\beta_{1}$, what ends the proof.\n\\end{enumerate}\n\n\n\\section{$T$ is a conservative translation}\n\nGiven logics $\\mathcal{L}_{1}$ and $\\mathcal{L}_{2}$ over the signatures $\\Sigma_{1}$ and $\\Sigma_{2}$, a function $\\mathcal{T}:F(\\Sigma_{1}, \\mathcal{V})\\rightarrow F(\\Sigma_{2},\\mathcal{V})$ is said to be a translation\\index{Translation}, originally defined in \\cite{Itala}, when, for every set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{1}$,\n\\[\\Gamma\\vdash_{\\mathcal{L}_{1}}\\varphi\\quad\\text{implies}\\quad\\mathcal{T}(\\Gamma)\\vdash_{\\mathcal{L}_{2}}\\mathcal{T}(\\varphi);\\]\na translation $\\mathcal{T}$ is said to be a conservative translation\\index{Translation, Conservative} whenever, for every set of formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma_{1}$,\n\\[\\mathcal{T}(\\Gamma)\\vdash_{\\mathcal{L}_{2}}\\mathcal{T}(\\varphi)\\quad\\text{implies}\\quad\\Gamma\\vdash_{\\mathcal{L}_{1}}\\varphi.\\]\n One may look at definition $2.4.1$ of \\cite{ParLog} for a reference for our definition, or the original work concerning conservative translations, \\cite{ConservativeTranslationsReference}: such a notion is recurring in a contemporary approach to logic, where systems may have been formulated in apparently distinct ways that prove to be, under translation, equivalent; coincidentally, we have already seen translations very briefly in Example \\ref{PTS}.\n\nWe shall prove that the function $T$ we previously defined is a translation, and furthermore, a conservative one in many cases. To prove the following lemma, and consequently the following theorem, remember a $\\Sigma$-homomorphism $\\sigma:F(\\Sigma, \\mathcal{V})\\rightarrow F(\\Sigma, \\mathcal{V})$ is determined by its action on $\\mathcal{V}$, given $F(\\Sigma, \\mathcal{V})$ is deterministic.\n\n\\begin{lemma}\\label{Existence of translated substitution}\nGiven a $\\Sigma_{\\LFI}$-homomorphism $\\sigma:F(\\Sigma_{\\LFI}, \\mathcal{V})\\rightarrow F(\\Sigma_{\\LFI}, \\mathcal{V})$, the $\\Sigma_{\\nbI}$-homomor\\-phism $\\overline{\\sigma}:F(\\Sigma_{\\nbI}, \\mathcal{V})\\rightarrow F(\\Sigma_{\\nbI}, \\mathcal{V})$ given by, for a propositional variable $p\\in\\mathcal{V}$, \n\\[\\overline{\\sigma}(p)=T(\\sigma(p))\\]\nsatisfies that, for any formula $\\alpha$ on $\\Sigma_{\\LFI}$, $T(\\sigma(\\alpha))=\\overline{\\sigma}(T(\\alpha))$.\n\\end{lemma}\n\n\\begin{proof}\nThe result is trivially true for formulas of order $0$, since they are invariant under $T$. So, proceeding inductively, assume the result holds for the formulas $\\alpha$ and $\\beta$:\n\\begin{enumerate}\n\\item for $\\#\\in\\{\\vee, \\wedge, \\rightarrow\\}$, \n\\[T(\\sigma(\\alpha\\#\\beta))=T(\\sigma(\\alpha)\\#\\sigma(\\beta))=T(\\sigma(\\alpha))\\# T(\\sigma(\\beta))=\\overline{\\sigma}(T(\\alpha))\\#\\overline{\\sigma}(T(\\beta))=\\]\n\\[\\overline{\\sigma}(T(\\alpha)\\# T(\\beta))=\\overline{\\sigma}(T(\\alpha\\#\\beta));\\]\n\\item $T(\\sigma(\\neg\\alpha))=T(\\neg\\sigma(\\alpha))=\\neg T(\\sigma(\\alpha))=\\neg \\overline{\\sigma}(T(\\alpha))=\\overline{\\sigma}(\\neg T(\\alpha))=\\overline{\\sigma}(T(\\neg\\alpha))$;\n\\item finally, remembering $\\overline{\\sigma}$ is a homomorphism,\n\\[T(\\sigma(\\circ\\alpha))=T(\\circ\\sigma(\\alpha))=T(\\sigma(\\alpha))\\uparrow\\neg T(\\sigma(\\alpha))=\\overline{\\sigma}(T(\\alpha))\\uparrow\\neg\\overline{\\sigma}(T(\\alpha))=\\]\n\\[\\overline{\\sigma}(T(\\alpha))\\uparrow\\overline{\\sigma}(\\neg T(\\alpha))=\\overline{\\sigma}(T(\\alpha)\\uparrow\\neg T(\\alpha))=\\overline{\\sigma}(T(\\circ\\alpha)),\\]\nwhat ends the proof.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theorem}\\label{Soundness of translations}\nIf $\\mathcal{L}$ is a logic over the signature $\\Sigma_{\\textbf{LFI}}$ with axiom schemata $\\Psi$ and $\\mathcal{L}^{*}$\\label{L*} is the logic over the signature $\\Sigma_{\\nbI}$ with axiom schemata $T(\\Psi)$, then $\\Gamma\\vdash_{\\mathcal{L}} \\varphi$ implies that $T(\\Gamma)\\vdash_{\\mathcal{L}^{*}} T(\\varphi)$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\alpha_{1}, \\dotsc , \\alpha_{n}$ be a demonstration of $\\varphi$ from $\\Gamma$, with $\\alpha_{n}=\\varphi$: we want to show that in this case $T(\\alpha_{1}), \\dotsc , T(\\alpha_{n})$ is a demonstration of $T(\\varphi)$; first of all, obviously $T(\\alpha_{n})=T(\\varphi)$. Then:\n\\begin{enumerate}\n\\item if $\\alpha_{i}$ is an instance of an axiom schema $\\psi$, $T(\\alpha_{i})$ is an instance of the axiom schema $T(\\psi)$, by an immediate application of Lemma \\ref{Existence of translated substitution};\n\\item if $\\alpha_{j}$ is in $\\Gamma$, $T(\\alpha_{j})\\in T(\\Gamma)$;\n\\item finally, given that our only rule of deduction is Modus Ponens, if $\\alpha_{k}$ is such that there exist $\\alpha_{i}$ and $\\alpha_{j}$ with $i, j<k$ and $\\alpha_{j}=\\alpha_{i}\\rightarrow \\alpha_{k}$ or $\\alpha_{i}=\\alpha_{j}\\rightarrow\\alpha_{k}$, then $T(\\alpha_{k})$ is such that either\n\\[T(\\alpha_{j})=T(\\alpha_{i}\\rightarrow\\alpha_{k})=T(\\alpha_{i})\\rightarrow T(\\alpha_{k})\\]\nor \n\\[T(\\alpha_{i})=T(\\alpha_{j})\\rightarrow T(\\alpha_{k}).\\]\n\\end{enumerate}\nThis finishes proving that, if $\\Gamma\\vdash_{\\mathcal{L}} \\varphi$, then $T(\\Gamma)\\vdash_{\\mathcal{L}^{*}} T(\\varphi)$.\n\\end{proof}\n\nThis proves how $T$ is always a translation.\n\nNotice how $\\nbI$ has as axiom schemata all axiom schemata of \\label{mbC*}$\\textbf{mbC}^{*}$, and therefore is capable of all deductions that $\\textbf{mbC}^{*}$ is capable, meaning that, if $\\Gamma\\vdash_{\\textbf{mbC}^{*}}\\varphi$, then $\\Gamma\\vdash_{\\nbI}\\varphi$. However, $\\nbI$ is strictly stronger than $\\textbf{mbC}^{*}$, as one can see from, to given one example, the formula\n\\[(\\alpha\\uparrow\\alpha)\\rightarrow(\\alpha\\rightarrow(\\alpha\\rightarrow\\beta)),\\]\nwhich is an instance of $\\textbf{Ip}$, but not of $T(\\textbf{bc1})$, is a tautology of $\\nbI$ but not of $\\textbf{mbC}^{*}$. Analogously, $\\nbIciw$, $\\nbIci$ and $\\nbIcl$ are strictly stronger than, respectively, \\label{mbCciw*}$\\textbf{mbCciw}^{*}$, \\label{mbCci*}$\\textbf{mbCci}^{*}$ and \\label{mbCcl*}$\\textbf{mbCcl}^{*}$. One more interesting example would be in $\\nbIcl$: although it is well known that $\\neg(\\alpha\\wedge\\neg\\alpha)$ does not imply $\\neg(\\neg\\alpha\\wedge\\alpha)$ in this logic, and this remains true in $\\mbCcl^{*}$ as we shall prove it, in $\\nbIcl$, through use of $\\textbf{cl}^{*}$, $\\textbf{Comm}$ and $\\textbf{Ip}$, one gets\n\\[\\vdash_{\\nbIcl}\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow\\neg(\\neg\\alpha\\wedge\\alpha).\\]\n\n\n\\subsection{$T$ is conservative for $\\textbf{mbC}$}\n\n\\begin{definition}\nA function $\\nu$ from the formulas of $\\textbf{mbC}$ to $\\{0,1\\}$ is said to be a \\index{Bivaluation for $\\textbf{mbC}$}bivaluation for $\\textbf{mbC}$ if it satisfies, for any formulas $\\alpha$ and $\\beta$,\n\\begin{enumerate}\n\n\\item $\\nu(\\alpha\\wedge\\beta)=1$ if and only if $\\nu(\\alpha)=1$ and $\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\vee\\beta)=1$ if and only if $\\nu(\\alpha)=1$ or $\\nu(\\beta)=1$;\n\\item $\\nu(\\alpha\\rightarrow\\beta)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\beta)=1$;\n\\item $\\nu(\\neg \\alpha)=0$ implies $\\nu(\\alpha)=1$;\n\\item $\\nu(\\circ\\alpha)=1$ implies $\\nu(\\alpha)=0$ or $\\nu(\\neg\\alpha)=0$.\n\\end{enumerate}\n\\end{definition}\n\nFor $\\Gamma\\cup\\{\\varphi\\}$ a set of formulas of $\\textbf{mbC}$, we define $\\Gamma\\vDash_{\\textbf{mbC}}\\varphi$ to mean that for every bivaluation $\\nu$ for $\\textbf{mbC}$, if $\\nu(\\Gamma)\\subseteq\\{1\\}$ then $\\nu(\\varphi)=1$.\n\n\\begin{theorem}\nFor formulas $\\Gamma\\cup\\{\\varphi\\}$ of $\\textbf{mbC}$,\n\\[\\Gamma\\vdash_{\\textbf{mbC}}\\varphi\\quad\\text{if and only if}\\quad\\Gamma\\vDash_{\\textbf{mbC}}\\varphi.\\]\n\\end{theorem}\n\nThis is a classical result, which can be found in section $2.2$ of \\cite{ParLog}, heavily inspired by Newton da Costa's valuation semantics for $C_{1}$ found in \\cite{Costa} and \\cite{Costa2}, that will serve us to prove that, if $T(\\Gamma)\\vdash_{\\textbf{mbC}^{*}}T(\\varphi)$, then $\\Gamma\\vdash_{\\textbf{mbC}}\\varphi$.\n\nBy contraposition, suppose $\\Gamma\\not\\vdash_{\\textbf{mbC}}\\varphi$: then, there exists a bivaluation $\\nu$ for $\\textbf{mbC}$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$ and $\\nu(\\varphi)=0$. We define a function $\\overline{\\nu}$ from the formulas of $\\nbI$ to $\\{0,1\\}$ by structural induction:\n\\begin{enumerate}\n\\item if $\\alpha$ is of order $0$, and is therefore a propositional variable, $\\overline{\\nu}(\\alpha)=\\nu(\\alpha)$;\n\\item $\\overline{\\nu}(\\alpha\\vee\\beta)=1$ if and only if $\\overline{\\nu}(\\alpha)=1$ or $\\overline{\\nu}(\\beta)=1$;\n\\item $\\overline{\\nu}(\\alpha\\wedge\\beta)=1$ if and only if $\\overline{\\nu}(\\alpha)=1$ and $\\overline{\\nu}(\\beta)=1$;\n\\item $\\overline{\\nu}(\\alpha\\rightarrow\\beta)=1$ if and only if $\\overline{\\nu}(\\alpha)=0$ or $\\overline{\\nu}(\\beta)=1$;\n\\item $\\overline{\\nu}(\\neg  T(\\alpha))=\\nu(\\neg \\alpha)$, and if $\\alpha$ is not a translation, $\\overline{\\nu}(\\neg \\alpha)=1$;\n\\item $\\overline{\\nu}(T(\\alpha)\\uparrow\\neg T(\\alpha))=\\nu(\\circ\\alpha)$, $\\overline{\\nu}(\\neg T(\\alpha)\\uparrow T(\\alpha))=\\nu(\\circ\\alpha)$, and, in all other cases, we make $\\overline{\\nu}(\\alpha\\uparrow\\beta)=0$.\n\\end{enumerate}\n\n\\begin{proposition}\nIf $\\alpha$ is a formula of $\\textbf{mbC}$, $\\overline{\\nu}(T(\\alpha))=\\nu(\\alpha)$.\n\\end{proposition}\n\n\\begin{proof}\nWe proceed by induction on the order of $\\alpha$: if it is $0$, $\\alpha$ is a propositional variable and $T(\\alpha)=\\alpha$, and by definition of $\\overline{\\nu}$ we have that $\\overline{\\nu}(T(\\alpha))=\\nu(\\alpha)$.\n\nNow assume the result holds for formulas of order smaller than that of $\\alpha$, and we have five cases to consider:\n\\begin{enumerate}\n\\item if $\\alpha=\\phi\\vee\\psi$, $\\overline{\\nu}(T(\\alpha))=1$ if and only if $\\overline{\\nu}(T(\\phi))=1$ or $\\overline{\\nu}(T(\\psi))=1$, which by hypothesis is equivalent to either $\\nu(\\phi)$ or $\\nu(\\psi)$ being equal to $1$, which occurs if and only if $\\nu(\\alpha)=1$;\n\\item if $\\alpha=\\phi\\wedge\\psi$, $\\overline{\\nu}(T(\\alpha))=1$ is equivalent to $\\overline{\\nu}(T(\\phi))$ and $\\overline{\\nu}(T(\\psi))$ equating $1$, which in turn is equivalent by induction hypothesis to $\\nu(\\phi)$ and $\\nu(\\psi)$ equating $1$, what happens if and only if $\\nu(\\alpha)=1$;\n\\item if $\\alpha=\\phi\\rightarrow\\psi$, $\\overline{\\nu}(T(\\alpha))=1$ if and only if $\\overline{\\nu}(T(\\phi))=0$ or $\\overline{\\nu}(T(\\psi))=1$, what by hypothesis is equivalent to $\\nu(\\phi)$ being $0$ or $\\nu(\\psi)$ being $1$, which happens if and only if $\\nu(\\alpha)=1$;\n\\item if $\\alpha=\\neg \\phi$, by definition of $\\overline{\\nu}$ we have that \n\\[\\overline{\\nu}(T(\\alpha))=\\overline{\\nu}(\\neg  T(\\phi))=\\nu(\\neg \\phi)=\\nu(\\alpha);\\]\n\\item if $\\alpha=\\circ\\phi$, we have that\n\\[\\overline{\\nu}(T(\\alpha))=\\overline{\\nu}(T(\\phi)\\uparrow\\neg T(\\phi))=\\nu(\\circ\\phi)=\\nu(\\alpha),\\]\nwhat ends the proof.\n\\end{enumerate}\n\\end{proof}\n\nIt is clear that, by its definition, $\\overline{\\nu}$ satisfies those conditions of being a bivaluation for $\\nbI$, found at the beginning of Section \\ref{Bivaluations for nbI}, related to disjunction, conjunction, and implication.\n\nIf $\\overline{\\nu}(\\neg \\alpha)=0$, the only possibility is that $\\alpha$ is the translation of some formula $\\phi$ in $\\textbf{mbC}$ and that $\\nu(\\neg \\phi)=0$; in this case $\\nu(\\phi)=1$, what implies \n\\[\\overline{\\nu}(\\alpha)=\\overline{\\nu}(T(\\phi))=\\nu(\\phi)=1.\\]\nIf $\\overline{\\nu}(\\alpha\\uparrow\\beta)=1$, this means that:\n\\begin{enumerate}\n\\item either $\\alpha$ is the translation of some formula $\\phi$, $\\beta=\\neg\\alpha$ and $\\nu(\\circ\\phi)=1$, which means that either $\\nu(\\phi)=0$, and therefore $\\overline{\\nu}(\\alpha)=0$, or $\\nu(\\neg \\phi)=0$, and then $\\overline{\\nu}(\\beta)=0$;\n\\item or $\\beta$ is the translation of some formula $\\phi$, $\\alpha=\\neg\\beta$ and $\\nu(\\circ\\phi)=1$, which again means that either $\\overline{\\nu}(\\alpha)=0$ or $\\overline{\\nu}(\\beta)=0$.\n\\end{enumerate}\n\nFinally, if $\\alpha$ is a translation of $\\phi$ and $\\beta=\\neg\\alpha$, $\\overline{\\nu}(\\alpha\\uparrow\\beta)=\\nu(\\circ\\phi)=\\overline{\\nu}(\\beta\\uparrow\\alpha)$, the same happening if $\\beta$ is a translation and $\\alpha=\\neg\\beta$. Otherwise, we have \n\\[\\overline{\\nu}(\\alpha\\uparrow\\beta)=0=\\overline{\\nu}(\\beta\\uparrow\\alpha).\\]\nThis finishes proving that $\\overline{\\nu}$ is a bivaluation for $\\nbI$; of course, $\\overline{\\nu}(T(\\Gamma))=\\nu(\\Gamma)\\subseteq\\{1\\}$ but $\\overline{\\nu}(T(\\varphi))=\\nu(\\varphi)=0$, what implies $T(\\Gamma)\\not\\vDash_{\\nbI}T(\\varphi)$; this is equivalent to $T(\\Gamma)\\not\\vdash_{\\nbI}T(\\varphi)$, what means that $T(\\Gamma)\\not\\vdash_{\\textbf{mbC}^{*}}T(\\varphi)$, since $\\nbI$ is strictly stronger than $\\textbf{mbC}^{*}$. Since $\\Gamma\\not\\vdash_{\\textbf{mbC}}\\varphi$ implies $T(\\Gamma)\\not\\vdash_{\\textbf{mbC}^{*}}T(\\varphi)$, $T(\\Gamma)\\vdash_{\\textbf{mbC}^{*}}T(\\varphi)$ implies $\\Gamma\\vdash_{\\textbf{mbC}}\\varphi$, and therefore \n\\[\\Gamma\\vdash_{\\textbf{mbC}}\\varphi\\quad\\text{if and only if}\\quad T(\\Gamma)\\vdash_{\\textbf{mbC}^{*}}T(\\varphi).\\]\n\n\n\\subsection{$T$ is conservative for $\\textbf{mbCciw}$, $\\textbf{mbCci}$ and $\\textbf{mbCcl}$}\n\n$\\textbf{mbCciw}^{*}$, $\\textbf{mbCci}^{*}$ and $\\textbf{mbCcl}^{*}$ are, respectively, the translations, under $T$, of $\\textbf{mbCciw}$, $\\textbf{mbCci}$ and $\\textbf{mbCcl}$, meaning:\n\\begin{enumerate}\n\\item $\\textbf{mbCciw}^{*}$ is obtained from $\\textbf{mbC}^{*}$ by addition of the axiom schema\n\\[\\tag{$\\textbf{ciw}^{*}$}(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg \\alpha);\\]\n\\item $\\textbf{mbCci}^{*}$ is obtained from $\\textbf{mbC}^{*}$ by adding\n\\[\\tag{$\\textbf{ci}^{*}$}\\neg (\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg \\alpha);\\]\n\\item $\\textbf{mbCcl}^{*}$ is obtained from $\\textbf{mbC}^{*}$ by adding\n\\[\\tag{$\\textbf{cl}^{*}$}\\neg (\\alpha\\wedge\\neg \\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha).\\]\n\\end{enumerate}\n\nWe already know by Theorem \\ref{Soundness of translations} that, for any of these logics $\\mathcal{L}$, if $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ then $T(\\Gamma)\\vdash_{\\mathcal{L}^{*}}T(\\varphi)$. To prove the reciprocal, as was done with $\\textbf{mbC}$, we will use bivaluations.\n\n\\begin{definition}\\label{Bivaluations for mbCciw, mbCci, ...}\nA bivaluation for $\\textbf{mbCciw}$, $\\textbf{mbCci}$ or \\index{Bivaluation for $\\textbf{mbCci}$}$\\textbf{mbCcl}$ (already defined, for $\\textbf{mbCciw}$ and $\\textbf{mbCcl}$, in Section \\ref{RNmatrix for mbCcl})  is a valuation for $\\textbf{mbC}$ satisfying also that, respectively:\n\\begin{enumerate}\n\\item $\\nu(\\circ\\alpha)=1$ if and only if $\\nu(\\alpha)=0$ or $\\nu(\\neg \\alpha)=0$;\n\\item the above condition, plus that $\\nu(\\neg \\circ\\alpha)=1$ implies $\\nu(\\alpha)=1$ and $\\nu(\\neg \\alpha)=1$;\n\\item the first condition, plus that $\\nu(\\neg (\\alpha\\wedge\\neg \\alpha))=1$ implies $\\nu(\\circ\\alpha)=1$.\n\\end{enumerate}\n\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ in the signature $\\Sigma_{\\textbf{LFI}}$, we say $\\Gamma$ proves $\\varphi$ according to bivaluations for $\\textbf{mbCciw}$, and write $\\Gamma\\vDash_{\\textbf{mbCciw}}\\varphi$, if for all valuations $\\nu$ for $\\textbf{mbCciw}$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$ we have that $\\nu(\\varphi)=1$; we define $\\vDash_{\\textbf{mbCci}}$ and $\\vDash_{\\textbf{mbCcl}}$ analogously.\n\\end{definition}\n\nThe following result is well established and can be found in sections $3.1$ and $3.3$ of \\cite{ParLog}.\n\n\\begin{theorem}\nGiven formulas $\\Gamma\\cup\\{\\varphi\\}$ over the signature $\\Sigma$,\n\\begin{enumerate}\n\\item $\\Gamma\\vdash_{\\textbf{mbCciw}}\\varphi$ if and only if $\\Gamma\\vDash_{\\textbf{mbCciw}}\\varphi$;\n\\item $\\Gamma\\vdash_{\\textbf{mbCci}}\\varphi$ if and only if $\\Gamma\\vDash_{\\textbf{mbCci}}\\varphi$;\n\\item $\\Gamma\\vdash_{\\textbf{mbCcl}}\\varphi$ if and only if $\\Gamma\\vDash_{\\textbf{mbCcl}}\\varphi$.\n\\end{enumerate}\n\\end{theorem}\n\n\nNow we have the tools necessary to prove that $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$ if and only if $T(\\Gamma)\\vdash_{\\mathcal{L}^{*}}T(\\varphi)$, for any $\\mathcal{L}\\in\\{\\textbf{mbCciw}, \\textbf{mbCci}, \\textbf{mbCcl}\\}$. We will proceed by contraposition, therefore suppose $\\Gamma\\not\\vdash_{\\mathcal{L}}\\varphi$, and there exists a bivaluation $\\nu$ for $\\mathcal{L}$ such that $\\nu(\\Gamma)\\subseteq\\{1\\}$ but $\\nu(\\varphi)=0$. We define a function $\\overline{\\nu}$ from the formulas over the signature $\\Sigma_{\\nbI}$ to $\\{0,1\\}$ by structural induction.\n\n\\begin{enumerate}\n\\item If $\\alpha$ is of order $0$, and therefore a propositional variable, $\\overline{\\nu}(\\alpha)=\\nu(\\alpha)$;\n\\item $\\overline{\\nu}(\\alpha\\vee\\beta)=1$ if and only if $\\overline{\\nu}(\\alpha)=1$ or $\\overline{\\nu}(\\beta)=1$;\n\\item $\\overline{\\nu}(\\alpha\\wedge\\beta)=1$ if and only if $\\overline{\\nu}(\\alpha)=1$ and $\\overline{\\nu}(\\beta)=1$;\n\\item $\\overline{\\nu}(\\alpha\\rightarrow\\beta)=1$ if and only if $\\overline{\\nu}(\\alpha)=0$ or $\\overline{\\nu}(\\beta)=1$;\n\\item $\\overline{\\nu}(\\neg T(\\alpha))=\\nu(\\neg \\alpha)$, and if $\\alpha$ is not a translation:\n\\begin{enumerate}\n\\item for $\\mathcal{L}=\\textbf{mbCci}$ and $\\alpha=\\beta\\uparrow\\neg\\beta$, if $\\overline{\\nu}(\\beta)=0$ or $\\overline{\\nu}(\\neg \\beta)=0$, $\\overline{\\nu}(\\neg \\alpha)=0$;\n\\item for $\\mathcal{L}=\\textbf{mbCcl}$ and $\\alpha=\\beta\\wedge\\neg \\beta$, if $\\overline{\\nu}(\\beta\\uparrow\\neg\\beta)=0$, $\\overline{\\nu}(\\neg \\alpha)=0$;\n\\end{enumerate}\nif $\\alpha$ is not a translation and we find ourselves in neither of the previous two described cases, $\\overline{\\nu}(\\neg \\alpha)=1$;\n\\item $\\overline{\\nu}(T(\\alpha)\\uparrow\\neg T(\\alpha))=\\nu(\\circ\\alpha)$ and $\\overline{\\nu}(\\neg T(\\alpha)\\uparrow T(\\alpha))=\\nu(\\circ\\alpha)$; if $\\alpha$ is not a translation, $\\overline{\\nu}(\\alpha\\uparrow\\neg\\alpha)$ and $\\overline{\\nu}(\\neg\\alpha\\uparrow\\alpha)$ are $1$ if, and only if, $\\overline{\\nu}(\\alpha)=0$ or $\\overline{\\nu}(\\neg \\alpha)=0$; and if $\\alpha$ is not a translation or $\\beta$ is not the negation of $\\alpha$, $\\overline{\\nu}(\\alpha\\uparrow\\beta)=0$.\n\\end{enumerate}\n\n\\begin{proposition}\nIf $\\alpha$ is a formula over the signature $\\Sigma_{\\textbf{LFI}}$, $\\overline{\\nu}(T(\\alpha))=\\nu(\\alpha)$.\n\\end{proposition}\n\nIt is clear that $\\overline{\\nu}$ satisfies the conditions for being a bivaluation for $\\nbI$ regarding disjunction, conjunction and implication. The image under $\\overline{\\nu}$ of the negation of a formula is $0$ under the following conditions:\n\\begin{enumerate}\n\\item if the formula is the translation of an $\\alpha$ and $\\nu(\\alpha)=0$, and since $\\nu$ is a bivaluation for $\\textbf{mbC}$, we find $\\overline{\\nu}(T(\\alpha))=\\nu(\\alpha)=1$;\n\\item if the formula is not a translation and of the form $\\beta\\uparrow\\neg\\beta$, $\\mathcal{L}=\\textbf{mbCci}$ and $\\overline{\\nu}(\\beta)=0$ or $\\overline{\\nu}(\\neg\\beta)=0$, and then we obtain that $\\overline{\\nu}(\\beta\\uparrow\\neg\\beta)=1$;\n\\item  if the formula is not a translation and of the form $\\beta\\wedge\\neg\\beta$, $\\mathcal{L}=\\textbf{mbCcl}$ and $\\overline{\\nu}(\\beta\\uparrow\\neg \\beta)=0$, in which case $\\overline{\\nu}(\\beta)=\\overline{\\nu}(\\neg\\beta)=1$ and therefore $\\overline{\\nu}(\\beta\\wedge\\neg\\beta)=1$.\n\\end{enumerate}\n\nTo summarize, in all cases such that $\\overline{\\nu}(\\neg\\alpha)=0$, we have $\\overline{\\nu}(\\alpha)=1$, as it would be expected of a bivaluation for $\\nbI$ regarding the paraconsistent negation. Finally, the image under $\\overline{\\nu}$ of a formula with the incompatibility operator as leading connective is $1$ under the following conditions:\n\\begin{enumerate}\n\\item if the formula is $T(\\alpha)\\uparrow\\neg T(\\alpha)$ or $\\neg T(\\alpha)\\uparrow T(\\alpha)$ and $\\nu(\\circ\\alpha)=1$, in which case $\\nu(\\alpha)=0$ or $\\nu(\\neg\\alpha)=0$, implying either $\\overline{\\nu}(T(\\alpha))=\\nu(\\alpha)=0$ or $\\overline{\\nu}(\\neg T(\\alpha))=\\nu(\\neg\\alpha)=0$;\n\\item if the formula is not a translation and of the form $\\alpha\\uparrow\\neg\\alpha$ or $\\neg\\alpha\\uparrow\\alpha$, meaning that either $\\overline{\\nu}(\\alpha)=0$ or $\\overline{\\nu}(\\neg\\alpha)=0$.\n\\end{enumerate}\n\nVery clearly, for any two formulas over the signature $\\Sigma_{\\nbI}$ we have that $\\overline{\\nu}(\\alpha\\uparrow\\beta)=\\overline{\\nu}(\\beta\\uparrow\\alpha)$, allowing us to conclude that $\\overline{\\nu}$ is a bivaluation for $\\nbI$, remaining for us to show that for each logic, $\\textbf{mbCciw}$, $\\textbf{mbCci}$ or $\\textbf{mbCcl}$, $\\overline{\\nu}$ is a bivaluation for, respectively, $\\nbIciw$, $\\nbIci$ and $\\nbIcl$ (refer back to Definition \\ref{definition of Bivaluation for nbIciw, nbIci, ...} if necessary).\n\n\\begin{enumerate}\n\\item For any of the three logics $\\textbf{mbCciw}$, $\\textbf{mbCci}$ or $\\textbf{mbCcl}$, it is clear that $\\overline{\\nu}(\\alpha\\uparrow\\neg\\alpha)=1$ if and only if $\\overline{\\nu}(\\alpha)=0$ or $\\overline{\\nu}(\\neg \\alpha)=0$, by definition of $\\overline{\\nu}$.\n\\item If $\\mathcal{L}=\\textbf{mbCci}$, and $\\overline{\\nu}(\\neg(\\alpha\\uparrow\\neg\\alpha))=1$, this means both $\\overline{\\nu}(\\alpha)$ and $\\overline{\\nu}(\\neg \\alpha)$ are $1$ (and therefore $\\overline{\\nu}(\\alpha\\wedge\\neg\\alpha)=1$), since otherwise, by definition of $\\overline{\\nu}$, we would have $\\overline{\\nu}(\\neg(\\alpha\\uparrow\\neg\\alpha))=0$.\n\\item Finally, when $\\mathcal{L}=\\textbf{mbCcl}$, if $\\overline{\\nu}(\\neg(\\alpha\\wedge\\neg \\alpha))=1$, we have that $\\overline{\\nu}(\\alpha\\uparrow\\neg \\alpha)=1$, since otherwise, by definition of $\\overline{\\nu}$, we would have $\\overline{\\nu}(\\neg (\\alpha\\wedge\\neg \\alpha))=0$.\n\\end{enumerate}\n \nSo $\\overline{\\nu}$ is a bivaluation for $\\nbIciw$, $\\nbIci$ or $\\nbIcl$, whenever $\\mathcal{L}$ is $\\textbf{mbCciw}$, $\\textbf{mbCci}$ or $\\textbf{mbCcl}$, such that $\\overline{\\nu}(T(\\Gamma))=\\nu(\\Gamma)\\subseteq\\{1\\}$ but $\\overline{\\nu}(T(\\varphi))=\\nu(\\varphi)=0$, what implies \n\\[T(\\Gamma)\\not\\vDash_{\\nbIciw}T(\\varphi), \\quad T(\\Gamma)\\not\\vDash_{\\nbIci}T(\\varphi)\\quad \\text{and}\\quad T(\\Gamma)\\not\\vDash_{\\nbIcl}T(\\varphi),\\]\nor what is equivalent,\n\\[T(\\Gamma)\\not\\vdash_{\\nbIciw}T(\\varphi), \\quad T(\\Gamma)\\not\\vdash_{\\nbIci}T(\\varphi)\\quad \\text{and}\\quad T(\\Gamma)\\not\\vdash_{\\nbIcl}T(\\varphi);\\]\nsince $\\nbIciw$, $\\nbIci$ and $\\nbIcl$ are strictly stronger than, respectively, $\\textbf{mbCciw}^{*}$, $\\textbf{mbCci}^{*}$ and $\\textbf{mbCcl}^{*}$, this implies that $T(\\Gamma)\\not\\vdash_{\\mathcal{L}^{*}}T(\\varphi)$, and therefore we have that $T(\\Gamma)\\vdash_{\\mathcal{L}^{*}}T(\\varphi)$ implies $\\Gamma\\vdash_{\\mathcal{L}}\\varphi$, or what is equivalent,\n\\[\\Gamma\\vdash_{\\mathcal{L}}\\varphi\\quad\\text{if and only if}\\quad T(\\Gamma)\\vdash_{\\mathcal{L}^{*}}T(\\varphi),\\]\nfor any $\\mathcal{L}\\in\\{\\textbf{mbCciw}, \\textbf{mbCci}, \\textbf{mbCcl}\\}$.\n\n\n\\newpage\n\\printbibliography[segment=\\therefsegment,heading=subbibliography]\n\\end{refsegment}\n\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\n\\renewcommand{\\chaptermark}[1]{\\markboth{#1}{#1}}\n\n\\chapter*{Conclusion}\n\\chaptermark{Conclusion}\n\\addcontentsline{toc}{chapter}{Conclusion} \n\nWe believe the main developments found in this work are, in its more algebraic side, weakly free multialgebras, and, in the logical one, RNmatrices and logics of incompatibility. There were other advances, of course, however if there are topics that may still offer further fruitful research, these are probably the ones.\n\nWe now summarize what we have hoped the ideas here presented can achieve, and future works planned based on them; for clarity, we divide our text between the two broad parts in which this thesis has been divided so far.\n\n\\subsection*{Multialgebras}\n\nOur main motivation, when studying multialgebras, remains to search for a general theory of their identities, much like universal algebra is to algebras themselves. Of course, this is no easy task, but we believe weekly free multialgebras are a step in the right direction. In Chapter \\ref{Chapter2} we have shown that these structures enjoy many characterizations similar to those of absolutely free algebras, and probably are the correct generalizations of objects without identities to the context of multialgebras. We have used this to offer simplified proofs of some standard, folkloric results in the theory of non-deterministic algebras, but we still are left wondering what should identities on a multialgebra be. There are many candidates, and the task ahead is to identify which serve what purposes; but the next logical step, once we have generalized absolutely free algebras, is to generalize the relatively free ones, a closely related endeavor since ideals of identities in universal algebra fully characterize relatively free objects.\n\nShowing, in Chapter \\ref{Chapter3}, the category of multialgebras is equivalent to a category of ordered, almost Boolean algebras seems to have mainly philosophical implications, in the following sense: non-deterministic semantics are seem by many, in what we consider an unfortunately regressive point of view, as an undesirable development in the problem of decision methods for logics, precisely because of the non-determinism. Here, we offer an alternative, not only for Nmatrices but also PNmatrices and closely related semantics: they may be recast as deterministic structures, as long as we consider, along their operations, also their orders. This invites one to recast already existing decision methods that rely on non-determinism as deterministic decision methods; alternatively, one can also analyze what the equivalences presented here, and others they may lead to being discovered, can say about the involved categories.\n\n\n\\subsection*{Paraconsistent Logic}\n\nChapter \\ref{Chapter4} introduces the semantics of restricted non-deterministic matrices. We start by showing how previous semantics found in the literature, created in order to solve specific problems and without an underlying theory, can be reinterpreted as RNmatrices, what already establishes that this new tool indeed is useful. To gather further evidence for that point, we show many logics of difficult treatment, as those paraconsistent systems between $\\mbCcl$ and $\\CILA$, da Costa's hierarchy as a whole, and logics of formal incompatibility, can be relatively easily characterized by RNmatrices, with accompanying decision methods. So we hope to have convinced the reader of restricted Nmatrices' utility, but the issue of determining a theory for these semantics is still very open: can a theory like the one Blok and Pigozzi designed for the process of algebraization of logics exist for non-deterministic algebraization, specifically in our case through RNmatrices? A less ambitious objective, however, is searching for other systems of demanding characterization that can be more easily studied with RNmatrices: in addition to paraconsistent ones, modal logics seem like good candidates.\n\nStill on the subject of RNmatrices, Chapter \\ref{Chapter5} investigates how to obtain, from them, decision methods, in this case truth tables and tableaux, for da Costa's systems. We ponder if these procedures can be systematized, producing for all restricted Nmatrices, in some up to now unspecified class, row-branching, row-eliminating truth tables and tableau calculi; even more, we are interested in knowing if other varieties of decision methods, such as sequents, may be extracted from these semantics. Of course, regarding the decision methods and restricted non-deterministic matrices for da Costa's $C_{n}$ logics themselves, here suggested to be $n+2$-valued in nature, one should be curious of how they place against  existing decision methods for these systems, such as bivaluations, different tableau calculi and effective Nmatrices: are they more efficient? Less efficient? We are yet to survey these questions, although we are certain of the importance of having produced new instruments with which to approach such challenging logics.\n\nFinally, even before it is ascertained whether a theory of non-deterministic algebraization is possible, Chapter \\ref{Chapter6} starts to look into other directions in which to extend RNmatrices' analysis: by looking at swap structures, a general methodology that can be very well the most successful for finding restricted Nmatrices, we are lead to the notion of a category of RNmatrices. In the case of the systems $C_{n}$, this category is shown to be astoundingly well-behaved; in this thesis a broad procedure for creating categories of restricted non-deterministic matrices is already outlined, and we are interested in the problem of whether this construction always returns nice categories. Restricted swap structures for da Costa's hierarchy also suggest strong model theoretic connections, and the combinatorial investigation of their snapshots is a first step in this direction: a second step would be to scrutinize those results in model theory, if any, hold for arbitrary restricted swap structures.\n\nChapter \\ref{Chapter7} broaches another of the more relevant contributions of this work, that of logics of incompatibility: yes, there have been previous attempts of formalizing the natural notion of incompatible properties, more prominently in Brandom's neo-pragmatic program for epistemology, but much as da Costa's systematization of contradiction brought new light into the subject of paraconsistency, we hope that a more controlled take on incompatibility could better develop the field. After defining some simpler systems, accounting for commutativity and propagation of incompatibility, we supply them with semantics, both based on bivaluations and RNamtrices, as well as decision methods constructed from the aforementioned RNmatrices; this gives additional evidence of the importance of restricted Nmatrices. In future developments, one hopes to define other systems, possibly modeling additional attributes of incompatible properties in natural language or including quantifiers and modalities.\n\nOne possible angle for studying logics of incompatibility is adding to them a negation, with which the incompatibility can interact, task we undertake in some depth in Chapter \\ref{Chapter8}. Most of the systems we define are based in others for paraconsistency, and are dealt with by using bivaluations and RNmatrices, obtaining thus decision methods again through row-branching, row-eliminating truth tables and tableaux. And, what is formally proven in Chapter \\ref{Chapter9} but heavily insinuated since their definitions, these logics extend those paraconsistent ones upon which they are built in a non-trivial way: that is, we have conservative translations from the latter into sublogics of the former; this means our logics of incompatibility are strictly, and non trivially, generalizing logics of formal inconsistency, and invites the interested to find the most appropriate translations for inconsistent systems, a line of future research. Furthermore, by providing characterizing RNmatrices for logics of incompatibility with negation, we feel compelled to prove that more classical semantics such as Blok and Pigozzi's algebraization, Nmatrices, and restricted matrices would not work. This means we have offered logics very simple to define, yet very difficult to characterize, and leads us to wonder about the interplay between competing generalizations of logical matrices: restricted matrices, to give one example, seem very promising although virtually unstudied.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{refsegment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\thispagestyle{plain}\n\\hspace{0pt}\n\\vfill\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{.\/Image3}\n\\caption*{Mountainous region of Parque Nacional do Itatiaia, Itatiaia, Brazil.\\\\Photographed by Guilherme Vicentin de Toledo, all rights reserved.}\n\\end{figure}\n\\vfill\n\\hspace{0pt}\n\n\\cleardoublepage\n\n\\phantomsection\n\n\n\n\\begin{refsegment}\n\\defbibfilter{notother}{not segment=\\therefsegment}\n\n\\chapter*{Lists}\n\\markboth{Lists}{}\n\\addcontentsline{toc}{chapter}{Lists}\n\nListed by order of appearance.\n\n\\section*{List of symbols}\n\n\\begin{longtable}{c | m{11cm} | c}\n\nSymbols & Meaning & Page \\\\ \\hline\n\n$\\mathcal{P}(A)$ & Powerset of the set $A$ & \\pageref{powerset}\\\\\n\n$\\varphi:\\mathcal{A}\\rightarrow\\mathcal{B}$ & Homomorphism from $\\mathcal{A}$ to $\\mathcal{B}$ & \\pageref{homomorphism}\\\\\n\n$F(\\Sigma, \\mathcal{V})$ & Formulas, on the signature $\\Sigma$, over the variables $\\mathcal{V}$ & \\pageref{FSigmaV} \\\\\n\n$\\textbf{F}(\\Sigma, \\mathcal{V})$ & $\\Sigma$-Algebra of formulas, on the signature $\\Sigma$, over the variables $\\mathcal{V}$ & \\pageref{tFSigmaV} \\\\\n\n$|\\alpha|$ & Order of the formula $\\alpha$ & \\pageref{order}\\\\\n\n$\\Sigma^{\\kappa}$ & Expanded signature & \\pageref{expandedsignature}\\\\\n\n$\\textbf{mF}(\\Sigma, \\mathcal{V}, \\kappa)$ & $\\Sigma$-Multialgebra of non-deterministic formulas, on the signature $\\Sigma$, over the variables $\\mathcal{V}$ and bounded by $\\kappa$ & \\pageref{mFSigmakappaV}\\\\\n\n$|X|$ & Cardinality of the set $X$ & \\pageref{|X|}\\\\\n\n$\\textbf{cdf}$-generated & Choice-dependent freely generated & \\pageref{cdfgenerated}\\\\\n\n$B(\\mathcal{A})$ & Build of the multialgebra $\\mathcal{A}$ & \\pageref{build}\\\\\n\n$G(\\mathcal{A})$ & Ground of the multialgebra $\\mathcal{A}$ & \\pageref{ground}\\\\\n\n$\\langle S\\rangle_{m}$ & The $m$-th set obtained in the process of defining $\\langle S\\rangle$ & \\pageref{mth generated}\\\\\n\n$\\langle S\\rangle$ & Set generated by $S$ & \\pageref{generated}\\\\\n\n$M(\\mathcal{A})$ & Supremum of the cardinalities of the operations on the multialgebra $\\mathcal{A}$ & \\pageref{M(A)}\\\\\n\n$o_{B}(a)$ & $B$-order of the element $a$ of a multialgebra with strong basis $B$ & \\pageref{Border}\\\\\n\n$\\mathcal{A}_{0}$ & Poset obtained from the poset $\\mathcal{A}$ by addition of a minimum element & \\pageref{A0}\\\\\n\n$\\mathcal{L}$ & Propositional language generated by $\\Sigma$ & \\pageref{language}\\\\\n\n$\\Gamma\\vdash\\varphi$ & $\\Gamma$ syntactically proves $\\varphi$ & \\pageref{vdash}\\\\\n\n$\\mathscr{L}$ & An arbitrary logic & \\pageref{logic}\\\\\n\n$\\alpha_{1}, \\dotsc , \\alpha_{n}|\\alpha$ & Rule of inference where $\\{\\alpha_{1},\\dotsc , \\alpha_{n}\\}$ deduces $\\alpha$ & \\pageref{ruleofinference}\\\\\n\n$\\mathfrak{A}$ & A set of axiom schemata & \\pageref{frakA}\\\\\n\n$\\mathfrak{R}$ & A set of rules of inference & \\pageref{frakR}\\\\\n\n$\\psi(p_{1}, \\dotsc , p_{n})$ & A formula $\\psi$ whose variables are among $\\{p_{1}, \\dotsc , p_{n}\\}$ & \\pageref{varformula}\\\\\n\n$\\circ\\alpha$ & Formula stating the consistency of $\\alpha$ & \\pageref{circ}\\\\\n\n$\\Gamma\\vDash_{\\mathcal{M}}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$ according to $\\mathcal{M}$ & \\pageref{semant.proves}\\\\\n\n$\\Gamma\\vDash_{\\mathbb{M}}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$ according to the class $\\mathbb{M}$ & \\pageref{vDashM}\\\\\n\n$\\textbf{2}(\\Sigma)$ & $\\Sigma$-multialgebra with universe $\\{0,1\\}$ whose operations always return $\\{0,1\\}$ & \\pageref{2Sigma}\\\\\n\n$\\textbf{2}(\\mathfrak{L})$ & RNmatrix, with multialgebra $\\textbf{2}(\\Sigma)$, characterizing $\\mathfrak{L}$ & \\pageref{2L}\\\\\n\n$K_{\\mathcal{M}}(\\Gamma)$ & Logical closure, according to $\\mathcal{M}$, of $\\Gamma$ & \\pageref{KMGamma}\\\\\n\n$K_{\\mathbb{M}}(\\Gamma)$ & Logical closure, according to the class $\\mathcal{M}$, of $\\Gamma$ & \\pageref{KMMGamma}\\\\\n\n$\\mathcal{K}_{\\textbf{L}}$ & Kearns RNmatrix for the logic $\\textbf{L}$ & \\pageref{KL}\\\\\n\n$N_{a}$ & Possible negations of $a$ in a Fidel structure & \\pageref{Na}\\\\\n\n$\\mathcal{M}^{s}$ & Static semantics for the Nmatrix $\\mathcal{M}$ & \\pageref{Ms}\\\\\n\n$\\mathcal{M}^{\\emptyset}$ & RNmatrix equivalent to the PNmatrix $\\mathcal{M}$ & \\pageref{Mo}\\\\\n\n$\\mathcal{M}_{\\textbf{mbCciw}}$ & Nmatrix characterizing $\\textbf{mbCciw}$ & \\pageref{MmbCciw}\\\\\n\n$\\mathcal{M}_{\\textbf{mbCcl}}$ & RNmatrix characterizing $\\textbf{mbCcl}$ & \\pageref{MmbCcl}\\\\\n\n$\\mathcal{M}_{\\textbf{CILA}}$ & RNmatrix characterizing $\\textbf{CILA}$ & \\pageref{MCILA}\\\\\n\n$\\alpha^{n}$ & $\\alpha^{0}=\\alpha$ and $\\alpha^{n+1}=\\neg(\\alpha^{n}\\wedge\\neg\\alpha^{n})$ & \\pageref{alpha^}\\\\\n\n$\\alpha^{(n)}$ & $\\alpha^{(1)}=\\alpha^{1}$ and $\\alpha^{(n+1)}=\\alpha^{n+1}\\wedge\\alpha^{(n)}$ & \\pageref{alpha^()}\\\\\n\n$\\alpha^{\\circ}$ & Equal to $\\alpha^{1}$ & \\pageref{alphacirc}\\\\\n\n$T_{n}$ & The snapshot $(1, 0, 1, \\dotsc , 1)$ & \\pageref{Tn}\\\\\n\n$F_{n}$ & The snapshot $(0, 1, 1, \\dotsc , 1)$ & \\pageref{Fn}\\\\\n\n$B_{n}$ & Set of snapshots for $C_{n}$ & \\pageref{Bn}\\\\\n\n$D_{n}$ & Set of designated snapshots for $C_{n}$ & \\pageref{Dn}\\\\\n\n$Boo_{n}$ & Set of Boolean snapshots for $C_{n}$ & \\pageref{Boon}\\\\\n\n$\\mathcal{A}_{C_{n}}$ & Swap structure for $C_{n}$ & \\pageref{ACn}\\\\\n\n$\\mathcal{F}_{C_{n}}$ & Set of restricted valuations for $C_{n}$ & \\pageref{FCn}\\\\\n\n$\\mathcal{RM}_{C_{n}}$ & RNmatrix characterizing $C_{n}$ & \\pageref{RMCn}\\\\\n\n$\\mathbb{T}_{n}$ & Tableau calculus for $C_{n}$ & \\pageref{mathbbTn}\\\\\n\n$\\Gamma\\vdash_{\\mathbb{T}_{n}}\\varphi$ & $\\Gamma$ proves $\\varphi$ according to the tableau calculus $\\mathbb{T}_{n}$ & \\pageref{vdashmathbbTn}\\\\\n\n$\\Gamma\\vDash_{C_{n}}^{\\mathcal{B}}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$, according to $\\mathcal{B}$-valuations & \\pageref{vDashBCn}\\\\\n\n$B_{n}^{\\mathcal{B}}$ & Set of snapshots, over $\\mathcal{B}$, for $C_{n}$ & \\pageref{BnB}\\\\\n\n$D_{n}^{\\mathcal{B}}$ & Set of designated snapshots, over $\\mathcal{B}$, for $C_{n}$ & \\pageref{DnB}\\\\\n\n$Boo_{n}^{\\mathcal{B}}$ & Set of Boolean snapshots, over $\\mathcal{B}$, for $C_{n}$ & \\pageref{BoonB}\\\\\n\n$\\mathcal{A}_{C_{n}}^{\\mathcal{B}}$ & Swap structure, over $\\mathcal{B}$, for $C_{n}$ & \\pageref{ACnB}\\\\\n\n$\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ & RNmatrix, over $\\mathcal{B}$, characterizing $C_{n}$ & \\pageref{RMCnB}\\\\\n\n$\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ & Set of restricted valuations, over $\\mathcal{B}$, for $C_{n}$ & \\pageref{FCnB}\\\\\n\n$\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}}^{RN}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$, according to $\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}$ & \\pageref{vDashRNRM}\\\\\n\n$\\Gamma\\vDash_{\\mathcal{RS}_{C_{n}}}^{RN}\\varphi$ & For every Boolean algebra $\\mathcal{B}$, $\\Gamma\\vDash_{\\mathcal{RM}_{C_{n}}^{\\mathcal{B}}}^{RN}\\varphi$ & \\pageref{vDashRNRS}\\\\\n\n$|\\mathcal{B}|$ & Universe of the Boolean algebra $\\mathcal{B}$ & \\pageref{|B|}\\\\\n\n$X_{m}$ & Generic set with $m$ elements & \\pageref{Xm}\\\\\n\n$\\pi_{i}$ & Projection from a product to its $i$-th coordinate & \\pageref{pii}\\\\\n\n$\\alpha\\uparrow\\beta$ & $\\alpha$ and $\\beta$ are incompatible & \\pageref{uparrow}\\\\\n\n$\\alpha\\downarrow\\beta$ & $\\alpha$ and $\\beta$ are compatible & \\pageref{downarrow}\\\\\n\n$\\Gamma\\vDash_{\\bI}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$, according to bivaluations for $\\textbf{bI}$ & \\pageref{vDashbI}\\\\\n\n$\\Gamma\\Vdash_{\\mathcal{F}}^{\\bI}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$, according to Fidel structures for $\\textbf{bI}$ & \\pageref{VdashFbI}\\\\\n\n$\\alpha\\equiv_{\\Gamma}^{\\bI}\\beta$ & $\\alpha$ and $\\beta$ are equivalent, according to $\\Gamma$, in $\\bI$ & \\pageref{equivbI}\\\\\n\n$A^{\\bI}_{\\Gamma}$ & The quotient set of $F(\\Sigma_{\\bI}, \\mathcal{V})$ by $\\equiv_{\\Gamma}^{\\bI}$ & \\pageref{AbIGamma}\\\\\n\n$\\mathcal{A}^{\\bI}_{\\Gamma}$ & The Lindenbaum-Tarski multialgebra of $\\bI$ associated to $\\Gamma$ & \\pageref{mAbIGamma}\\\\\n\n$\\textbf{2}_{\\bI}$ & Two-valued Fidel structure for $\\bI$ & \\pageref{2bI}\\\\\n\n$\\Gamma\\vDash_{\\bIpr}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$, according to bivaluations for $\\textbf{bIpr}$ & \\pageref{vDashbIpr}\\\\\n\n$\\Gamma\\Vdash_{\\mathcal{F}}^{\\bIpr}\\varphi$ & $\\Gamma$ semantically proves $\\varphi$, according to Fidel structures for $\\textbf{bIpr}$ & \\pageref{VdashFbIpr}\\\\\n\n$\\textbf{2}_{\\bIpr}$ & Two-valued Fidel structure for $\\bIpr$ & \\pageref{2bIpr}\\\\\n\n$\\Gamma\\vDash_{\\nbI}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to bivaluations for $\\nbI$ & \\pageref{vDashnbI}\\\\\n\n$\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbI}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to Fidel structures for $\\nbI$ & \\pageref{VdashFnbI}\\\\\n\n$\\alpha\\equiv_{\\Gamma}^{\\nbI}\\beta$ & $\\alpha$ and $\\beta$ are equivalent, according to $\\Gamma$, in $\\nbI$ & \\pageref{equivnbI}\\\\\n\n$\\mathcal{A}^{\\nbI}_{\\Gamma}$ & The Lindebaum-Tarski multialgebra of $\\nbI$ associated to $\\Gamma$ & \\pageref{AnbIGamma}\\\\\n\n$\\textbf{2}_{\\nbI}$ & Two-valued Fidel structure for $\\nbI$ & \\pageref{2nbI}\\\\\n\n$\\Gamma\\vDash_{\\nbIciw}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to bivaluations for $\\nbIciw$ & \\pageref{vDashnbIciw}\\\\\n\n$\\Gamma\\vDash_{\\nbIci}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to bivaluations for $\\nbIci$ & \\pageref{vDashnbIci}\\\\\n\n$\\Gamma\\vDash_{\\nbIcl}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to bivaluations for $\\nbIcl$ & \\pageref{vDashnbIcl}\\\\\n\n$\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbIciw}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to Fidel structures for $\\nbIciw$ & \\pageref{VdashFnbIciw}\\\\\n\n$\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbIci}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to Fidel structures for $\\nbIci$ & \\pageref{VdashFnbIci}\\\\\n\n$\\Gamma\\Vdash_{\\mathcal{F}}^{\\nbIcl}\\varphi$ & $\\Gamma$ proves $\\varphi$, according to Fidel structures for $\\nbIcl$ & \\pageref{VdashFnbIcl}\\\\\n\n$\\nabla$ & Trivial congruence on a multialgebra $\\mathcal{A}$, equal to $A\\times A$ & \\pageref{nabla}\\\\\n\n$\\Delta$ & Trivial congruence on a multialgebra $\\mathcal{A}$, equal to $\\{(a,a) : a\\in A\\}$ & \\pageref{Delta}\\\\\n\n$T$ & Translation from the $\\textbf{LFI}$'s to logics of incompatibility & \\pageref{T}\\\\\n\n\n\n\n\\end{longtable}\n\n\n\n\n\n\n\n\n\\newpage\n\n\\section*{List of logics and axioms}\n\n\\begin{longtable}{c | m{11cm} | c}\n\nSymbols & Meaning & Page \\\\ \\hline\n\n$\\textbf{CPL}$ & Classical propositional logic & \\pageref{CPL}\\\\\n\n$\\textbf{LFI}$ & Logics of formal inconsistency & \\pageref{LFI}\\\\\n\n$\\textbf{mbC}$ & One of the simplest $\\textbf{LFI}$'s & \\pageref{mbC}\\\\\n\n$\\textbf{mbCciw}$ & Logic obtained from $\\textbf{mbC}$ by addition of $\\textbf{ciw}$ & \\pageref{mbCciw}\\\\\n\n$\\textbf{ciw}$ & Axiom scheme $\\circ\\alpha\\vee(\\alpha\\wedge\\neg\\alpha)$ & \\pageref{ciw}\\\\\n\n$\\textbf{mbCci}$ & Logic obtained from $\\textbf{mbC}$ by addition of $\\textbf{ci}$ & \\pageref{mbCci}\\\\\n\n$\\textbf{ci}$ & Axiom scheme $\\neg \\circ\\alpha\\rightarrow(\\alpha\\wedge\\neg \\alpha)$ & \\pageref{ci}\\\\\n\n$\\textbf{mbCcl}$ & Logic obtained from $\\textbf{mbC}$ by addition of $\\textbf{cl}$ & \\pageref{mbCcl}\\\\\n\n$\\textbf{cl}$ & Axiom scheme $\\neg(\\alpha\\wedge\\neg \\alpha)\\rightarrow\\circ\\alpha$ & \\pageref{cl}\\\\\n\n$\\textbf{B}[\\{\\textbf{i1}, \\textbf{i2}\\}]$ & System, defined by Avron, equivalent to $\\textbf{mbCci}$ & \\pageref{Bi1i2}\\\\\n\n$\\textbf{Bi}$ & System, defined by Avron, equivalent to $\\textbf{mbCci}$ & \\pageref{Bi}\\\\\n\n$\\textbf{Bl}$ & System, defined by Avron, equivalent to $\\textbf{mbCcl}$ & \\pageref{Bl}\\\\\n\n$\\textbf{CILA}$ & An $\\textbf{LFI}$ equivalent to da Costa's $C_{1}$ & \\pageref{CILA}\\\\\n\n$\\textbf{cf}$ & Axiom scheme $\\neg\\neg\\alpha\\rightarrow\\alpha$ & \\pageref{cf}\\\\\n\n$\\textbf{Ci}$ & Logic obtained from $\\textbf{mbCci}$ by addition of $\\textbf{cf}$ & \\pageref{Ci}\\\\\n\n$C_{\\omega}$ & An intuitionistic and paraconsistent system by da Costa & \\pageref{Comega}\\\\\n\n$C_{n}$ & The $n$-th logic in da Costa's hierarchy & \\pageref{Cn}\\\\\n\n$\\textbf{bc}_{n}$ & Axiom scheme $\\alpha^{(n)}\\rightarrow(\\alpha\\rightarrow(\\neg\\alpha\\rightarrow\\beta))$ & \\pageref{bcn}\\\\\n\n$\\textbf{p}\\#_{n}$ & Axiom scheme $(\\alpha^{(n)}\\#\\beta^{(n)})\\rightarrow(\\alpha\\vee\\beta)^{(n)}$ & \\pageref{phashn}\\\\\n\n$\\textbf{LIp}$ & Logic of formal incompatibility & \\pageref{LIp}\\\\\n\n$\\textbf{bI}$ & A basic logic of incompatibility, with commutativity & \\pageref{bI}\\\\\n\n$\\textbf{Ip}$ & Axiom scheme $(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\rightarrow(\\beta\\rightarrow \\gamma))$ & \\pageref{Ip}\\\\\n\n$\\textbf{Comm}$ & Axiom scheme $(\\alpha\\uparrow\\beta)\\rightarrow(\\beta\\uparrow\\alpha)$ & \\pageref{Comm}\\\\\n\n$\\textbf{bI}^{-}$ & The logic $\\textbf{bI}$ without commutativity & \\pageref{bI-}\\\\\n\n$\\textbf{bIpr}$ & Logic obtained from $\\bI$ by addition of $\\textbf{pr}_{\\vee}$ and $\\textbf{pr}_{\\wedge}$ & \\pageref{bIpr}\\\\\n\n$\\textbf{pr}_{\\wedge}$ & Axiom scheme $[(\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma)]\\rightarrow[(\\alpha\\vee\\beta)\\uparrow\\gamma]$ & \\pageref{prwedge}\\\\\n\n$\\textbf{pr}_{\\vee}$ & Axiom scheme $[(\\alpha\\uparrow\\gamma)\\wedge(\\beta\\uparrow\\gamma)]\\rightarrow[(\\alpha\\vee\\beta)\\uparrow\\gamma]$ & \\pageref{prvee}\\\\\n\n$\\textbf{Ex}$ & Axiom scheme $(\\alpha\\wedge\\beta\\rightarrow\\bot_{\\alpha\\beta})\\rightarrow(\\alpha\\uparrow\\beta)$ & \\pageref{Ex}\\\\\n\n$\\bI\\textbf{Ex}$ & Logic obtained from $\\bI$ by addition of $\\textbf{Ex}$ & \\pageref{bIEx}\\\\\n\n$\\textbf{ciw}^{\\uparrow}$ & Axiom scheme $(\\alpha\\uparrow\\beta)\\vee(\\alpha\\wedge\\beta)$ & \\pageref{ciwuparrow}\\\\\n\n$\\bI\\textbf{ciw}^{\\uparrow}$ & Logic obtained from $\\bI$ by addition of $\\textbf{ciw}^{\\uparrow}$ & \\pageref{bIciw}\\\\\n\n$\\nbI$ & Logic obtained from $\\bI$ by addition of the axiom scheme $\\alpha\\vee\\neg\\alpha$ & \\pageref{nbI}\\\\\n\n$\\textbf{ci}^{\\uparrow}$ & Axiom scheme $\\neg(\\alpha\\uparrow\\beta)\\rightarrow(\\alpha\\wedge\\beta)$ & \\pageref{ciuparrow}\\\\\n\n$\\textbf{cl}^{\\uparrow}$ & Axiom scheme $\\neg(\\alpha\\wedge\\beta)\\rightarrow(\\alpha\\uparrow\\beta)$ & \\pageref{cluparrow}\\\\\n\n$\\nbIciw$ & Logic obtained from $\\nbI$ by addition of the axiom scheme $\\textbf{ciw}^{*}$ & \\pageref{nbIciw}\\\\\n\n$\\textbf{ciw}^{*}$ & Axiom scheme $(\\alpha\\uparrow\\neg\\alpha)\\vee(\\alpha\\wedge\\neg\\alpha)$ & \\pageref{ciw*}\\\\\n\n$\\nbIci$ & Logic obtained from $\\nbI$ by addition of the axiom scheme $\\textbf{ci}^{*}$ & \\pageref{nbIci}\\\\\n\n$\\textbf{ci}^{*}$ & Axiom scheme $\\neg(\\alpha\\uparrow\\neg\\alpha)\\rightarrow(\\alpha\\wedge\\neg\\alpha)$ & \\pageref{ci*}\\\\\n\n$\\nbIcl$ & Logic obtained from $\\nbI$ by addition of the axiom scheme $\\textbf{cl}^{*}$ & \\pageref{nbIcl}\\\\\n\n$\\textbf{cl}^{*}$ & Axiom scheme $\\neg(\\alpha\\wedge\\neg\\alpha)\\rightarrow(\\alpha\\uparrow\\neg\\alpha)$ & \\pageref{cl*}\\\\\n\n$\\textbf{cc}^{*}$ & Axiom scheme $(\\alpha\\uparrow\\neg\\alpha)\\uparrow\\neg(\\alpha\\uparrow\\neg\\alpha)$ & \\pageref{cc*}\\\\\n\n$\\mathcal{L}^{*}$ & Translation of the $\\textbf{LFI}$ logic $\\mathcal{L}$ to an incompatibility logic & \\pageref{L*}\\\\\n\n$\\textbf{mbC}^{*}$ & Translation of $\\textbf{mbC}$ & \\pageref{mbC*}\\\\\n\n$\\textbf{mbCciw}^{*}$ & Translation of $\\textbf{mbCciw}$ & \\pageref{mbCciw*}\\\\\n\n$\\textbf{mbCci}^{*}$ & Translation of $\\textbf{mbCci}$ & \\pageref{mbCci*}\\\\\n\n$\\textbf{mbCcl}^{*}$ & Translation of $\\textbf{mbCcl}$ & \\pageref{mbCcl*}\\\\\n\n\n\\end{longtable}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\n\n\\section*{List of categories and functors}\n\n\\begin{longtable}{c | m{11cm} | c}\n\nSymbols & Meaning & Page \\\\ \\hline\n\n$\\textbf{MAlg}$ & Category of $\\Sigma$-multialgebras equipped with homomorphisms & \\pageref{MAlg}\\\\\n\n$\\textbf{MAlg}_{=}$ & Category of $\\Sigma$-multialgebras equipped with full homomorphisms & \\pageref{MAlg=}\\\\\n\n$\\textbf{MMAlg}$ & Category of $\\Sigma$-multialgebras equipped with multihomomorphisms & \\pageref{MMAlg}\\\\\n\n$\\textbf{MMAlg}_{=}$ & Category of $\\Sigma$-multialgebras equipped with full multihomomorphisms & \\pageref{MMAlg=}\\\\\n\n$\\textbf{MSet}$ & Category of sets equipped with multifunctions & \\pageref{MSet}\\\\\n\n$\\textbf{MG}(\\Sigma)$ & Category of $\\Sigma$-multialgebras equipped with classes of equivalence of ground-preserving homomorphisms & \\pageref{MGSigma}\\\\\n\n$\\textbf{CABA}$ & Category of complete, atomic Boolean algebras, equipped with continuous homomorphisms & \\pageref{CABA}\\\\\n\n$\\textbf{Alg}(\\Sigma)$ & Category of $\\Sigma$-algebras equipped with homomorphisms & \\pageref{AlgSigma}\\\\\n\n$\\mathcal{P}_{\\mathcal{B}}$ & Functor from $\\textbf{MAlg}(\\Sigma)$ to $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ & \\pageref{Functor PB}\\\\\n\n$\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ & Category of $(\\Sigma, \\leq)$-algebras equipped with $(\\Sigma, \\leq)$-homomorphisms & \\pageref{AlgBSigma}\\\\\n\n$P$ & Endofunctor on $\\textbf{MAlg}(\\Sigma)$ that is part of a monad &\\pageref{P}\\\\\n\n$\\mathbb{A}$ & Functor from $\\textbf{Alg}_{\\mathcal{B}}(\\Sigma)$ to $\\textbf{MAlg}(\\Sigma)$ & \\pageref{A}\\\\\n\n$\\textbf{Set}^{*}$ & Category of non-empty sets equipped with functions & \\pageref{Set*}\\\\\n\n$\\textbf{CABo}$ & Category of complete, atomic and bottomless Boolean algebras equipped with continuous, atoms-preserving functions & \\pageref{CABo}\\\\\n\n$\\textbf{PMAlg}(\\Sigma)$ & Category of partial multialgebras equipped with homomorphisms & \\pageref{PMAlgSigma}\\\\\n\n$\\textbf{Alg}_{CABA}(\\Sigma)$ & Category of $\\Sigma$-algebras, with a Boolean algebra structure, equipped with homomorphisms that are continuous and atoms-preserving & \\pageref{AlgCABASigma}\\\\\n\n$\\textbf{MAlg}_{\\mathcal{B}}(\\Sigma)$ & Category of $(\\Sigma, \\leq)$-algebras equipped with continuous almost-homomorphisms & \\pageref{MAlgBSigma}\\\\\n\n$\\textbf{RSwap}_{C_{n}}$ & Category of swap structures for $C_{n}$, equipped with homomorphisms commuting with $\\mathcal{F}_{C_{n}}^{\\mathcal{B}}$ & \\pageref{RSwapCn}\\\\\n\n$\\textbf{BA}$ & Category of non-degenerate Boolean algebras, equipped with homomorphisms & \\pageref{BA}\\\\\n\n$\\mathcal{A}_{n}$ & Functor from $\\textbf{BA}$ to $\\textbf{RSwap}_{C_{n}}$ & \\pageref{An}\\\\\n\n$\\textbf{Boo}_{n}$ & Functor, inverse of $\\mathcal{A}_{n}$ & \\pageref{tBoon}\\\\\n\n\n\n\n\n\\end{longtable}\n\n\\newpage\n\n\\section*{List of signatures}\n\n\\begin{longtable}{c | m{11cm} | c}\n\nSymbols & Meaning & Page \\\\ \\hline\n\n$\\Sigma$ & Arbitrary signature & \\pageref{signature}\\\\\n\n$\\Sigma_{\\textbf{Lat}}$ & Signature with $\\vee$ and $\\wedge$ as symbols & \\pageref{SigmaLat}\\\\\n\n$\\Sigma_{\\textbf{Lat}}^{0}$ & Signature with $0, \\vee$ and $\\wedge$ as symbols & \\pageref{SigmaLat0}\\\\\n\n$\\Sigma_{\\textbf{Lat}}^{1}$ & Signature with $1, \\vee$ and $\\wedge$ as symbols & \\pageref{SigmaLat1}\\\\\n\n$\\Sigma_{\\textbf{Lat}}^{0,1}$ & Signature with $0, 1, \\vee$ and $\\wedge$ as symbols & \\pageref{SigmaLat01}\\\\\n\n$\\Sigma_{\\textbf{Imp}}$ & Signature with $1, \\vee, \\wedge$ and $\\rightarrow$ as symbols & \\pageref{SigmaImp}\\\\\n\n$\\Sigma_{\\textbf{Hey}}$ & Signature with $0, 1, \\vee, \\wedge$ and $\\rightarrow$ as symbols & \\pageref{SigmaHey}\\\\\n\n$\\Sigma_{\\textbf{Boo}}$ & Signature with $0, 1, \\neg, \\vee, \\wedge$ and $\\rightarrow$ as symbols & \\pageref{SigmaBoo}\\\\\n\n$\\Sigma_{s}$ & Signature with an unary $s$ as only symbol & \\pageref{Sigmas}\\\\\n\n$\\Sigma_{\\textbf{CPL}}$ & Signature with $\\bot, \\top, \\sim, \\vee, \\wedge$ and $\\rightarrow$ as connectives & \\pageref{SigmaCPL}\\\\\n\n$\\Sigma_{\\textbf{LFI}}$ & Signature with $\\neg, \\circ, \\vee, \\wedge$ and $\\rightarrow$ as connectives & \\pageref{SigmaLFI}\\\\\n\n$\\Sigma^{\\top}$ & Signature obtained from $\\Sigma$ by addition of the $0$-ary connective $\\top$ & \\pageref{Sigmatop}\\\\\n\n$\\Sigma_{\\textbf{C}}$ & Signature with $\\neg, \\vee, \\wedge$ and $\\rightarrow$ as connectives & \\pageref{SigmaC}\\\\\n\n$\\Sigma_{\\textbf{bI}}$ & Signature with $\\vee, \\wedge, \\rightarrow$ and $\\uparrow$ as connectives & \\pageref{SigmabI}\\\\\n\n$\\Sigma_{\\bI}^{\\textbf{CPL}}$ & Signature with $\\bot, \\top, \\sim, \\vee, \\wedge, \\rightarrow$ and $\\uparrow$ as connectives &\\pageref{SigmabICPL}\\\\\n\n$\\Sigma_{\\nbI}$ & Signature with $\\neg, \\vee, \\wedge, \\rightarrow$ and $\\uparrow$ as connectives & \\pageref{SigmanbI}\\\\\n\n$\\Sigma_{\\nbI}^{\\textbf{CPL}}$ & Signature with $\\bot, \\top, \\sim, \\neg, \\vee, \\wedge, \\rightarrow$ and $\\uparrow$ as connectives & \\pageref{SigmanbICPL}\\\\\n\n$\\Sigma_{\\textbf{LFI}}^{\\textbf{CPL}}$ & Signature with $\\bot, \\top, \\circ, \\sim, \\neg, \\vee, \\wedge$ and $\\rightarrow$ as connectives & \\pageref{SigmaLFICPL}\\\\\n\n\n\\end{longtable}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{refsegment}\n\n\n\n\n\n\\printindex\n\n\\addcontentsline{toc}{chapter}{Bibliography}\n\\nocite{*}\n\\printbibliography\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction}\nOver the last few decades, various quantum systems, including superconducting\ncircuits, neutral atoms, trapped ions, and spins \\cite{Clarke:2008ul, HAFFNER2008155, Greiner:2002wt},\nhave shown exciting progress in controlling quantum effects for applications in\nquantum\nsensors \\cite{Gross:2010uf}, simulators \\cite{Bloch:2012ty}, and\ncomputers \\cite{Kielpinski:2002ua}.\nIn these setups, quantum operations are implemented using external fields or pulses\nwhich are\ngenerated and influenced by several electronic and optical devices.\nFor high-fidelity and uptime applications, this requires high performance of, e.g., population transfers\nand quantum gates, while suppressing interactions with the\nenvironment as well as decoherence. By shaping temporal and spatial profiles of external\nfields and pulses, the time-dependent system Hamiltonian steers the\nquantum dynamics towards the targeted outcome.\n\nExperimental distortions of the applied pulses may reduce\nthe effectiveness and robustness of the desired quantum operation \\cite{PhysRevA.84.022307,PhysRevLett.117.260501}.\nMethods have\nbeen developed to characterize distortions based on the\nimpulse response or transfer function of the experimental system\n\\cite{Skinner2010,PhysRevA.84.022307, PhysRevLett.110.040502,\nSPINDLER201249,Kaufmann2013,Spindler2017,Rife1989419,PhysRevLett.117.260501,le2022analytic}.\nThese approaches for estimating field distortions\nwork well for distortions with a linear transfer function.\nThis work, however, addresses the more general case\nwith substantial non-linear distortions originating from the experimental\nhardware.\n\nThe description of the distortions can be challenging without\nknowing the exact characteristics of the experimental hardware.\nAlso, approximating a significant non-linearity\nusing a linear model will result in\nmodel coefficients and control pulses that\nare not robust against experimental distortions and suffer from a loss in fidelity. To\naccount for this problem, we introduce a mathematical model and an estimation method\nwhich rely on limited experimental data\nand can characterize the system behavior up to a non-linearity of finite order. To streamline our presentation, we focus\non quadratic non-linearities, but more general non-linearities\ncan be treated similarly.\nWe illustrate our estimation approach\nwith numerical data for a single-Rydberg atom excitation experiment in the presence of significant\nnon-linearities\nand we highlight how our approach can calibrate for and suppress\nlarge distortions.\nWe\ndescribe an effective approach for estimating the coefficients of this non-linear\nmodel and correct the pulses accordingly.\nWe emphasize that our approach is independent\nof a specific experimental setup and can therefore be applied to various (spatially or temporally)\nfield-tunable phenomena on different quantum platforms.\n\nOur estimation method for distortions is particularly effective in combination\nwith methods from quantum optimal control\n\\cite{Brif_2010,Glaser:2015tm,schulte2018,dalessandro2021introduction,Koch2022} and\nit yields optimized pulses for highly efficient gates while accounting\nfor estimated distortions. To this end, we provide an analytical\nexpression for estimating the Jacobian of the transfer function for\nquadratic distortions, which can be further generalized to higher orders.\nWe also validate this combined approach with our Rydberg atom excitation example.\nIn the context of quantum control,\nany inaccuracy in the system Hamiltonian can severely affect the performance of pulses produced by optimal control.\nGiven a reasonably accurate model, control fields might also suffer from discretization effects,\nelectronic distortions, and bandwidth limitations (mostly assumed to be linear).\nAccounting for these distortions by including the linear transfer\nfunction within the dynamics, as well as its combined gradient, has been incorporated in\nrelated optimization work\n\\cite{PhysRevA.84.022307,hincks2015controlling,le2022analytic,PhysRevA.88.035601,PhysRevApplied.15.034080}.\nAnother strategy\nfor minimizing non-linear pulse distortions\nis to avoid high frequencies altogether in control pulses \\cite{sorensen2020optimization, PhysRevA.98.022119}.\n\nStarting from initial applications \\cite{4307754,PhysRevA.37.4950},\noptimal control methods\nhave been extensively used\nin quantum computing, quantum simulation, and quantum\ninformation processing\n\\cite{Glaser:2015tm,Koch2022,PhysRevB.74.161307,khani2009optimal, goerz2014robustness, doi:10.1126\/science.aax9743}.\nAnalytic results applicable to smaller quantum systems\nshape our understanding for the limits\nto population transfers and quantum gates (see\n\\cite{Glaser:2015tm,Koch2022,Khaneja2001,KGB02,BCL+02,VHC02,Zeier2004,CHKO06,KHSY07,\nYZK08,ZYK:2008,CHKO11,NZN12,CK12,VZGS14,PhysRevA.92.053414,freeman1998spin,\nguery2019shortcuts,theis2018counteracting} and references therein).\nIncreasing the efficiency of quantum operations by numerically optimizing and\nfine-tuning control parameters can rely on\nopen-loop or\nmodel-based optimal control\nmethods \\cite{Khaneja2005,Nigmatullin_2009,Tosner2009,de2011second,PhysRevA.84.022307,\nMachnes2011,Bergholm2019,PhysRevA.68.062308,PhysRevLett.89.188301,goerz2019krotov}.\nOur work on the estimation of distortions can be seen in the context of model-based approaches,\nwhich might rely on an accurate gradient calculation of the analytical cost function\nand thus on the knowledge of the Hamiltonian of the system \\cite{Glaser:2015tm, PhysRevA.84.022307}.\nThis knowledge might be available in naturally occurring qubits (such as atomic, molecular, or optical systems),\nbut may also be estimated in engineered (solid-state) technologies.\nSimilarly, closed-loop (i.e.~adaptive feed-forward) control methods \\cite{doi:10.1063\/1.459386,GOODWIN20189,BARDEEN1997151,PhysRevA.84.022326,PhysRevLett.106.190501,\nPhysRevA.92.062343,doi:10.1126\/science.aax9743,M_ller_2022}\nare used in situ to reduce adverse experimental effects on the control pulses,\nwhile direct (real-time) feedback and reservoir engineering methods\ncan also be used where appropriate to counteract control uncertainties \\cite{Solomon:2011vf,motzoi2016backaction}.\n\nThe paper is organized as follows: Section~\\ref{sec:level2} sketches\nthe control setup for optimizing quantum experiments and describes the\nconventional method for estimating the transfer function and its inclusion\nin the optimization. In Sec.~\\ref{sec:level3}, we detail our non-linear estimation method\nusing non-linear kernels. We also describe how to derive the transformation\nmatrix and its gradient. The non-linear effects\non quantum operations are shown with a numerical example of Rydberg atom excitations in\nSec.~\\ref{sec:level4}. We apply the estimation methodology to\nour numerical Rydberg example in Sec.~\\ref{sec:level5} and discuss\nrequirements on the available measurement data.\nFinally, we consider different numerical optimization methods in combination\nwith our estimation method\nin Sec.~\\ref{sec:level6} (see also Appendix~\\ref{app:optim})  and conclude in\nSec.~\\ref{sec:level7}.\n\n\\section{\\label{sec:level2}Time-dependent Control Problems}\nWe aim to efficiently transferring the population from an initial quantum state to a final target state.\nThe evolving state of a quantum system is described by its density operator\n$\\rho(t)$ and the corresponding equation of motion\nis written for coherent dynamics as\n\\begin{equation}\\label{eqn:2.1}\n\\dot{\\rho}= -i[H(t), \\rho] + \\mathcal{L}(\\rho).\n \\end{equation}\nThe form of the Lindblad term  $\\mathcal{L}(\\rho)$ is discussed in Sec.~\\ref{sec:level4} while\nthe Hamiltonian can be expressed as\n\\begin{equation}\\label{eqn:2.2}\nH(t) = H_d + \\sum_{i}{u_{i}(t)\\, H_{i}}.\n\\end{equation}\nThe free-evolution or drift component is given by $H_{d}$, while\n$H_i$ denotes the control Hamiltonians which are\nmultiplied with time-dependent control pulses\n$u_{i}(t)$. More precisely, our goal is to transfer\na quantum system from a given initial pure state with density operator $\\rho_{i}$\nto a target pure-state density operator $\\rho_t$ in time $T$ by varying the control\npulses $u_{i}(t)$ while minimizing the cost function\n\\begin{equation}\\label{eqn:2.3}\nC=1-\\abst{\\langle \\rho_t | \\rho(T)\\rangle}^2 = 1- \\abst{\\mathrm{Tr}[\\rho_t^ \\dagger \\rho(T)]}^2,\n\\end{equation}\nwhere $\\mathrm{Tr}(M)$ denotes the trace of a matrix $M$.\nThis cost function measures the difference between the target-state\ndensity operator $\\rho_t$ and the final-state density operator $\\rho(T)$.\nIn this work, we employ gradient-based optimization methods, which are\ndescribed and discussed in Section~\\ref{sec:level6}\nand Appendix~\\ref{app:optim}.\n\n\\begin{figure}[t]\n    \\includegraphics[]{distortion_horizontal}\n\\caption{Quadratic estimation of distorted pulses\n[Eq.~\\eqref{eqn:3.2}]\nis preferable to linear\nestimation [Eq.~\\eqref{eqn:2.4}]:\n(a) A pulse is numerically distorted (solid line); later the distortion is\nestimated up to linear (dashed line) and quadratic terms (dotted line).\nThe quadratic estimation better matches the actual distorted pulse when compared to the linear estimation.\n(b) Numerically computed errors for different types of distortions [including the distortion C plotted in (a)]\ngenerated by Eqs.~\\eqref{eqn:4.4}--\\eqref{eqn:4.5} are plotted for both the linear and the quadratic\nestimation. The error\nis defined in Eq.~\\eqref{eqn:5.4} and describes the difference between the actual distorted\nand the estimated pulse.\\label{fig:5.3}}\n\\end{figure}\n\n\nThe experimental realization of control pulses $u_{i}(t)$ relies on several devices,\nwhich might introduce systematic distortions and reduce the overall control efficiency.\nIt is our objective to determine these systematic distortions\nin order to adapt the control pulses during the optimization\nand counteract any adverse effects.\nFor a linear distortion, we can calculate its transfer function\n\\begin{equation}\\label{eqn:2.4}\nT(\\omega)=\\frac{Y(\\omega)}{X(\\omega)}\n\\end{equation}\nin the Fourier domain as the ratio of the Fourier\ntransform of the input and output pulses $x(t)$ and $y(t)$, i.e.\\ before and\nafter the distortion has taken place.\nAlternatively, we can calculate the impulse response $\\mathcal{I}(t)$ of the system which relates\nthe input and output pulse in the time domain using the convolution\n\\begin{equation}\\label{eqn:2.5}\ny(t)=(x*\\mathcal{I})(t) = \\int_{-\\infty}^{\\infty}x(\\tau)\\,\\mathcal{I}(t{-}\\tau)d\\tau.\n\\end{equation}\nFigure~\\ref{fig:5.3} highlights that a linear model might not be sufficient\nfor estimating experimental distortions as it cannot account for\nnon-linear effects. Non-linear\neffects are demonstrated in Fig.~\\ref{fig:5.3}(a) by passing one estimated example pulse\nthrough a numerically generated\ndistortion [see Eqs.~\\eqref{eqn:4.4}--\\eqref{eqn:4.5}].\nWhen estimating the distortion coefficients using a linear model,\nthe resulting distorted pulse does not match in Fig.~\\ref{fig:5.3}(a)\nwith the actual distorted pulse. However, the quadratic estimation with\na non-linear model (as described in Sec.~\\ref{sec:level3}) precisely\nrecovers the actual distorted pulse. Non-linear models\nare, e.g., preferable for Rydberg excitations which are detailed\nwith realistic experimental parameters in Sec.~\\ref{sec:level4}.\n\n\\section{\\label{sec:level3}Non-linear Estimation Method }\n\nWe provide now a general approach for estimating\nnon-linear distortions in a controlled quantum system\nand explain how this estimation approach can be incorporated\ninto the synthesis of robust optimal control pulses.\n\n\n\\subsection{Truncated Volterra series method\\label{sec:level3a}}\nWe characterize non-linear distortions using the truncated Volterra\nseries method \\cite{Mathews2000}. The Volterra series is a mathematical description of\nnon-linear behaviors for a wide range of systems \\cite{Cheng2017}. In analogy to Eq.~\\eqref{eqn:2.5}, we can write the general\nform of the Volterra series as\n\\begin{equation}\\label{eqn:3.1}\ny(t)=h^{(0)}+\\sum_{n=1}^{P}\\int_{a}^{b}\\!\\!\\!\\ldots \\!\n\\int_{a}^{b}h^{(n)}(\\tau_{1},\\ldots,\\tau_{n})\\prod_{j=1}^{n}x(t{-}\\tau_{j})d\\tau_{j}\n\\end{equation}\nwhere $x(t)$ is assumed to be zero for $t< 0$ as we consider general, non-periodic signals.\nThe output function $y(t)$ can be expressed as a sum of the higher-order\nfunctionals of the input function $x(t)$ weighted by the corresponding Volterra kernels\n$h^{(n)}$. These kernels can be regarded as higher-order impulse responses of the\nsystem. The Volterra series in Eq.~\\eqref{eqn:3.1} is truncated to the\norder $P<\\infty$ and it is called doubly finite if\n$a$ and $b$ are also finite. For a causal system, the output $y(t)$ can\nonly depend on the input $x(t{-}\\tau_j)$ for earlier times (i.e.\\ $t\\geq \\tau_j$)\nwhich results in $a\\geq0$; recall that $x(t{-}\\tau_j)=0$ for $\\tau_j>t$. The Volterra series\ncan therefore also model memory effects (which are assumed to be\nof finite length) and it is not restricted to instantaneous effects.\n\n\n\nThe discretized form of the Volterra series truncated to second order (i.e.\\ $P=2$)\nis given by (\\cite[Eq.~2.25]{Mathews2000})\n\\begin{equation}\ny_n =h^{(0)}+\\sum_{j=0}^{R-1}h^{(1)}_j\\, x_{n{-}j}\n+\\sum_{k,\\ell=0}^{R-1}\nh^{(2)}_{k\\ell}\\, x_{n{-}k} \\, x_{n{-}\\ell},\\label{eqn:3.2}\n\\end{equation}\nThe discrete output entries $y_n$ have $N$ time steps with $n \\in \\{0,\\ldots,N{-}1\\}$\nwhich are obtained {from $L$ discrete input entries $x_q$ where $x_q=0$\nfor $q<0$. Note that\n$N = L +R -1 \\geq L$, where\n$R\\geq 1$} denotes the\nassumed memory length of the distortion.\nThe memory length $R$ quantifies how the response\nat the current time step depends\non the input of previous time steps, i.e., $R$ bounds the number\nof previous time steps that can affect the current one.\nVolterra kernel coefficients of\nthe zeroth, first, and second order are represented by\n$h^{(0)}$, $h^{(1)}_j$, and $h^{(2)}_{k\\ell}$.\nThe matrix given by $h^{(2)}_{k\\ell}$ is symmetric.\nWe are characterizing the transfer function by estimating\nthe kernel coefficients in Eq.~\\eqref{eqn:3.2}.\nThe number $M$ of the to-be-estimated coefficients scales quadratically with\nthe memory length $R$ (in general, the number of coefficients\nscales with $R^P$). Although the Volterra estimation can be\nextended to any higher order $P>2$, we will focus in this work on\nthe quadratic case.\n\nFor the estimation process, we assume that we are provided with\na training data set consisting of input-output pulse pairs $(x(t),y(t))$ from an experimental device\n(or a sequence of devices) which causes the distortion. Next, we discuss how given the\ntraining data, we can estimate the kernel coefficients in Eq.~\\eqref{eqn:3.2}\nby minimizing some error measures (such as the mean square error)\nbetween the modeled output and the measured output.\n\n\\subsection{Truncated Volterra series via least squares\\label{sec:level3b}}\nWe can choose from different methods to estimate the Volterra series.\nThe most widely used ones are the crosscorrelation method of\nLee and Schetzen \\cite{doi:10.1080\/00207176508905543} and\nthe  exact orthogonal method of Korenberg \\cite{Korenberg:1988uv}.\nWe choose the latter due to its simplicity and\nas it does not require an infinite-length input.\nWe can write Eq.~\\eqref{eqn:3.2} as\n\\begin{align}\\label{eqn:3.3}\ny_n&=\\sum_{m=0}^{M-1}u_{nm}\\,k_m\n\\intertext{or equivalently as the matrix equation $Y = U K$ or}\n\\label{eqn:3.5}\n\\begin{bmatrix} y_0\\\\ y_1 \\\\[-1mm] \\vdots \\\\ y_{N-1} \\end{bmatrix}\n&=\n\\begin{bmatrix}\nu_{00} & u_{01} & \\cdots & u_{0,M-1}  \\\\\nu_{10} & u_{11} & \\cdots & u_{1,M-1}  \\\\[-1mm]\n\\vdots & \\vdots         &  & \\vdots  \\\\\nu_{N-1,0} & u_{N-1,1} & \\cdots & u_{N-1,M-1}      \\\\\n\\end{bmatrix} \\begin{bmatrix} k_0 \\\\ k_1  \\\\[-1mm] \\vdots \\\\k_{M-1}\n\\end{bmatrix},\n\\end{align}\nwhere $K$ is defined in Eq.~\\eqref{eqn:K} below.\nWe follow the convention that the entries\nof a given matrix (or vector) $D$ are\nrepresented by $d_{ij}$ (or $ d_i$).\nHere, $n\\in\\{0,\\ldots,N{-}1\\}$ and $m\\in \\{0,\\ldots,M{-}1\\}$ where\n\\begin{equation}\nM= 1{+} R {+} R(R{+}1)\/2\n\\label{eqn:3.11}\n\\end{equation}\ndenotes the number\nof coefficients that need to be estimated to describe\nthe quadratic Volterra series.\nIn particular, $u_{nm}$ are obtained from the input pulses\nvia (recall again $x_q=0$ for $q<0$)\n\\[\nu_{nm} =\n\\begin{cases}\n    1& \\text{for }\\;  m=0,\\\\\n    x_{n{-}m{+}1} & \\text{for }\\;  m\\in \\{1,\\ldots,R\\},\\\\\n    {x_{n{-}a}\\, x_{n{-}b}} & \\text{for }\\;  m\\in \\{R{+}1,\\ldots,M{-}1\\},\n\\end{cases}\n\\]\n{where $(a,b)$ with $0\\leq a \\leq b \\leq R{-}1$ is the $(m{-}R{-}1)$th element\nin the lexicographically ordered sequence from $(0,0)$ to $(R{-}1,R{-}1)$.}\nAs the quadratic distortion coefficients $h^{(2)}_{k\\ell}$ are symmetric, only the upper (or lower)\ntriangular entries need to be considered. {The column vector\n\\begin{equation}\nK=[h^{(0)},\nh^{(1)}_0,\\ldots,h^{(1)}_{R-1}, h^{(2)}_{00},\\ldots,h^{(2)}_{R-1,R-1}]^T\n\\label{eqn:K}\n\\end{equation}\nconsists of all the Volterra kernels, where $k_m = h^{(2)}_{ab}$ for\n$R{+}1 \\leq m \\leq M{-}1$ and $(a,b)$\nis chosen as above.\n\nThe example of $R=2$, $L=3$, $N=L+R-1=4$, and  $M=6$\nresults in (with $x_q=0$ for $q<0$)}\n\\begin{align}\n\\begin{bmatrix} y_0\\\\ y_1 \\\\y_2 \\\\ y_{3} \\end{bmatrix}\n&=\n\\begin{bmatrix}\n1 & x_0 & x_{-1} & x_{0}x_{0}&x_{0}x_{-1}&x_{-1}x_{-1}  \\\\\n1 & x_1 & x_{0} & x_{1}x_{1}&x_{1}x_{0}&x_{0}x_{0}\\\\\n1 & x_2 & x_{1} & x_{2}x_{2}&x_{2}x_{1}&x_{1}x_{1} \\\\\n1 & x_3 & x_{2} & x_{3}x_{3}&x_{3}x_{2}&x_{2}x_{2} \\\\\n\\end{bmatrix} \\begin{bmatrix} h^{(0)} \\\\ h^{(1)}_0  \\\\ h^{(1)}_1  \\\\h^{(2)}_{00}\\\\ h^{(2)}_{01}\\\\h^{(2)}_{11}\n\\end{bmatrix}.\n\\end{align}\n\n\n\nFor the estimation of the distortions, we need to determine the values of $K$\nby solving the matrix equation \\eqref{eqn:3.5} with the method of least squares.\nWe assume now that the output data vector $Y$ has been measured in an experimental setup.\nWe can also concatenate multiple output pulses into a single vector to form $Y$,\nwhich allows us to perform the estimation using multiple short pulses with different\ncharacteristics as compared to a single long pulse. This provides the freedom of choosing\nthe format for our training data while observing experimental constraints.\nIn addition to taking a single long pulse or a set of short pulses,\nwe can also repeatedly use the same set of pulses to reduce the measurement error.\n\n\nAs the matrix $U$ contains higher-order terms of the input $x_n$, different\ncolumns of $U$ are highly correlated with each other. This leads to the problem\nof solving a linear regression model with a correlated basis set, i.e., the input variables are dependent\non each other. The precision of the estimation is adversely affected and less robust\nwhen naively applying the method of least squares to solve the matrix equation \\eqref{eqn:3.5}.\nWe resolve this problem by first orthogonalizing\nthe columns of the matrix $U$. The orthogonalization transforms the input variables (stacked in columns of $U$)\nsuch that they are independent of each other. After orthogonalizing $U$ to $V$,\nEq.~\\eqref{eqn:3.5} is transformed to\n\\begin{equation}\\label{eqn:3.6}\nY=VW.\n\\end{equation}\nNow we can solve the modified matrix equation \\eqref{eqn:3.6}\nusing the method of least squares to robustly obtain the values of the vector $W$.\nFinally, if the Gram-Schmidt method is\nused for orthogonalization, then one can convert $W$ to $K$ by recursive methods (as explained in \\cite{Korenberg:1988uv})\nto extract the Volterra kernels $h^{(0)}$, $h^{(1)}_{j}$, and $h^{(2)}_{k\\ell}$. In this work, we use the\nQR factorization method which directly provides the values for $K$ \\cite{HornJohnson:1985,GolubVanLoan:1996}.\n\n\\subsection{Gradient of the input response function\\label{sec:gradientdistortion}}\n\nAssuming that we have successfully estimated the transfer function,\nwe want to include this information in our gradient-based optimization.\nThis would allow us to also go beyond\nthe piecewise-constant control basis of GRAPE by\nincluding arbitrarily deformed controls, generalizing further along the\nlines of Ref.~\\cite{PhysRevA.84.022307}.\nWe provide now an analytic expression for the corresponding gradient\n(i.e.\\ Jacobian) to build upon the earlier work discussed in Appendix~\\ref{app:optim}.\n\nWe apply the commutativity of the convolution (i.e.\\ $f*g=g*f$), e.g., by\nchanging the integration variable from $\\tau$ to $z=t-\\tau$ in Eq.~\\eqref{eqn:2.5}.\nUsing a slight generalization, Eq.~\\eqref{eqn:3.2} can be rewritten\nas\\footnote{Note that using Eq.~\\eqref{eqn:3.7} for the estimation in\nSecs.~\\ref{sec:level3a}-\\ref{sec:level3b} would require\na number of  coefficients given by $N\\times M$ instead of only $M$ and is therefore not recommended.}\n\n\\begin{equation}\ny_n = h^{(0)} + \\sum_{j=0}^{L-1} h^{(1)}_{n{-}j}\\, x_j\n+\\sum_{k,\\ell=0}^{L-1}  h^{(2)}_{n{-}k,n{-}\\ell}\\, x_{k}\\, x_{\\ell},\n\\label{eqn:3.7}\n\\end{equation}\nwhere the upper summation bound $L{-}1$ differs from $R{-}1$ in Eq.~\\eqref{eqn:3.2},\ni.e.~integrating over the length of the input instead of the length of the kernel.\nFrom Eq.~\\eqref{eqn:3.7}, we specify for each time step (indexed by $n$) a\nscalar $K^{(0)}=h^{(0)}$, a column vector $K^{(1)}_n$ with\nentries $[K^{(1)}_n]_j=h^{(1)}_{n{-}j}$, and a\nmatrix $K^{(2)}_n$ with entries $[K^{(2)}_n]_{k\\ell}=h^{(2)}_{n{-}k,n{-}\\ell}$ for\n$j, k,\\ell \\in \\{0,\\ldots,L{-}1\\}$. With this notation, we can write Eq.~\\eqref{eqn:3.7}\nas a matrix equation\n\\begin{equation}\\label{eqn:3.8}\ny_n=K^{(0)} + {X}^T K^{(1)}_n+{X}^T K^{(2)}_n {X},\n\\end{equation}\nwhere the column vector $X=(x_0,\\ldots,x_{L-1})^T$ has length $L$.\nThe corresponding partial derivatives are\ngiven by\n\\begin{equation}\\label{eqn:3.9}\n\\frac{\\partial {y_n}}{\\partial {X}}= K^{(1)}_n+ {X}^T [K^{(2)}_n+(K^{(2)}_n)^T],\n\\end{equation}\nwhich simplifies for a symmetric quadratic kernel to\n\\begin{equation}\\label{eqn:3.10}\n\\frac{\\partial {y_n}}{\\partial {X}}= K^{(1)}_n+ 2{X}^T K^{(2)}_n\n\\end{equation}\nWe can calculate\n$\\partial {y_n}\/\\partial{X}$ for all $n$ and then determine the Jacobian.\nEventually, the gradient of the cost function \\eqref{eqn:2.3} is obtained\nusing the chain rule as, e.g., in \\cite{PhysRevA.84.022307} and as discussed in Appendix~\\ref{app:optim}.\n\n\\section{\\label{sec:level4}Non-linear distortions during Rydberg excitations}\n\n\n\nWe illustrate our scenario of non-linear distortions during controlled quantum dynamics with robust state-to-state\ntransfers in a single Rydberg atom experiment. In recent years, Rydberg\natoms have been proven to be a promising platform for quantum\nsimulation \\cite{Weimer:2010vq} and quantum\ncomputation \\cite{RevModPhys.82.2313}. One of the most distinctive features of\nthese atoms in quantum experiments is their strong and tunable\ndipole-dipole interactions  \\cite{Jau:2016tg,Browaeys_2016}.\nFor larger Rydberg atom arrays as for quantum simulators, excitation\nprotocols (and more general operations)\nfrom the ground state to the Rydberg state are crucial.\nWe consider a gradient-based optimization of control pulses\n(without feedback) for tailored\nexcitation pulses as outlined in Sec.~\\ref{sec:level2}, (see also\nSec.~\\ref{sec:level6}\nand Appendix~\\ref{app:optim}).\n\n\\begin{figure}[t]\n\\includegraphics{energy}\n\\caption{(a) Path of the input control pulse\nfrom the computer code via an arbitrary waveform generator (AWG)\nand an acousto-optic modulator (AOM) to the atom. Before the atom,\nthe output control pulse can be measured using a photodiode (PD).\n(b) Three-level excitation scheme for a single Rydberg atom (see text).\\label{fig:4.2}}\n\\end{figure}\n\n\nThe Lindblad master equation for the time evolution of the system is given by Eq.~\\eqref{eqn:2.1}.\nFollowing \\cite{PhysRevA.97.053803}, the model Hamiltonian for a single\nRydberg atom is equal to\n\\begin{align}\nH(t) &= \\Omega_b(t) \\frac{\\ket{g}\\bra{p}+\\ket{p}\\bra{g}}{2}+\n\\Omega_r(t) \\frac{\\ket{p}\\bra{r}+\\ket{r}\\bra{p}}{2} \\nonumber \\\\\n&- \\Delta \\ket{p}\\bra{p} - \\delta \\ket{r}\\bra{r}.\n\\label{eqn:4.2}\n\\end{align}\nThe Rabi frequency $\\Omega_b(t)$ of the blue laser excites the atom from the\nground state $\\ket{g}$ to the intermediate state $\\ket{p}$ and the Rabi frequency $\\Omega_r(t)$\nexcites the atom from $\\ket{p}$ to the desired Rydberg state $\\ket{r}$ (see Fig.~\\ref{fig:4.2}(b)).\nIn terms of Eq.~\\eqref{eqn:2.2}, $\\Omega_b(t)$ and $\\Omega_r(t)$ constitute the\ntime-dependent control pulses. Moreover, $\\Delta$ and $\\delta$ are the single-photon and the\ntwo-photon resonance detuning, which will be for simplicity assumed to be zero\n($\\Delta=0$ MHz and $\\delta=0$ MHz).\nThe Lindblad operator \\cite{Lindblad:1976} reads as \\cite{PhysRevA.97.053803}\n\\begin{equation}\\label{eqn:4.3}\n{\\mathcal{L}(\\rho) =\\sum\\limits_{j\\in\\{d,g,g'\\}}\n(V_j \\rho V_j^{\\dagger}) -\\tfrac{1}{2}(V_j^{\\dagger}V_j  \\rho + \\rho V_j^{\\dagger} V_j)}\n\\end{equation}\nwhere $V_g = \\sqrt{\\Gamma_{g}} \\ket{g}\\bra{p}$,\n$V_{g'}= \\sqrt{\\Gamma_{g'}} \\ket{g'}\\bra{p}$,\nand $V_d = \\sqrt{\\Gamma_{d}} \\ket{r}\\bra{r}$ are the Kraus operators.\nHere,\n$\\Gamma_{g}={\\Gamma}\/{3}$ and $\\Gamma_{g'}={2\\Gamma}\/{3}$\ndenote the probability for spontaneous emission from $\\ket{p}$ to the\nground state $\\ket{g}$ or to $\\ket{g'}$ which represents all other ground-state sublevels.\nRealistic experimental parameters $\\Gamma=2\\pi\\times 1.41$ MHz and\n$\\Gamma_d=2\\pi\\times 0.043$ MHz have been provided by the Browaeys group,\nwhere  $\\Gamma_d$ is the Doppler effect.\nIn a real experiment, the gradients of the controls are restricted\ndue to bandwidth limitations.\nIn particular, the controls cannot have derivatives larger than a certain\nrise speed given by the experimental setup.\nIn our simulations, we take realistic values for the rise times of $0.1\\mu s$ and $0.15 \\mu s$ for\nthe red and blue laser pulses respectively (which translate into rise speeds).\n\n\n\nLet us now discuss how systematic distortions can be introduced in this experimental\nplatform during the processing and forwarding of the control signals which finally act\non the atom(s). The path of the control signals is sketched in Fig.~\\ref{fig:4.2}(a).\nStarting from some computer program, the input pulse (modulated with a fixed carrier\nfrequency) is passed through an arbitrary waveform generator (AWG) to produce\nthe radio-frequency pulse. This pulse is then used as an input for an\nacousto-optic modulator (AOM) which modulates the intensity of a laser beam.\nThe final laser pulse is then applied to the atom(s) to perform the excitation.\nThe AOM can shape pulses using optical effects such as\ndispersion \\cite{doi:10.1063\/1.2409868,doi:10.1063\/1.5020796}.\nIn this experimental setup, one can measure the laser signal before\nit acts on the atom(s) using a photo diode. In summary, one can choose the input\npulse and measure the output pulse; multiple measured input-output pulse\npairs serve as training data, which is used to determine systematic distortions.\n\nIn our simulation, we excite the Rydberg atom using the system\nHamiltonian from Eq.~\\eqref{eqn:4.2} by applying our optimized\ninput control pulses. After\nthat, we introduce quadratic distortions to the control pulses and\nrepeat the simulation. The discrete linear and quadratic distortions are\nprepared from Gaussian distributions described by\n\\begin{align}\\label{eqn:4.4}\nh_{1}(t) &= \\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp[{-\\tfrac{(t{-}\\mu)^2}{2{\\sigma_{1}}^2}}], \\\\\n\\label{eqn:4.5}\nh_{2}(t_{1},t_{2}) & =\nJ\\exp[{-\\frac{(t_{1}{-}\\mu_{1})^2+(t_{2}{-}\\mu_{2})^2}{2{\\sigma_{2}}^2}}].\n\\end{align}\nThe memory length of the discretized dimensionless distortion is $R$.\nFor the distortions A, B, and C, we have chosen $R=50$,\nstandard deviations $\\sigma_1$\nof $1$, $6$, and $11$, and $\\sigma_2$ of $4.25$, $6.37$, and $8.50$.\nSimilarly, for the distortions D, E, and F, we have varied $R$ between\n$20$, $40$, and $60$ while fixing\n$\\sigma_{1}=1$ and $\\sigma_{2}=4.25$. The amplitude term $J$ has been\nkept  constant at $5 \\times 10^{-6}$ in all cases.\nThe example distortion C is shown in Figs.~\\ref{fig:5.2}(a1) and \\ref{fig:5.2}(b1).\nThroughout this work, the zeroth order kernel is set to $h^{0}=0.1$.\n\n\n\nWe observe optimized controls with a simulated\nRydberg excitation error in the range from $0.06$  to $0.008$ for\ndifferent pulse lengths (see Fig.~\\ref{fig:4.1}). As expected, longer total durations\nfor the excitation lead to smaller simulated errors. But longer pulse durations\nmight lead to further decoherence effects in the experimental implementation\n(particularly when combined with additional experimental steps).\nWe, therefore, aim at reducing the length of the pulses\n(e.g.\\ to a pulse duration around $0.3\\mu s$) with reduced excitation errors.\nIn Fig.~\\ref{fig:4.1}(a), we notice a uniform increase in the error\nmagnitude when we increase the standard deviation of the Gaussian kernels of\nEqs.~\\eqref{eqn:4.4}-\\eqref{eqn:4.5} for the distortions A to C.\nThe standard deviation is kept constant in Fig.~\\ref{fig:4.1}(b), but we increase\nthe memory length for the distortions D to F which also results\nin bigger excitation error.\nThe increased excitation errors suggest that optimized control pulses\nwould be susceptible to distortions when applied in the Rydberg atom\nexperiments (and particularly for short pulse lengths). In Sec.~\\ref{sec:level5},\nwe present estimation results building on Sec.~\\ref{sec:level3}\nfor the considered types of distortions.\n\n\\begin{figure}\n\\includegraphics{excitation_error}\n\\caption{Reduced excitation efficiencies of optimized\ncontrol pulses\ndue to non-linear distortions in a simulated\nsingle Rydberg atom for distortions with\n(a) an increasing variance but constant memory length (A-C) and\n(b) a constant\nvariance but increasing memory length (D-F); refer to Sec.~\\ref{sec:level4}.\\label{fig:4.1}}\n\\end{figure}\n\n\n\\begin{figure*}\n\\includegraphics{kernels}\n\\caption{Estimation of both the linear and quadratic components for a non-linear distortion: (a) The linear component (a1) of the distortion C\nis compared with its estimated value (a2). The amplitude and\ntime steps are dimensionless. (a3) The mean absolute scaled error\n[as defined in Eq.~\\eqref{eqn:5.4}]\nbetween the actual and the estimated values\nis calculated for various types of distortions A-F [see Eqs.~\\eqref{eqn:4.4}--\\eqref{eqn:4.5}].\n(b) Quadratic component similar as in (a).\n\\label{fig:5.2}}\n\\end{figure*}\n\n\n\\section{\\label{sec:level5}Numerical estimation results}\n\n\n\nWe report in this section on different simulated estimation results\nwhich describe the characteristics and precision of applying\nthe Truncated Volterra series method while also comparing\nmultiple types of input control pulses used in the estimation.\nWe also perform the optimization for a single Rydberg\nexcitation again by including the distortions in the algorithm.\nIn each analysis, the estimated results are compared with the actual ones\nusing the mean absolute scaled error (MASE) measure\n\\begin{equation}\\label{eqn:5.4}\n\\text{MASE}=\\frac{1}{N}\\sum_{i=1}^{N}\n\\abs{\\frac{z^{\\text{true}}_i}{\\norm{z^{\\text{true}}}}-\\frac{z^{\\text{est}}_i}{\\norm{z^{\\text{est}}}}}\n\\end{equation}\nwhere $z^{\\text{true}}$ is the actual value, $z^{\\text{est}}$ is the estimated value, and $\\norm{z}$ is the\nFrobenius norm of the observable $z$ of length $N$. The\nMASE is numerically more stable compared to the mean relative error, which can be very large\nwhen the measured and the actual values are very small.\n\n\n\n\\subsection{\\label{subsec:level1}Estimation of distortions}\n\nWe start with the results presented in Fig.~\\ref{fig:5.2}\nwhere numerical distortions are estimated by relying on\na single randomly generated control pulse with $4000$ time steps. We apply\ndifferent distortions to the pulse and employ the resulting\ninput-output pulse pairs in the estimation. In order to provide\na more realistic analysis, we add an additional noise term to\nthe output pulse\n\\begin{equation}\\label{eqn:5.3}\ny_{\\text{noise}}=y_{\\text{output}}+ \\tfrac{1}{\\sigma\\sqrt{2\\pi}}\n\\exp[{-\\tfrac{(t{-}\\mu)^2}{2{\\sigma_{}}^2}}],\n\\end{equation}\nwhere the noise is drawn from a normal distribution\nwith mean $\\mu=0$ and standard deviation $\\sigma=10^{-4}$.\nFigures~\\ref{fig:5.2}(a1)-(b1) display the linear and quadratic\ncontribution of the distortion C. The corresponding\nestimated contributions are shown in Figs.~\\ref{fig:5.2}(a2)-(b2)\nwhich match closely with values in Figs.~\\ref{fig:5.2}(a1)-(b1).\nThe results also emphasize that precisely knowing\nthe memory length of the distortion (which is here $R=50$)\nis not required as redundant coefficients are automatically set to\nzero during the estimation for\na sufficiently large $R$ (here set to $60$).\nThe estimation process has been repeated for\nmultiple distortions of type A to F and we observe in Figs.~\\ref{fig:5.2}(a3)-(b3)\nlow estimation errors of approximately $10^{-7}$ to $10^{-8}$.\n\n\n\nWe now also compare the estimation method of Sec.~\\ref{sec:level3}\nwith a linear estimation method in the time domain which relies on\na linear impulse response [cf.\\ Eq.~\\eqref{eqn:2.5}]. We omit here\nthe very similar linear estimation in the frequency domain. We again\nuse the distortion types A to F from Sec.~\\ref{sec:level4} for this\ncomparison and apply them again to a random-noise pulse of $4000$ steps to obtain\ninput-output pulse pairs for the estimation.\nFigure~\\ref{fig:5.3}(a) shows the effect of the\ntrue and estimated distortion C when applied to\nan example pulse of $0.4$$\\mu s$ duration. The example pulse is\nstretched under the distortion to a final duration of $0.65$$\\mu s$.\nThe linear estimation is considerably less\nprecise when compared to the quadratic estimation. This effect\nis confirmed in Fig.~\\ref{fig:5.3}(b) which plots the estimation errors\nfor the different distortion types A-F. Naturally, this also validates that\nthe chosen distortion types contain some non-linearity which is not\naccounted for by a linear estimation.\n\n\\subsection{\\label{subsec:level2}Orthogonalization}\n\nOne important step of the estimation method is orthogonalization\nand we have discussed its significance in Sec.~\\ref{sec:level3}.\nTo further highlight the benefits of orthogonalizing the basis functionals,\nwe test the estimation by directly\nsolving the matrix equation\n\\begin{equation}\\label{eqn:5.1}\n{U^TY=U^TUK},\n\\end{equation}\nwhere $U$ is the matrix of the non-orthogonalized and correlated\nbasis functionals, $K$ is the to-be-estimated\nvector of linear\nor nonlinear kernel coefficients and $Y$ is the\nmeasured output vector.\nWe compare the results with coefficients we get from solving\nthe matrix equation \\eqref{eqn:3.6} with the orthogonalized basis set.\n\nIn this analysis, along with the benefit of orthogonalization, we also demonstrate how the estimation depends on\nthe number $M$ of the to-be-estimated coefficients for the distortion,\nthe amount of training data, and the presence of noise in the output pulse.\nFigures~\\ref{fig:5.4}(a)-(b) discuss the case without added noise.\nThe nonlinear distortion with $\\sigma_{1}=0.1, \\sigma_{2}= 0.42$ and $R=5$ is estimated\nusing spline input pulses as the training data.\nEach test and training pulse has 500 time steps and a unique frequency.\nFor a fixed number of spline pulses, we observe\nin Fig.~\\ref{fig:5.4}(a) an increasing estimation error for an\nincreasing number of coefficients $M$ (or memory length $R$\nas $M\\propto{R^2}$).\nFor each $M$, we apply the estimation results on\n$50$ different spline pulses which serve as test data.\nThe corresponding mean error is plotted as a line and\nthe $95\\%$ confidence interval is shown\nas a shaded region around the mean.\n\\begin{figure}\n\\includegraphics{with_without_ortho}\n\\caption{Comparison of the simulated estimation of\nnon-linear distortions without and with orthogonalization\nsolving respectively Eq.~\\eqref{eqn:5.1} and  \\eqref{eqn:3.6}:\n(a) Using a fixed number of noiseless training data for spline\ninput pulses, the relative error rises with an increasing number of coefficients\n$M$. The plotted\nline shows the mean error and the shaded area\nindicates the spread between the $95\\%$ confidence interval found from applying\nthe estimation results to $50$ different test pulses.\nOrthogonalization is advantageous for\na larger number of coefficients. (b) Average estimation errors for different\ntraining data sets (see text) highlight the importance of increasing the frequency\ncontent of the available data. The averaging is performed over the full\nrange of all number of coefficients $M$ in (a).\n(c)-(d) As in (a)-(b), but the added noise in the output pulses of the data\nrequires a higher frequency content for comparable error rates.\n\\label{fig:5.4}}\n\\end{figure}\nIn Fig.~\\ref{fig:5.4}(b), we gradually increase the number of training\npulses used in the estimation.\nFor each fixed number of pulses, we perform the estimation on all the values of $M$\nas shown in Fig.~\\ref{fig:5.4}(a).\nHence each point in Fig.~\\ref{fig:5.4}(b) is averaged over $500$ results.\nIn all cases, the estimation\nbenefits from being performed with orthogonalization. Also, extending\nthe amount of training data points by adding more spline pulses with different\nfrequencies improves the estimation precision as seen in Fig.~\\ref{fig:5.4}(b).\nFor Figs.~\\ref{fig:5.4}(c)-(d) in the presence\nof a noise term in the output pulse with a standard deviation of $10^{-9}$, we\nobserve higher estimation errors which need to be compensated with\nadditional training data points. One can also reduce correlations\npresent in the training data by considering a random input pulse as its\nautocorrelation is zero. However, even a completely random input pulse\nresults in correlations in $U$ from Eqs.~\\eqref{eqn:3.5} and \\eqref{eqn:5.1}\nwhich contains various non-linear terms of the same input vector\n\\cite[p.~165]{Mathews2000}. In summary, Fig.~\\ref{fig:5.4} illustrates the positive\neffect of orthogonalization on the error rates in the estimation of the distortion.\n\n\n\\begin{figure}[t]\n\\includegraphics{bandwidth_comparison}\n\\caption{Simulated estimation errors of distortions with multiple types\nof data:\n(a) training data with more frequency content (such as random-noise pulses)\nperform better, even as the number of to-be-estimated coefficients $M$ increases.\n(b) A lower error can be achieved  by increasing the frequency content of\nthe training data. For cosine and Gaussian pulses, the frequency content\nis increased by adding more pulses with different frequencies, whereas\nthe number of knots is increased within a single pulse for splines.\nSpectrally rich random-noise pulses are highly effective while keeping the data requirements low.\n(c)-(d) Similar to (a) and (b), but noisy training data\nincreases the overall error, while random-noise pulses are the most robust.\nThe estimation setup is similar to Fig.~\\ref{fig:5.4}.\n\\label{fig:5.5}}\n\\end{figure}\n\n\n\\subsection{\\label{subsec:level3}Frequency requirements}\n\n\n\n\n\n\nWe investigate different types of training data and their performance\nin the estimation following the setup of Fig.~\\ref{fig:5.5}. We can order\ndifferent training data types according to their increasing frequency content,\nwith Gaussian pulses having the minimum frequency and random-noise pulses\nhaving the maximum. Here, the frequency content describes the spectral content\nof the training data while its value depends on the type of pulses used\n(see Fig.~\\ref{fig:5.4}(b) and (d).\nThere are different errors for spline and cosine pulses depending on the amount of data.\nFor a fixed number of pulses, the estimation error grows with an\nincreasing number of coefficients $M$ [see Fig.~\\ref{fig:5.5}(a)]. Gaussian input pulses\nare most strongly affected by this, while this effect is essentially negligible\nin the case of random-noise pulses.\nThis illustrates the importance of spectrally rich input training pulses, which is further\nemphasized in Fig.~\\ref{fig:5.5}(b) where the estimation error is plotted, relative\nto the frequency content. For different types of input pulses, the frequency\ncontent is increased differently: we add more pulses with different standard\ndeviations for Gaussian pulses, we add more pulses with different frequencies\nfor cosine pulses, we add more\nrandom knots to a single spline. Since a random-noise pulse has a very large bandwidth,\nwe aim at increasing the frequency content by increasing the number of random-noise pulses which\nonly slightly reduces the estimation error.\nFigure~\\ref{fig:5.5}(b) highlights that the frequency content is crucial for the\nestimation and even a single random-noise pulse is highly effective due to its\nhigh-frequency content.\nSplines start to outperform the cosine pulses as soon as they\nattain higher frequency content than the latter.\nSimilar conclusions hold under noise as shown\nin Fig.~\\ref{fig:5.5}(c)-(d) while the overall estimation error increases\nfor the different input pulse types when compared to the noiseless case.\nThe data suggests that a high-frequency content in the training pulses\nmight prevent overfitting noise, which is important when\nworking with real experimental data. Also under noise,\nrandom-noise input pulses are most effective in the estimation due to their high frequency content.\n\n\n\n\\section{\\label{sec:level6}Application in optimal control}\nStarting from early developments in the field, various theoretical\nand experimental aspects of quantum control have been discussed\nin the recent review \\cite{Koch2022}.\nThe overall aim of quantum control is to shape a set of\nexternal field pulses that drive a quantum system and perform a given quantum process efficiently.\nWhile the analytical way of finding the\ncontrol parameters works for special cases, one can use\nhighly developed numerical tools in the context of\noptimal control theory. One solves the Schr\u00f6dinger or master equation iteratively and\nproduces pulse shapes that perform the desired time evolution.\nQuantum optimal control is broadly divided into at least the two categories of\nopen-loop and closed-loop. Open-loop\nmethods can be gradient-based or not.\nOpen-loop control is based on the available information about the\nHamiltonian of the system and hence it suffers when the system parameters are\nnot completely known such as in the case of an engineered quantum system (such as solid state systems) or when\nthe model cannot be solved precisely as in the case of many-body dynamics \\cite{PhysRevLett.109.153603}.\nThese limitations might be overcome by means of closed-loop optimal control where the control parameters are updated\nbased on the earlier measurements results \\cite{PhysRevA.84.022326,PhysRevLett.106.190501}.\nClosed-loop quantum optimal control can be implemented via\nboth gradient-based and gradient-free algorithms \\cite{PhysRevLett.118.150503, PhysRevLett.112.240504, PhysRevApplied.7.041001}.\nIn some cases, hybrid approaches have also been suggested \\cite{PhysRevLett.112.240503}.\nBut in the case where the system Hamiltonian is well known, open-loop control\nprovides more freedom to precisely tune the controls depending on experimental constraints and\ngenerally explore a wider range of control solutions.\nMoreover, it also gives a better understanding of the system and works well with systems where\nfast measurements are not feasible or very noisy, in contrast to closed-loop methods which may require many\nmeasurements to converge.\n\n\\begin{figure}\n\\includegraphics{excitation_error_tr}\n\\caption{Reduced excitation errors after correcting for the distortion with a\ngradient-based optimization relying on the trust-region method.\n(a) distortion C: significant reduced errors, (d) distortion F: mostly recovers\nthe ideal case.\n\\label{fig:5.6}}\n\\end{figure}\n\n\n\n\nTo take full advantage of the open-loop control method and to provide more robust pulses,\none can also characterize the experimental system completely or at least partially. Here, we highlight how\nthe estimation method from Sec.~\\ref{sec:level3} can be employed in an open-loop control setting\nto minimize the cost function $C$ in Eq.~\\eqref{eqn:A.2}\nby relying on the corresponding gradients as computed via\nEqs.~\\eqref{eqn:3.9} and \\eqref{eqn:A.9}.\nWe refer to Appendix~\\ref{app:optim} for details.\nThis compensates for distortions\nand decreases the error. Figure~\\ref{fig:5.6} shows test minimizations of the cost function\nusing the trust-region constrained algorithm \\cite{doi:10.1137\/1.9780898719857}, which\ncan perform constraint minimization with linear or non-linear constraints on the control pulses.\nTrust-region methods allow us to explicitly observe bandwidth limitations of the control hardware such\nas limited rise speeds as discussed in Sec.~\\ref{sec:level4} by enforcing the corresponding\npulse constraints.\nSince distortions C and F defined by Eqs.~\\eqref{eqn:4.4} and \\eqref{eqn:4.5}, have the strongest effects on the Rydberg excitation error (see Fig.~\\ref{fig:4.1}),\nwe correct the control pulses affected by them in the simulations.\nWe limit our test to pulses with shorter durations ranging from 0.1$\\mu s$ to 0.4$\\mu s$\nas they are less susceptible to decoherence and hence might be more suitable for the excitation process.\nWe compare the excitation error\nproduced from the corrected pulse with the ideal and the distorted pulse excitation error.\nIn particular, Figure~\\ref{fig:5.6} shows that the effect of the distortion C can be significantly\nreduced, but it cannot be completely corrected\ndue to a large standard deviation and long memory length in the distortion.\nThe distortion F has a small standard deviation combined with a\nlong memory length which still produces strong effects on the control pulse but\nwith a generally weaker distortion. In this case, the effect of the distortion can\nbe almost completely corrected.\n\n\n\n\nThe estimation of transfer functions in order to correct for distortions has one\nadditional benefit. The experimental hardware given by, e.g., AWGs and AOMs usually has\nbandwidth limitations which translate into limited rise speeds as discussed in\nSec.~\\ref{sec:level4}. In the process of characterizing the experimental devices\nvia their transfer function, we also estimate the effects of these bandwidth limitations.\nThe estimated transfer function is then applied during the optimization,\nwhich mirrors the effects in the experimental platform and implicitly enforces limitations\non the bandwidth or rise time.\nAssuming that the bandwidth-limiting effect of the estimated transfer function\nis pronounced enough,\nthis allows us to use the limited memory\nBroyden\u2013Fletcher\u2013Goldfarb\u2013Shanno (L-BFGS) algorithm to perform the\nminimization of the cost function \\cite{de2011second}. L-BFGS usually offers\na more efficient optimization but it cannot explicitly\naccount for general linear or non-linear constraints.\nIn the corresponding\noptimizations, we only enforce simple box constraints\nto limit the amplitude of the controls while using the extended L-BFGS or L-BFGS-B algorithm\n\\cite{53712fe04a3448cfb8598b14afab59b3}.\nThe results are shown in Fig.~\\ref{fig:5.7} where L-BFGS-B improves the excitation efficiency\nmore effectively than the trust-region method (which needs to also explicitly enforce\nthe constraints on the rise speeds).\nIn summary, combining the estimation of distortions with gradient-based optimizations\ncan often effectively compensate for these non-linear distortions during an open-loop\noptimization.\n\n\n\n\n\\section{\\label{sec:level7}Conclusion}\nWe have proposed a method for estimating\nnon-linear pulse distortions originating from\nexperimental hardware. Hardware limitations affect the performance\nof optimal control pulses as highlighted using numerical data for\nsingle Rydberg atom excitations. In this case, the errors are increased\nfor distorted control pulses beyond purely linear effects.\nWe provide a general model for\ndescribing the complex characteristics of these non-linear effects.\nTo incorporate estimated distortions into\nopen-loop optimizations, we have detailed\na formula to determine\nthe Jacobian of the transfer function.\n\n\\begin{figure}\n\\includegraphics{excitation_error_bfgs}\n\\caption{Reduced excitation error for  L-BFGS-B when\ncompared to the trust-region method, even though\nL-BFGS-B does not explicitly enforce constraints on\nthe control pulses (such as limited rise speeds).\nBut the correctly estimated transfer function\nwill implicitly account for pulse constraints.\nAlso, L-BFGS-B  is more effective\nin the optimization.\n\\label{fig:5.7}}\n\\end{figure}\n\n\nWe tested and validated our\nproposed method by\nefficiently estimating different numerical quadratic distortions with varying strength\nand duration. We have also shown that linear estimation methods\ncannot effectively handle non-linear transfer functions. From our detailed analysis and tests,\nwe deduce that the orthogonalization (as described\nin Sec.~\\ref{subsec:level2}) is key for a robust estimation.\nA robust least-squares estimation is effective only after\nthe orthogonalization is applied to the matrix containing the training data\nas its correlated columns would otherwise interfere with the estimation.\nAnother critical requirement for effectively performing\nthe estimation is training data with enough frequency content.\nLarge frequency content such as in random-noise pulses better\ncaptures the non-linear features of transfer functions, particularly in the\npresence of measurement noise.\n\nSince the estimation method is independent\nof any particular type of device characteristics, it can easily be adapted to a\nwide range of experimental platforms.\nCombining our estimation method with\nexisting numerical optimization techniques can improve the quality and\nrobustness of quantum operations. Our work thereby addresses a key challenge\nof enhancing the accuracy and robustness\nof experimental quantum technology platforms.\n\n\\begin{acknowledgments}\nThe authors acknowledge funding from the EU H2020-FETFLAG-03-2018 under\ngrant agreement No 817482 (PASQuanS). We also appreciate support from the German Federal Ministry of Education\nand Research through the funding program quantum technologies---from basic research to market\nunder the project FermiQP, 13N15891. We would like to thank\nAntoine Browaeys, Daniel Barredo, Thierry Lahaye, Pascal Scholl, and Hannah Williams for\nthe illuminating discussions about the Rydberg system as well as providing detailed\nexperimental parameters. R.Z. would like to thank\nJian Cui for initial discussions about the Rydberg setup.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\lblsec{intro}\n\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\vspace{-0.5cm}%\n    \\includegraphics[scale=0.58, trim=230 106 150 20, clip]{figures\/figure_2_dataset_distillation.pdf}\\vspace{-0.85cm}\n    \\caption{Dataset distillation aims to generate a small synthetic dataset for which a model trained on it can achieve a similar test performance as a model trained on the whole real train set.}%\n    \\lblfig{dataset_distillation}\\vspace{-5pt}\n\\end{figure}\n\nIn the seminal 2015 paper, Hinton et al.~\\cite{hinton2015distilling} proposed {\\em model distillation}, which aims to distill the knowledge of a complex model into a simpler one.  \n{\\em Dataset distillation}, proposed by Wang et al.~\\cite{dd}, is a related but orthogonal task: rather\nthan distilling the model, the idea is to distill the dataset. \nAs shown in Figure~\\ref{fig:dataset_distillation}, the goal is to distill the knowledge from a large training dataset into a very small set of synthetic training images (as low as one image per class) such that training a model on the distilled data would give a similar test performance as training one on the original dataset. Dataset distillation has become a lively research topic in machine learning~\\cite{bohdal2020flexible,sucholutsky2021soft,dc,dsa,nguyen2020dataset,nguyen2021dataset,dm} with various applications, such as continual learning, neural architecture search, and privacy-preserving ML. Still, the problem has so far been of mainly theoretical interest, since most prior methods focus on toy datasets, like MNIST and \\mbox{CIFAR}, while struggling on real, higher-resolution images. In this work, we present a new approach to dataset distillation that not only outperforms previous work in performance, but is also applicable to large-scale datasets, as shown in \\reffig{teaser}.\n\n\nUnlike classical data compression, dataset distillation aims for a small synthetic dataset that still retains adequate task-related information so that models trained on it can generalize to unseen test data, as shown in \\reffig{dataset_distillation}. Thus, the distilling algorithm must strike a delicate balance by heavily compressing information without completely obliterating the discriminative features.  \nTo do this, dataset distillation methods attempt to discover exactly which aspects of the real data are critical for learning said discrimination. Several methods consider end-to-end training~\\cite{dd,nguyen2020dataset,nguyen2021dataset} but often require huge compute and memory and suffer from inexact relaxation~\\cite{nguyen2020dataset,nguyen2021dataset} or training instability of unrolling many iterations~\\cite{dd,maclaurin2015gradient}. To reduce the optimization difficulty, other methods~\\cite{dc,dsa} focus on short-range behavior, enforcing a single training step on distilled data to match that on real data. However, error may accumulate in evaluation, where the distilled data is applied over many steps. We confirm this hypothesis experimentally in \\refsec{shortlongrange}.\n\n\nTo address the above challenges, we sought to directly imitate the long-range training dynamics of networks trained on real datasets. In particular, we match segments of parameter trajectories trained on synthetic data with segments of pre-recorded trajectories from models trained on real data and thus avoid being short-sighted (\\ie focusing on single steps) or difficult to optimize (\\ie modeling the full trajectories). \nTreating the real dataset as the gold standard for guiding the network's training dynamics, we can consider the induced sequence of network parameters to be an \\emph{expert trajectory}. If our distilled dataset were to induce a network's training dynamics to follow these expert trajectories, then the synthetically trained network would land at a place close to the model trained on real data (in the parameter space) and achieve similar test performance. %\n\n\\input{figText\/overview}\n\n\nIn our method, our loss function \\textit{directly} encourages the distilled dataset to guide the network optimization along a similar trajectory (\\Cref{fig:method_traj_match}). We first train a set of models from scratch on the real dataset and record their expert training trajectories. We then initialize a new model with a random time step from a randomly chosen expert trajectory and train for several iterations on the \\textit{synthetic} dataset. Finally, we penalize the distilled data based on how far this synthetically trained network deviated from the expert trajectory and back-propagate through the training iterations. Essentially, we transfer the knowledge from many expert training trajectories to the distilled images. \n\nExtensive experiments show that our method handily outperforms existing dataset distillation methods as well as coreset selection methods on standard datasets, including CIFAR-10, CIFAR-100, and Tiny ImageNet. For example, we achieve 46.3\\% with a \\emph{single} image per class and 71.5\\% with 50 images per class on CIFAR-10, compared to the previous state of the art (28.8\\% \/ 63.0\\% from \\cite{dsa, dm} and 36.1\\% \/ 46.5\\% from \\cite{nguyen2021dataset}).\nFurthermore, our method also generalizes well to larger data, allowing us to see high $128\\times128$-resolution images distilled from ImageNet \\cite{deng2009imagenet} for the first time. Finally, we analyze our method through additional ablation studies and visualizations. %\n\\href{https:\/\/github.com\/GeorgeCazenavette\/mtt-distillation}{Code} and models are also available on our \\href{https:\/\/georgecazenavette.github.io\/mtt-distillation\/}{webpage}.\n\\lblfig{intro}\n\\section{Related Work}\n\n\n\n\\myparagraph{Dataset Distillation.}\nDataset distillation was first introduced by Wang et al.~\\cite{dd}, who proposed expressing the model weights as a function of distilled images and optimized them using gradient-based hyperparameter optimization~\\cite{maclaurin2015gradient}, which is also widely used in meta-learning research \\citep{finn2017model,nichol2018first}. %\nSubsequently, several works significantly improved the results by learning soft labels \\cite{bohdal2020flexible,sucholutsky2021soft}, amplifying learning signal via gradient matching \\cite{dc}, adopting augmentations~\\cite{dsa}, and optimizing with respect to the infinite-width kernel limit \\cite{nguyen2020dataset,nguyen2021dataset}. Dataset distillation has enabled various applications including continual learning~\\cite{dd,dc,dsa}, efficient neural architecture search~\\cite{dc,dsa}, federated learning~\\cite{goetz2020federated,zhou2020distilled,sucholutsky2020secdd}, and privacy-preserving ML~\\cite{sucholutsky2020secdd,li2020soft} for images, text, and medical imaging data. As mentioned in the introduction, our method does not rely on single-step behavior matching~\\cite{dc,dsa},  costly unrolling of full optimization trajectories~\\cite{dd,sucholutsky2021soft}, or large-scale Neural Tangent Kernel computation~\\cite{nguyen2020dataset,nguyen2021dataset}. Instead, our method achieves long-range trajectory matching by transferring the knowledge from pre-trained experts. %\n\n\nConcurrent with our work, the method of Zhao and Bilen~\\cite{dm} completely disregards optimization steps, instead focusing on distribution matching between synthetic and real data. While this method is applicable to higher-resolution datasets (\\eg Tiny ImageNet) due to reduced memory requirements, it attains inferior performance in most cases (\\eg when compared to previous works~\\cite{dc,dsa}). In contrast, our method simultaneously reduces memory costs while outperforming existing works~\\cite{dc,dsa} and the concurrent method~\\cite{dm} on both standard benchmarks and higher-resolution datasets. \n\nA related line of research learns a generative model to synthesize training data \\cite{such2020generative,masarczyk2020reducing}. However, such methods do not generate a small-size dataset, and are thus not directly comparable with dataset distillation methods.\n\n\n\n\\myparagraph{Imitation Learning.} \nImitation learning attempts to learn a good policy by observing a collection of expert demonstrations \\cite{osa2018algorithmic,peng2018deepmimic,peng2021amp}. Behavior cloning trains the learning policy to act the same as expert demonstrations. Some more sophisticated formulations involve on-policy learning with labeling from the expert \\cite{ross2011reduction}, while other approaches avoid any label at all, \\eg via distribution matching\n\\cite{ho2016generative}. Such methods (behavior cloning in particular) have been shown to work well in offline settings \\cite{fu2020d4rl,gulcehre2020rl}. \nOur method can be viewed as imitating a collection of expert network training trajectories, which are obtained via training on real datasets. Therefore, it can be considered as doing imitation learning over optimization trajectories.\n\n\\myparagraph{Coreset and Instance Selection.}\nSimilar to dataset distillation, coreset~\\cite{tsang2005core,har2007smaller,bachem2017practical,sener2018active,chen2010super} and instance selection~\\cite{olvera2010review} aim to select a subset of the entire training dataset, where training on this small subset achieves good performance. Most of such methods do not apply to modern deep learning, but new formulations based on bi-level optimization have shown promising results on applications like continual learning \\cite{borsos2020coresets}. Related to coreset, other lines of research aim to understand which training samples are    ``valuable'' for modern machine learning, including measuring single-example accuracy \\citep{lapedriza2013all} and counting misclassification rates \\citep{toneva2018empirical}. In fact, dataset distillation is a generalization of such ideas, as the distilled data do not need to be realistic or come from the training set.\n\n\n\n\n\n\n\n\\section{Method}\n\\lblsec{method}\n\\emph{Dataset Distillation} refers to the curation of a small, synthetic training set $\\mathcal{D}_\\mathsf{syn}$ such that a model trained on this synthetic data will have similar performance on the real test set as a model trained on the large, real training set $\\mathcal{D}_\\mathsf{real}$. In this section, we describe our method that directly mimics the long-range behavior of real-data training, matching multiple training steps on distilled data to \\textit{many} more steps on the real data.\n\n\n\nIn \\refsec{expert}, we discuss how we obtain expert trajectories of networks trained on real datasets. In \\refsec{matching}, we describe a new dataset distillation method that explicitly encourages the distilled dataset to induce similar long-range network parameter trajectories as the real dataset, resulting in a synthetically-trained network that performs similarly to a network trained on real data. Finally, \\refsec{memory} describes our techniques to reduce memory consumption.\n\n\n\\subsection{Expert Trajectories}\n\\lblsec{expert}\nThe core of our method involves using \\emph{expert trajectories} $\\tau^*$ to guide the distillation of our synthetic dataset. By expert trajectories, we mean the time sequence of parameters $\\{\\theta^*_t\\}_0^T$ obtained during the training of a neural network on the \\emph{full, real dataset}. To generate these expert trajectories, we simply train a large number of networks on the real dataset and save their snapshot parameters at every epoch. We call these sequences of parameters ``expert trajectories'' because they represent the theoretical upper bound for the dataset distillation task: the performance of a network trained on the full, real dataset. Similarly, we define student parameters $\\hat{\\theta}_t$ as the network parameters trained on synthetic images at the training step $t$. Our goal is to distill a dataset that will induce a similar trajectory (given the same starting point) as that induced by the real training set such that we end up with a similar model.\n\nSince these expert trajectories are computed using only real data, we can pre-compute them before distillation. All of our experiments for a given dataset were performed using the same pre-computed set of expert trajectories, allowing for rapid distillation and experimentation.\n\n\\input{sections\/algorithm}\n\\subsection{Long-Range Parameter Matching}\n\\lblsec{matching}\nOur distillation process learns from the generated sequences of parameters making up our expert trajectories $\\{\\theta^*_t\\}_0^T$. Unlike previous work, our method directly encourages the long-range training dynamics induced by our synthetic dataset to match those of networks trained on the real data.\n\nAt each distillation step, we first sample parameters from one of our expert trajectories at a random timestep $\\theta^*_t$ and use these to initialize our student parameters $\\hat{\\theta}_t \\coloneqq \\theta^*_t$. Placing an upper bound $T^+$ on $t$ lets us ignore the less informative later parts of the expert trajectories where the parameters do not change much.\n\nWith our student network initialized, we then perform $N$ gradient descent updates on the student parameters with respect to the classification loss of the \\emph{synthetic} data:\n\\begin{equation}\n    \\hat{\\theta}_{t+n+1} = \\hat{\\theta}_{t+n} - \\alpha \\nabla \\ell(\\mathcal{A}(\\mathcal{D}_\\mathsf{syn}); \\hat{\\theta}_{t+n}),\n\\end{equation}\nwhere $\\mathcal{A}$ is the differentiable augmentation technique~\\cite{stylegan2ada,zhao2020image,tran2020towards,diffaug} used in previous work~\\cite{dsa}, and $\\alpha$ is the (trainable) learning rate used to update the student network. Any data augmentation used during distillation must be differentiable so that we can back-propagate through the augmentation layer to our synthetic data. Our method does not use differentiable \\textit{Siamese} augmentation since there is no real data used during the distillation process; we are only applying the augmentations to synthetic data at this time. However, we do use the same types of differentiable augmentations on real data during the generation of the expert trajectories.\n\nFrom this point, we return to our expert trajectory and retrieve the expert parameters from $M$ training updates after those used to initialize the student network $\\theta^*_{t+M}$. Finally, we update our distilled images according to the weight matching loss: i.e., the normalized squared $L_2$ error between the updated student parameters $\\hat{\\theta}_{t+N}$ and the known future expert parameters $\\theta^*_{t+M}$:\n\\begin{equation}\n    \\mathcal{L} = \\frac{\\|\\hat{\\theta}_{t+N} - \\theta^*_{t+M}\\|^2_2}{\\|\\theta^*_{t} - \\theta^*_{t+M}\\|_2^2}, %\n    \\lbleq{weight_matching}\n\\end{equation}%\nwhere we normalize the $L_2$ error by the expert distance traveled so that we still get a strong signal from later training epochs where the expert does not move as much. This normalization also helps self-calibrate the magnitude difference across neurons and layers. We have also experimented with other choices of loss functions such as a cosine distance, but find our simple $L_2$ loss works better empirically. We also tried to match the network's output logits between expert trajectory and student network but did not see a clear improvement. We speculate that backpropagating from the network output to the weights introduces additional optimization difficulty.  %\n\n\nWe then minimize this objective to update the pixels of our distilled dataset, along with our trainable learning rate $\\alpha$, by back-propagating through all $N$ updates to the student network. The optimization of trainable learning rate $\\alpha$ serves as automatic adjusting for the number of student and expert updates (hyperparameters $M$ and $N$).  We use SGD with momentum to optimize $\\mathcal{D}_\\mathsf{syn}$ and $\\alpha$ with respect to the above objective. \\refalg{main} illustrates our main algorithm. \n\\input{figText\/table_sota_new}\n\n\\subsection{Memory Constraints}\n\\lblsec{memory}\nGiven that we are back-propagating through many gradient descent updates, memory consumption quickly becomes an issue when our distilled dataset is sufficiently large, as we have to jointly optimize all the images of all the classes at each optimization step. To reduce memory consumption and ease the learning problem, previous methods distill one class at a time~\\cite{dc, dsa, dm}, but this may not be an ideal strategy for our method since the expert trajectories are trained on all classes simultaneously.\n\nWe could potentially circumvent this memory constraint by sampling a new mini-batch at every distillation step (the outer loop in Line \\ref{step:outer} of \\refalg{main}). \nUnfortunately,  this comes with its own issues, as redundant information could be distilled into multiple images across the synthetic dataset, degrading to catastrophic mode collapse in the worst case.\n\nInstead, we can sample a new mini-batch $b$ for every update of the \\emph{student network} (i.e., the inner loop in  Line \\ref{step:inner} of \\refalg{main}) such that all distilled images will have been seen by the time the final weight matching loss (\\refeq{weight_matching}) is calculated. The mini-batch $b$ still contains images from different classes but has much fewer images per class. In this case, our student network update then becomes%\n\\vspace{-5pt}\\begin{equation}\n\\begin{split}\n      & b_{t+n} \\sim \\mathcal{D}_\\mathsf{syn}\\\\\n    & \\hat{\\theta}_{t+n+1} =\\hat{\\theta}_{t+n} - \\alpha \\nabla \\ell(\\mathcal{A}(b_{t+n}); \\hat{\\theta}_{t+n}).\n\\end{split}\\vspace{-10pt}\n\\end{equation}%\nThis method of batching allows us to distill a much larger synthetic dataset while ensuring some amount of heterogeneity among the distilled images of the same class. \n\n\n\n\n\n\n\\section{Experiments}\n\\lblsec{expr}\nWe evaluate our method on various datasets, including \\begin{itemize}[topsep=1pt, partopsep=9pt, itemsep=-1pt, parsep=0.5ex]\n    \\item $32\\times32$ CIFAR-10 and CIFAR-100 (\\refsec{low-res}), two commonly used datasets in dataset distillation literature,\n    \\item $64\\times 64$ Tiny ImageNet (\\refsec{tiny}), a recent benchmark by the concurrent work~\\cite{dm}, and \n    \\item our new $128 \\times 128$ ImageNet subsets (\\refsec{imagenet}).\n\\end{itemize}\nWe provide additional visualizations and ablation studies in the supplementary material. \n\n\n\\myparagraph{Evaluation and Baselines.} %\nWe evaluate various methods according to the standard protocol: training a randomly initialized neural network from scratch on \\emph{distilled} data and evaluating on the validation set. %\n\nTo generate the distilled images for our method, we employ the distillation process detailed in the previous section and Algorithm \\ref{alg:alg}, using the same suite of differentiable augmentations as done in previous work~\\cite{dsa,dm}. The hyperparameters used for each setting (real epochs per iteration, synthetic updates per iteration, image learning rate, etc.) can be found in the supplemental material. \n\n\n\n\n\nWe compare to several recent methods including Dataset Distillation~\\cite{dd} (\\texttt{DD}), Flexible Dataset Distillation ~\\cite{bohdal2020flexible} (\\texttt{LD}), Dataset Condensation~\\cite{dc} (\\texttt{DC}), and Differentiable Siamese Augmentation~\\cite{dsa} (\\texttt{DSA}), along with a method based on the infnite-width kernel limit~\\citep{nguyen2020dataset,nguyen2021dataset} (\\texttt{KIP}) and concurrent works Distribution Matching~\\cite{dm} (\\texttt{DM}) and Aligning Features~\\cite{wang2022cafe} (\\texttt{CAFE}). We also compare our methods with instance selection algorithms including random selection (\\texttt{random}), herding methods~\\cite{chen2010super} (\\texttt{herding}), and example forgetting~\\cite{toneva2018empirical} (\\texttt{forgetting}). \n\n\\myparagraph{Network Architectures.}\nStaying with precedent~\\cite{dc, dsa, dm, nguyen2021dataset}, we mainly employ a simple ConvNet architecture designed by Gidaris and Komodakis~\\cite{gidaris2018dynamic} for our distillation tasks. The architecture consists of several convolutional blocks, each containing a\n$3\\times 3$ convolution layer with 128 filters, Instance normalization~\\cite{ulyanov2016instance}, RELU, and $2\\times2$ average pooling with stride 2. After the convoluation blocks, a single linear layer produces the logits. The exact number of such blocks is decided by the dataset resolution and is specified below for each dataset.\nStaying with this simple architecture allows us to directly analyze the effectiveness of our core method and remain comparable with previous works.\n\n\\subsection{Low-Resolution Data (32$\\times$32)}\n\\lblsec{low-res}\n\nFor low-resolution tasks, we begin with the 32$\\times$32 CIFAR-10 and CIFAR-100 datasets~\\cite{CIFAR10}. For these datasets, we employ ZCA whitening as done in previous work~\\cite{nguyen2020dataset, nguyen2021dataset}, using the Kornia \\cite{kornia} implementation with default parameters.\nStaying with precedent, we use a depth-3 ConvNet taken directly from the open-source code~\\cite{dc, dsa}.\n\nAs seen in \\reftbl{sota}, our method significantly outperforms all baselines in every setting. In fact, on the one image per class setting, we  improve the next best method (\\texttt{DSA} \\cite{dsa}) to almost twice test accuracy, on both datasets. For CIFAR-10, these distilled images can be seen in \\reffig{CIFAR-10}.  CIFAR-100 images are visualized in the supplementary material\n\nIn  \\reftbl{transfer_nn}, we also compare with a recent method \\texttt{KIP}~\\cite{nguyen2020dataset,nguyen2021dataset}, where the distilled data is learned with respect to the Neural Tangent Kernel. Because \\texttt{KIP} training is agnostic to actual network width, we test their result on both a ConvNet of the same width as us and other methods (128) and a ConvNet of larger width (1024) (which is shown in \\texttt{KIP} paper \\citep{nguyen2021dataset}).  Based on the the infinite-width network limit, \\texttt{KIP} may exhibit a gap with practical finite-width networks. Our method does not suffer from this limitation and generally achieves better performance.  In all settings, our method, trained on the 128-width network, outperforms \\texttt{KIP} results evaluated on both widths, except for just one setting where \\texttt{KIP} is applied on the much wider 1024-width network. \n\nAs noted in the previous methods \\citep{dd}, we also see significant diminishing returns when allowing more images in our synthetic datasets. For instance, on CIFAR-10, we see an increase from 46.3\\% to 65.3\\% classification accuracy when increasing from 1 to 10 images per class, but only an increase from 65.3\\% to 71.5\\% when increasing the number of distilled images per class from 10 to 50.\n\nIf we look at the one image per class visualizations in \\reffig{CIFAR-10} (top), we see very abstract, yet still recognizable, representations of each class. When we limit the task to just one synthetic image per class, the optimization is forced to squeeze as much of the class's distinguishing information as possible into just one sample. When we allow more images in which to disperse the class's information, the optimization has the freedom to spread the class's discriminative features among the multiple samples, resulting in a diverse set of structured images we see in \\reffig{CIFAR-10} (bottom) (e.g., different types of cars and horses with different poses).\n\\input{figText\/kip_to_nn}\n\\input{figText\/crossarch}\n\n\\myparagraph{Cross-Architecture Generalization.}\nWe also evaluate how well our synthetic data performs on architectures different from the one used to distill it on the CIFAR-10, 1 image per class task. In \\reftbl{cross}, we show our baseline \\linebreak ConvNet performance and evaluate on ResNet~\\cite{resnet}, VGG~\\cite{vgg}, and AlexNet~\\cite{alexnet}. \n \nFor \\texttt{KIP}, instead of the Kornia \\cite{kornia} ZCA implementation, we use the authors' custom ZCA implementation for evaluation of their method. Our method is solidly the top performer on all the transfer models except for AlexNet where we lie within one standard deviation of \\texttt{DSA}. This could be attributed to our higher baseline performance, but it still shows that our method is robust to changes in architecture.\n\n\\begin{figure}\n    \\centering\n    \\vspace{-6pt}\n    \\includegraphics[width=0.88\\linewidth]{figures\/CIFAR10_1_ipc.pdf}\\\\[-0.6ex]\n    {\\smaller 1 image per class} \\vspace{0.5cm}\\\\\n        \\vspace{-0.34cm}\n\n    \\includegraphics[width=0.88\\linewidth, trim=0 0 0 20, clip]{figures\/CIFAR10.pdf}\\\\[-0.6ex]\n    {\\smaller 10 images per class}\n         \\vspace{-7pt}\n    \\caption{CIFAR-10: The 1 image per class images are more abstract but also more information-dense while the 10 images per class images are more expressive and contain more structure.}\n    \\lblfig{CIFAR-10}\n         \\vspace{-7pt}\n\\end{figure}\n\n\n\\subsection{Short-Range vs.~Long-Range Matching}\n\\lblsec{shortlongrange}\n\nUnlike some prior works (\\texttt{DC} and \\texttt{DSA}), our method performs long-range parameter matching, where $N$ training steps on distilled data match a much larger $M$ steps on real data. Methods that optimize over entire training processes (\\eg \\texttt{DD} and \\texttt{KIP}) can be viewed as even longer range matching. However, their performances fall short of our method (\\eg in \\reftbl{sota}), likely due to related instabilities or inexact approximations. Here, we experimentally confirm our hypothesis that long-range matching achieved by larger $M$ and $N$ in our method is superior to the short-range counterparts (such as small $M$ and $N$ and \\texttt{DSA}).\n\n\nIn \\reffig{mn-alpha} (left), we evaluate our method on different settings of $M$ and $N$. Really short-range matching (with $N=1$ and small $M$) generally exhibits worse performance than long-range matching, with the best performance attained when both $N$ and $M$ are relatively large. Furthermore, as we increase $N$, the power of  $N$ combined steps (on distilled data) becomes stronger and can approximate longer-range behavior, leading to the optimal $M$ values shifting to greater values correspondingly.\n\n\\input{figText\/ablation_plots2}\n\nIn  \\reffig{mn-alpha} (right), we evaluate our method and a short-range matching work (\\texttt{DSA}) on their abilities to approximate real training behavior over short and long ranges. Starting from a set of initial parameters, we set the target parameters to be the result of $\\Delta_t$ training steps on real data (\\ie the long-range behavior that distilled data should mimic). A small (or large) $\\Delta_t$ means evaluating matching over a short (or long) range. For both methods, we test how close they can train the network (using distilled data) from the same initial parameters to the target parameters. \\texttt{DSA} is only optimized to match short-range behavior, and thus errors may accumulate during longer training. Indeed, as $\\Delta_t$ grows larger, \\texttt{DSA} fails to mimic the real data behavior over longer ranges. In comparison, our method is optimized for long-range matching and thus performs much better.\n\n\n\n\n\n\\input{figText\/imagenet_table}\n\n\n\\subsection{Tiny ImageNet (64$\\times$64)}\n\\lblsec{tiny}\n\nIntroduced to the dataset distillation task by the concurrent work, Distribution Matching (\\texttt{DM})~\\cite{dm}, we also show the effectiveness of our algorithm on the 200 class, 64$\\times$64 Tiny ImageNet~\\cite{tiny} dataset (a downscaled subset of ImageNet \\cite{deng2009imagenet}). %\nTo account for the higher image resolution, we move up to a depth-4 ConvNet, similar to \\texttt{DM}~\\cite{dm}.\n\nMost dataset distillation methods (other than \\texttt{DM}) are unable to handle this larger resolution due to extensive memory or time requirement, as the \\texttt{DM} authors also observed \\citep{dm}. In \\reftbl{sota}, our method consistently outperforms the only viable such baseline, \\texttt{DM}. Notably, on the 10 images per class task, our method improves the concurrent work \\texttt{DM} from  12.9\\% and 23.2\\%. A subset of our results is shown in \\reffig{tinyimagenet}. The supplementary material contains the rest of the images. %\n\n\nAt 200 classes and 64$\\times$64 resolution, Tiny ImageNet certainly poses a much harder task than previous datasets. Despite this, many of our distilled images are still recognizable, with a clear color, texture, or shape pattern.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/Tiny.pdf}\n         \\vspace{-7pt}\n    \\caption{Selected samples distilled from Tiny ImageNet, one image per class. Despite the higher resolution, our method still produces high-fidelity images. (Can you guess which classes these images represent? Check your answers in the footnote!\\protect\\footnotemark)}\n    \\lblfig{tinyimagenet}%\n    \\vspace{-7pt}\n\\end{figure}\n\n\\footnotetext{\\smaller{Answers for \\reffig{tinyimagenet}: \\textbf{First Row:} African Elephant, Jellyfish, Kimono, Lampshade, Monarch} \\textbf{Second Row:} Organ, Pizza, Pretzel, Teapot, Teddy}\n\\subsection{ImageNet Subsets (128$\\times$128)}\n\\lblsec{imagenet}\n\\input{figText\/imagenet_fig}\nNext, we push the boundaries of dataset distillation even further by running our method on yet higher resolution images in the form of 128$\\times$128 subsets of ImageNet~\\cite{deng2009imagenet}. Again, due to the higher resolution, we increase the depth of our architecture and use a depth-5 ConvNet for the 128$\\times$128 ImageNet subsets.\n\n\nImageNette (assorted objects) and ImageWoof (dog breeds) are existing subsets~\\cite{imagenette} designed to be easy and hard to learn respectively. We also introduce ImageFruit (fruits), ImageMeow (cats), ImageSquawk (birds), and ImageYellow (yellow-ish things) to further illustrate our algorithm. %\n\nSimilar to Tiny ImageNet, most dataset distillation baselines do not scale up to our ImageNet subset settings.\nAs the code of \\texttt{DM}~\\cite{dm} is not publicly available now, we choose to only compare to the networks trained on the full dataset. We wish to show that our method transfers well to large images and still produces meaningful results at a higher resolution. Validation set accuracies are presented in \\reftbl{imagenet}.\n\nWhile all of the generated images are devoid of high-frequency noise, the tasks still differ in the type of distilled image they induce. For tasks where all the classes have a similar structure but unique textures like ImageSquawk (\\reffig{imagenet}), the distilled images may not have much structure but instead store discriminating information in the textures.\n\nConversely, for tasks where all classes have similar color or textures like ImageYellow (\\reffig{imagenet}), the distilled images seem to diverge from their common trait and accentuate the structure or secondary color that makes them unique. Specifically, note the differences between the distilled ``Banana'' images for the ImageFruit and ImageYellow (bottom row, \\reffig{imagenet}). Although the expert trajectory-generating networks saw the same ``Banana'' training images, the distilled images differ between the two tasks. The distilled ``Banana'' for the ImageYellow task is clearly much ``greener'' than the equivalent image for the ImageFruit task. This implies that the expert networks identify different features by which to identify classes based on the other classes in the task.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion and Limitations}\n\\lblsec{discussion}\nIn this work, we introduced a dataset distillation algorithm by means of directly optimizing the synthetic data to induce similar network training dynamics as the real data.  The main difference between ours and prior approaches is that we are neither limited to the short-range single-step matching nor subject to instability and compute intensity of optimizing over the full training process. Our method balances these two regimes and shows improvement over both. \n\nUnlike prior methods, ours is the first to scale to $128\\times 128$ ImageNet images, which not only allows us to gain interesting insights of the dataset (\\eg in \\refsec{imagenet}) but also may serve as an important step towards practical applications of dataset distillation on real-world datasets. \n\n\\myparagraph{Limitations.} Our use of pre-computed trajectories allows for significant memory saving, at the cost of additional disk storage and computational cost for expert model training. The computational overhead of training and storing expert trajectories is\nquite high. For example, CIFAR experts took $\\sim$3 seconds per epoch (8 GPU hours total for all 200 CIFAR experts) while each ImageNet (subset) expert took $\\sim$11 seconds per epoch (15 GPU hours total for all 100 ImageNet experts). Storage-wise, each CIFAR expert took up $\\sim$60MB of storage while each ImageNet expert took up $\\sim$120MB.\n\n\\section{Appendix}\\subsection{Additional Visualizations}\nWe first include some additional visualizations here. CIFAR-100 (1 image per class) can be seen in \\reffig{cifar-100}. \nAll of Tiny ImageNet (1 image per class) is broken up into Figures \\ref{fig:tiny1} and \\ref{fig:tiny2}. We specifically show the 10 best and worst-performing distilled classes in Figures \\ref{fig:tinygood} and \\ref{fig:tinybad} respectively.\nWe include 10 image per class visualizations of all our 128$\\times$128 ImageNet subsets in Figures \\ref{fig:nette_10}-\\ref{fig:yellow_10}.\n\n\n\n\\subsection{Additional Quantitative Results}\n\n\n\\myparagraph{Analysis of learned learning rates \\boldmath{$\\alpha$}.}\nIn \\reffig{lr}, we explore the effect of our learnable synthetic step size $\\alpha$. The left plot confirms that we learn different values of $\\alpha$ for different combinations of $M$ and $N$. The logic here is that different numbers of synthetic steps $N$ require a different step size $\\alpha$ to cover the same distance as $M$ real steps. The right plot illustrates the practical benefits of our adaptive learning rate; instead of yet another hyper-parameter to tune, our adaptive learning rate works from a wide range of initializations.\n\n\n\n\\myparagraph{Effects of ZCA Whitening}\nNote that \\texttt{DC}, \\texttt{DSA}, and \\texttt{DM} do not use ZCA normalization, while \\texttt{KIP} started using ZCA as it was a ``crucial ingredient for [their] strong results.'' We report our results w\/o ZCA in \\reffig{zca} (Left).\nWe find that ZCA normalization is \\textit{not} critical to our performance. \nHowever, the expert models trained without ZCA normalization take significantly longer to converge. Thus, when distilling using these models as experts, we must use a larger value of $T^+$ (and therefore save more model snapshots). When we use a larger value of $T^+$ for non-ZCA distillations, we get results comparable to or even better than those of the ZCA distillations.  \nIn short, ZCA helps expert convergence but does not notably improve distillation performance. \n\n\\input{figText\/zca}\n\n\\subsubsection{Additional Ablation Studies}\n\n\n\\myparagraph{Initialization, normalization, and augmentation.}\nIn the main paper, we show ablations over several hyper-parameters. Here, we study the role of initialization, data normalization, and data augmentation for CIFAR-100 (1 image per class) in \\reftbl{ablation}. For initialization in particular, recall that we typically initialize our synthetic images with real samples. Here, we evaluate initializing with Gaussian noise instead. Visualizations of these distilled sets can be seen in Figures \\ref{fig:noise}-\\ref{fig:noaug}. We also include a visualization of a set distilled with only one expert trajectory in \\reffig{1exp}.\n\\input{figText\/ablation_tab}\n\n\\input{figText\/ablation_plots}\n\\input{figText\/ablation_plots3}\n\n\\myparagraph{Performance w.r.t. the number of expert trajectories.}\nSince they effectively make up our method's ``training set,'' it is reasonable to assume that having more expert trajectories would lead to better performance. We see that this is indeed the case for the CIFAR-100, 1 image per class setting in \\reffig{experts} (left). However, what's most interesting is the sharp, logarithmic increase in validation accuracy w.r.t. the number of experts. We note the most amount of improvement when increasing from  1 to 20 experts but see almost complete saturation by the time we reach 200. Given how high-dimensional the parameter space of a neural network is, it is remarkable that we can achieve such high performance with so few expert trajectories. \\vspace{10pt}\n\n\\myparagraph{Performance w.r.t. expert time-step range.}\nWhen we initialize our student networks, we do so at a randomly selected time-step from an expert trajectory. We find that it is important to put an upper bound on this starting time-step (\\reffig{experts}, right). If the upper bound is too high, the synthetic data receives gradients from points where the experts movements are small and uninformative. If it is too low, the synthetic data is never exposed to mid and later points in the trajectories, missing out on a significant portion of the training dynamics.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/CIFAR100.pdf}\n    \\caption{CIFAR-100, 1 Image Per Class}\n    \\lblfig{cifar-100}\n    \\vspace{-9pt}\n\\end{figure}\n\n\\subsection{Experiment Details}\n\\myparagraph{Hyper-Parameters.}\nIn \\reftbl{hparams}, we enumerate the hyper-parameters used for our results reported in the main text. Limited compute forced us to batch our synthetic data for some of the larger sets. The ``ConvNet'' architectures are as explained in the main text.\n\\input{figText\/table_hparams}\n\n\\myparagraph{Compute resources.} We had a relatively limited compute budget for our experiments, using any GPU we could access. As such, our experiments were run on a mixture of RTX2080ti, RTX3090, and RTX6000 GPUs. The largest amount of VRAM we used for a single experiment was 144GB over 6xRTX6000 GPUs.\n\n\\myparagraph{Training Time.} Distillation time varied based on dataset and type and number of GPUs used. Regardless of dataset or compute resources, time per distillation iteration scaled linearly with the number of synthetic steps $N$. For CIFAR-100, 1 image per class with $N=20$, we had an average time of 0.6 seconds per distillation step when using a single RTX3090. We ran our experiments for 10000 distillation steps but saw the most improvement within the first 1000.\n\nOur distillation time, in general, is comparable to \\texttt{DC\/DSA}, as they also utilize a bi-level optimization. In the 10 img\/class setting (for example), \\texttt{DC\/DSA} trains on the synthetic data for 50 epochs on the between each update. We include a sample distillation curve in \\reffig{zca} (Right). Both experiments were run on RTX3090. Note that  \\texttt{KIP} requires over 1,000 GPU hours.\n\nRegarding the distillation time for learning different sets on CIFAR10\/100 and TinyImageNet, we report them in \\reftbl{time}. Note that most improvement occurs within the first 1k iterations, but we continue training for 10k.\n\\input{figText\/time}\n\n\\myparagraph{KIP to NN}\nIn the \\texttt{KIP} paper, results are presented for images distilled using the neural tangent kernel method and then evaluated by training a modified width-1024 ConvNetD3. Aside from the increased width of the finite model, the ConvNet architecture used in the \\texttt{KIP} paper also has an additional 1-layer convolutional stem.\n\nUsing the training notebook provided with the \\texttt{KIP} paper, we perform an exhaustive search over a reasonable set of hyper-parameters for the KIP to width-128 NN problem: \\texttt{checkpoint} $\\in$ \\{112, 335, 1000\\}, \\texttt{weight\\_decay} $\\in $ \\{0, 0.0001, 0.001, 0.01\\}, \\texttt{aug} $\\in$ \\{\\texttt{True}, \\texttt{False}\\}, \\texttt{zca} $\\in$ \\{\\texttt{True}, \\texttt{False}\\}, \\texttt{label\\_learn} $\\in$ \\{\\texttt{True}, \\texttt{False}\\}, and \\texttt{norm} $\\in$ \\{\\texttt{none}, \\texttt{instance}\\}. The architecture originally used for KIP to NN in the \\texttt{KIP} paper contained no normalization layers. However, we found that with the smaller width, this model could not even converge on the synthetic training data for CIFAR-100, so we added instance normalization layers as found in the ConvNets we and \\texttt{DC}, \\texttt{DSA}, and \\texttt{DM} use.\n\nIn \\reftbl{kip_hyperparams}, we include the optimal hyper-parameters from this search that were used to obtain the KIP to NN (128\\nobreakdash-width) values reported in the main text.\n\n\\input{figText\/kip_to_nn_params}\n\n\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/Tiny_Good.pdf}\n    \\vspace{-20pt}\n    \\caption{Most-correct distilled images for Tiny ImageNet \\smaller{($\\geq 30\\%$)}}\n    \\label{fig:tinygood}\n    \\vspace{-8pt}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/Tiny_Bad.pdf}\n    \\vspace{-8pt}\n    \\caption{Least-correct distilled images for Tiny ImageNet \\smaller{($\\leq 4\\%$)}}\n    \\label{fig:tinybad}\n    \\vspace{-8pt}\n\\end{figure}\n\n\n\\clearpage\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/cifar100-noise.pdf}\n    \\caption{CIFAR-100, Initialized as Random Noise}\n    \\label{fig:noise}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/cifar100-nozca.pdf}\n    \\caption{CIFAR-100, No ZCA Whitening}\n    \\label{fig:nozca}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/cifar100-noaug.pdf}\n    \\caption{CIFAR-100, No Differentiable Augmentation}\n    \\label{fig:noaug}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/cifar100-single.pdf}\n    \\caption{CIFAR-100, Only 1 Expert Trajectory}\n    \\label{fig:1exp}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/tiny_all_1.pdf}\n    \\caption{Tiny ImageNet, 1 Image Per Class (Classes 1-100)}\n    \\label{fig:tiny1}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/tiny_all_2.pdf}\n    \\caption{Tiny ImageNet, 1 Image Per Class (Classes 101-200)}\n    \\label{fig:tiny2}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/imagenet_10\/imagenette_all.pdf}\n    \\caption{ImageNette, 10 Images Per Class}\n    \\label{fig:nette_10}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/imagenet_10\/imagewoof_all.pdf}\n    \\caption{ImageWoof, 10 Images Per Class}\n    \\label{fig:woof_10}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/imagenet_10\/imagesquawk_all.pdf}\n    \\caption{ImageSquawk, 10 Images Per Class}\n    \\label{fig:squawk_10}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/imagenet_10\/imagemeow_all.pdf}\n    \\caption{ImageMeow, 10 Images Per Class}\n    \\label{fig:meow_10}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/imagenet_10\/imagefruit_all.pdf}\n    \\caption{ImageFruit, 10 Images Per Class}\n    \\label{fig:fruit_10}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/imagenet_10\/imageyellow_all.pdf}\n    \\caption{ImageYellow, 10 Images Per Class}\n    \\label{fig:yellow_10}\n\\end{figure*}\n\\vfill\\break\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAs one of the most intriguing features of quantum mechanics, quantum entanglement has been regarded as a key resource to detect and understand properties of complex many-body systems. For instance, recently a great deal of effort has been devoted to establishing a deep understanding about quantum phase transitions (QPTs) via the examination of their entanglement behaviors~\\cite{osterloh2002scaling,vidal2003entanglement,lambert2004entanglement,gu2004entanglement,le2008entanglement,pereira2016effects,wu2004quantum,wu2006linking,yang2005reexamination,braun2017quantum,vidal2006concurrence}. Quantum phase transition is defined as a transition between distinct ground states of quantum many-body systems when a controlled parameter in the Hamiltonian crosses a critical point~\\cite{sachdev2011quantum}. Compared with the great development of studying QPTs in equilibrium systems, the understanding of the non-equilibrium dynamical phase transitions (DPTs) is still inadequate~\\cite{polkovnikov2011colloquium}. Although some investigations have been done to study the properties of DPTs~\\cite{lo1990ising,jung1990scaling,ryu1996dynamical,sides1998kinetic,yuzbashyan2006relaxation,sciolla2010quantum,diehl2010dynamical,calabrese2011quantum,mitra2012time,heyl2013dynamical,heyl2014dynamical,klinder2015dynamical}, only few works linked it with entanglement~\\cite{lin2016non,jurcevic2017direct,Zhang2017observation}. Recently, the observation of many-body DPTs with up to 10 trapped ion qubits has shown that DPTs in the simulated Ising models can control entanglement production~\\cite{jurcevic2017direct}. Another elegant experiment with a quantum simulator composed of up to 53 ion qubits with long-range Ising interactions has also uncovered the connection between DPTs and many-body correlations~\\cite{Zhang2017observation}. Inspired by those achievements, we investigate theoretically an extension of this connection in quantum systems other than Ising model. For instance, a rich variety of quantum phases in spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) has been investigated theoretically and experimentally~\\cite{dalibard2011colloquium, galitski2013spin, zhou2013unconventional, goldman2014light, zhai2015degenerate, zhang2016properties}. A proposal of simulating spin DPT by ultra-cold atoms has been reported in this system~\\cite{poon2016quantum}. We are wondering whether this kind of DPT can be characterized by the behavior of two-mode entanglement in the synthetic spin space. \n\nIn this paper, by introducing an external perturbation (switching on an additional lattice potential) in the Hamiltonian of a spin-1\/2 BEC of $^{87}$Rb atoms with 1D synthetic spin-orbit coupling, we study the behavior of dynamical two-mode entanglement in spin space and use it to characterize spin DPT. A previous analysis~\\cite{poon2016quantum} shows that the additional lattice potential can drive a periodic evolution of the system in spin space, and there exists a DPT between magnetized and unmagnetized states at a critical lattice depth. By examining the entropic entanglement measure~\\cite{hines2003entanglement, xie2006quantum, byrnes2012quantum}, we show that the periodic motion in spin space leads to the periodic evolution of two-mode entanglement, and the time-averaged entropic entanglement over an oscillation period reaches a maximal value at the DPT. \n\nAlthough the entropy of entanglement provides an excellent indicator of DPT in such a system, it works only for pure states at zero temperature and requires reconstruction of the quantum states via tomography in experiments. To overcome these difficulties, we also use a correlation-based entanglement criterion that is suitable for measuring to detect DPT and study the influence of thermal excitations. Specifically, a criterion was originally introduced by Hillery and Zubairy (HZ)~\\cite{hillery2006entanglement} and developed for double-well BEC systems~\\cite{he2011einstein, he2012einstein}.  We show that the HZ criterion is an excellent proxy for entropic entanglement measure, hence can be used to characterize the DPT. Moreover, we find that the thermal effects will change the critical point of the DPT, which can also be confirmed by the shifts of the maximum of time-averaged HZ entanglement parameter. \n\nThe remainder of this paper is organized as follows. In Sec.~\\ref{Sec:Model}, we introduce the model of DPT induced by external perturbation in BEC with SOC. We then characterize the two-mode entanglement of the system across the DPT via von Neumann entropy and HZ criterion in Sec.~\\ref{Sec:entropy} and Sec.~\\ref{Sec:HZ}, respectively. The results show that the behavior of entanglement can infer the existence of DPT. In Sec.~\\ref{Sec:thermal}, we discuss finite temperature effect and show that the entanglement is robust to thermal excitations and can also be used to signature DPT. Finally we summarize our results and discuss the experimental feasibility in Sec.~\\ref{Sec:conclusion}.\n\n\\section{Model}\\label{Sec:Model}\n\nWe consider a two-component BEC of $^{87}$Rb atoms with one-dimensional (1D) synthetic spin-orbit coupling. As discussed in Ref.~\\cite{poon2016quantum}, such a system features a quantum spin DPT in the presence of an additional lattice potential as external perturbation. By labeling the two atomic components as (pseudo-)spin up and down, the Hamiltonian reads (with natural units $\\hbar=m=1$)\n\\begin{eqnarray}\n\tH&=&H_{0}+H_{\\rm int} ,\\nonumber\\\\\n\tH_{0}&=&\\sum_{s,s'=\\uparrow,\\downarrow}\\int d^3r \\psi_s^\\dagger \\left( -\\frac{\\nabla_r^2}{2}+ik_0\\partial_x\\sigma_z+\\frac{\\Omega}{2}\\sigma_x \\right)_{ss'}\\psi_{s'} , \\nonumber\\\\\n\tH_{\\rm int}&=&\\int d^3r \\frac{g_s}{2}\\left(\\psi_{\\downarrow}^{\\dagger}\\psi_{\\downarrow}^{\\dagger}\\psi_{\\downarrow}\\psi_{\\downarrow}+\\psi_{\\uparrow}^{\\dagger}\\psi_{\\uparrow}^{\\dagger}\\psi_{\\uparrow}\\psi_{\\uparrow}\\right) \\nonumber\\\\\n\t&&+\\int d^3r g_a \\psi_{\\downarrow}^{\\dagger}\\psi_{\\uparrow}^{\\dagger}\\psi_{\\uparrow}\\psi_{\\downarrow}.\n\\end{eqnarray}\nHere $H_0$ is the single-particle Hamiltonian including 1D SOC along the $x$ direction, and $H_{\\rm int}$ is the interaction term. The field operators $\\psi_{s}$ ($\\psi_{s}^{\\dagger}$) annihilates (creates) an atom with spin $s=\\uparrow, \\downarrow$, $\\Omega$ is the Raman coupling strength, and $k_0$ is determined by the Raman laser's wave vector. Note that we assume a spin-symmetric interaction with $g_{\\uparrow\\uparrow}=g_{\\downarrow\\downarrow}=g_{s}$ and $g_{\\uparrow\\downarrow}=g_{a}$, and the two-photon detuning $\\delta=0$ for simplicity. \n\nBy diagonalizing $H_{0}$, the eigenenergies of two subbands are given by $E_{k}^{\\pm}={k^2}\/{2}\\pm \\sqrt{k_0^2k_x^2+{\\Omega^2}\/{4}}$. For $\\Omega<2k_0^2$, the lower subband has a double-minimum structure at $k_x=\\pm k_{\\rm min}$ with $k_{\\rm min}=k_0\\sqrt{1-\\Omega^2\/4k_0^4}$. For $\\Omega>2k_0^2$, it has only a single minimum at $k_{\\rm min}=0$. This structure is the origin of phase transitions among magnetized plane-wave phase, unpolarized stripe phase, and zero-momentum normal phase.\n\nWhen the interaction is concerned, we can assume that the ground-state wave function of the condensate takes the form\n\\begin{equation}\n\t\\Psi=\\sqrt{n}\\left[ \\alpha\\left( \n\t\t\t\\begin{array}{c}\n\t\t\t\t\\cos\\theta \\\\ -\\sin\\theta\n\t\t\t\\end{array}\n\t\\right)e^{ik_mx}+\\beta\\left( \n\t\t\\begin{array}{c}\n\t\t\t\\sin\\theta \\\\ -\\cos\\theta\n\t\t\\end{array}\n\t\\right)e^{-ik_mx} \\right] ,\n\t\\label{ground-state}\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are arbitrary complex numbers satisfying $|\\alpha|^2+|\\beta|^2=1$, and $\\tan2\\theta=\\Omega\/(2k_0k_m)$. In general, the momenta of many-body eigenstates $\\pm k_m$ are different from the minimal positions of single-particle dispersion $\\pm k_{\\rm min}$. But in our parameter region where $\\Omega\/k_0^2<0.6$ and $g_s n \\approx 1.0k_0^2$ ($n=n_{\\uparrow}+n_\\downarrow$ represents the condensate density), we can safely set $k_m\\simeq k_{\\rm min}$. The remarkable property is that there exist two critical Raman couplings~\\cite{li2012quantum}, given by\n\\begin{eqnarray}\n\t\\Omega_{c1}&=&2(k_0^2-2G_2) ,\\\\\n\t\\Omega_{c2}&=&2\\sqrt{(k_0^2+G_1)(k_0^2-2G_2)\\frac{2G_2}{G_1+2G_2}} ,\n\\end{eqnarray}\nwhere $\\Omega_{c1}>\\Omega_{c2}$ and $G_{1}={n}(g_{s}+g_{a})\/4$, $G_{2}={n}(g_{s}-g_{a})\/4$. When $\\Omega<\\Omega_{c2}$, the ground state is a superposition of states with $k_x=\\pm k_{m}$ ($|\\alpha|^2=|\\beta|^2=1\/2$), which is referred as the stripe phase. For $\\Omega_{c2}<\\Omega<\\Omega_{c1}$, there exist two degenerate states with $k_x=k_m$ ($|\\alpha|=1$, $|\\beta|=0$) or $k_x=-k_m$ ($|\\alpha|=0$, $|\\beta|=1$), called the magnetized phase. If the Raman coupling is large enough that $\\Omega>\\Omega_{c1}$, the ground state is at $k_x=0$, giving the normal phase~\\cite{dalibard2011colloquium, galitski2013spin, zhou2013unconventional, goldman2014light, zhai2015degenerate, zhang2016properties, li2012quantum, martone2012anisotropic}.\n\nFor a magnetized phase with, e.g., $k_x=k_m$ ($|\\alpha|=1, |\\beta|=0$), an additional external potential is switched on at $t=0$~\\cite{poon2016quantum},\n\\begin{equation}\n\tV_{\\rm ex}(r,t)=\\left\\{\n\t\t\\begin{array}{lc}\n\t\t\t0, & t<0,\\\\\n\t\t\tV_{0}\\int d^3r \\cos^2k_m x(\\psi_{\\uparrow}^{\\dagger}\\psi_{\\uparrow}+\\psi_{\\downarrow}^{\\dagger}\\psi_{\\downarrow}), & t>0,\n\t\t\\end{array}\n\t\\right. \n\t\\label{Vex}\n\\end{equation}\nwhere $V_0$ is the strength of the perturbation. $V_{\\rm ex}$ can drive resonant couplings between the two degenerate magnetized phases $\\psi_R$ and $\\psi_L$ at $k_{m}$ and $-k_m$, respectively, as indicated schematically in Fig.~\\ref{fig:BEC_ground_state}. The normalized time-dependent condensate wave function takes the form \n\\begin{equation}\n\t|\\Psi_{\\rm BEC}(r,t)\\rangle = \\frac{1}{\\sqrt{N!}} \\left[ \\alpha^{\\ast}(t)\\psi_{R}^{\\dagger}+\\beta^{\\ast}(t)\\psi_{L}^{\\dagger} \\right]^{N}|vac\\rangle ,\n\t\\label{wave-function}\n\\end{equation}\nwhere $N$ is the total number of atoms, $|vac\\rangle$ denotes the vacuum state, and $\\psi_{R}=(\\cos\\theta\\psi_{\\uparrow}-\\sin\\theta\\psi_{\\downarrow})e^{ik_mx}$, $\\psi_{L}=(\\sin\\theta\\psi_{\\uparrow}-\\cos\\theta\\psi_{\\downarrow})e^{-ik_mx}$ are the field operators.\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{figure1}\n\t\\caption{The external perturbation $V_{\\rm ex}$ described by Eq.~(\\ref{Vex}) can induce resonant couplings between the two magnetized phases, marked as $\\psi_{R}$ and $\\psi_{L}$ at $k_m$ and $-k_m$, respectively. }\n\t\\label{fig:BEC_ground_state}\n\\end{figure}\n\nThe quantum spin dynamics of the system under the perturbation given in Eq.~(\\ref{Vex}) has been studied in Ref.~\\cite{poon2016quantum}. It is found that a critical external perturbation strength\n\\begin{equation}\n\tV_{0,{\\rm crit}}=\\frac{2(E_{s}-E_{m})}{\\sin2\\theta} \n\t\\label{V0c}\n\\end{equation}\nis required to have a full transition from one magnetized phase to another. Specifically, when the perturbation strength is below $V_{0,{\\rm crit}}$, the maximum number of atoms that can transit from the initial state with $k_x=k_m$ to the other degenerate state at $k_x=-k_m$ is less than half of the total atom number. On the other hand, when the perturbation strength $V_{0}>V_{0,{\\rm crit}}$, all atoms can fully transit during time evolution. Thus, the critical perturbation strength $V_{0,{\\rm crit}}$ reveals a quantum DPT. We also stress that the critical point $V_{0,{\\rm crit}}$ depends on the condensate interaction energy $g_sn$ via $E_{s}=2G_{1}\\cos^{2}\\theta\\sin^{2}\\theta$ and $E_{m}=G_{2}\\cos^{2}2\\theta$, which characterize the interaction energies for the stripe phase and magnetized phase, respectively. \n\nThe transition mentioned above can be characterized by an order parameter defined as the time average of spin polarization $\\langle s_z(t)\\rangle= |\\alpha(t)|^2-|\\beta(t)|^2$ ($s_z$ is the Pauli matrix acting on the pseudospin space spanned by the two magnetized states) over an oscillation period $T_R$~\\cite{poon2016quantum}\n\\begin{equation}\n\t\\bar{M}=\\frac{1}{T_{R}}\\int_{0}^{T_{R}}\\langle s_{z}(t)\\rangle dt .\n\t\\label{order-parameter}\n\\end{equation}\nIt is straightforward to check that $\\bar{M}> 0$ when the perturbation strength is lower than the critical point and the system is in the dynamical magnetized phase, and $\\bar{M}\\equiv 0$ when the system is dynamically non-magnetized with perturbation strength exceeding the critical point.\n\n\\section{Entropy of entanglement}\\label{Sec:entropy}\n\nPrevious works have studied entanglement of pure states of bipartite systems using entropy of entanglement, which is the von Neumann entropy of the reduced density operator of either of the subsystems. In this section we first study whether the quantum spin DPT at zero temperature can be characterized by the entropic entanglement measure. As the two degenerate ground states of the magnetized phase (labeled by $\\psi_{R}$ and $\\psi_{L}$) can be considered as an analogue of a double-well BEC in momentum space, we can adopt the same method which is commonly used in spatially separated double-well BEC~\\cite{spekkens1999spatial,he2012einstein,he2011einstein,vidal2007entanglement}, and treat $\\psi_{R}$ and $\\psi_{L}$ as two modes within the two-mode approximation~\\cite{spekkens1999spatial}. Our results show that the behavior of the entropic entanglement can act as a indicator of the quantum spin DPT in this system.\n\nBy applying the two-mode approximation, the time-dependent wave function in Eq.~(\\ref{wave-function}) can be expanded in term of Fock states as\n\\begin{equation}\n\t|\\Psi_{\\rm BEC}(t)\\rangle = \\sum_{n=0}^{N} \\sqrt{C_{N}^{n}} \\alpha^{\\ast n}\\beta^{\\ast N-n}|n,N-n\\rangle_{R,L} ,\n\t\\label{two-mode_wave_function}\n\\end{equation} \nwhere $C_{N}^{n}$ is the binomial coefficient, and the Fock basis are defined as\n\\begin{equation}\n\t|n,N-n\\rangle_{R,L}=\\frac{(\\psi_{R}^{\\dagger})^{n}}{\\sqrt{n!}}\\frac{(\\psi_{L}^{\\dagger})^{N-n}}{\\sqrt{(N-n)!}}|vac\\rangle .\n\t\\label{number_state}\n\\end{equation}\n\nThe dynamics can be described by defining the field operator $\\psi(t)=\\alpha(t)\\psi_{R}+\\beta(t)\\psi_{L}$, and considering Heisenberg equation\n\\begin{equation}\n\ti\\frac{d\\psi(t)}{dt}=\\left[ \\psi(t), H_{0}+H_{\\rm int}+V_{\\rm ex} \\right],\n\t\\label{Heisenberg}\n\\end{equation}\nleading to the following equation of motion\n\\begin{equation}\n\ti\\frac{d}{dt}\\left( \n\t\t\\begin{array}{c}\n\t\t\t \\alpha(t) \\\\ \\beta(t)\n\t\t\\end{array}\n\\right)=H_{\\rm eff}\\left( \n\t\\begin{array}{c}\n\t\t\\alpha(t) \\\\ \\beta(t)\n\t\\end{array}\n\\right).\n\t\\label{motion}\n\\end{equation}\nHere, the effective two-mode Hamiltonian is given by\n\\begin{eqnarray}\n\tH_{\\rm eff}&=&E_{k}^{-}+\\frac{V_{0}}{2}+G_{1}+V_{p}s_{x}+E_{m}(|\\alpha|^{2}-|\\beta|^{2})s_{z} \\nonumber\\\\\n\t&&+2E_{s}\\left[ {\\rm Re}(\\alpha\\beta^{\\ast})s_{x}-{\\rm Im}(\\alpha\\beta^{\\ast})s_{y} \\right],\n\t\\label{Heff}\n\\end{eqnarray}\nwhere $s_{x,y,z}$ are spin matrices spanned by the two magnetized states, and $V_{p}=V_{0}\\cos\\theta\\sin\\theta\/2$ represents the coupling strength induced by external perturbation.\n\nTheoretically, the von Neumann entropy $E_{\\rm vn}=-{\\rm Tr}\\left[ \\rho_{R}\\log\\rho_{R} \\right]$ can be used to evaluate the entanglement between two subsystems, where $\\rho_{R}={\\rm Tr}_{L}[\\rho_{R,L}]$ is the reduced density operator for $\\psi_R$ and $\\rho_{R,L}=|\\Psi_{\\rm BEC}\\rangle\\langle\\Psi_{\\rm BEC}|$. Considering the specific form of the time-dependent wave function $|\\Psi_{\\rm BEC}\\rangle$ described in Eq.~(\\ref{two-mode_wave_function}), the entropy of entanglement between $\\psi_R$ and $\\psi_L$ thus reads\n\\begin{equation}\n\tE_{\\rm vn}=-\\sum_{n=0}^{N}C_{N}^{n}|\\alpha|^{2n}|\\beta|^{2(N-n)}\\log_2\n\t\\left[C_{N}^{n}|\\alpha|^{2n}|\\beta|^{2(N-n)}\\right].\n\t\\label{entanglement_entropy}\n\\end{equation}\nHere, $E_{\\rm vn}=0$ for separable product states, and $E_{\\rm max}=\\log_2(N+1)$ for maximally entangled states when all atoms are equally represented~\\cite{hines2003entanglement}. As a measure of the entanglement, we plot in Fig.~\\ref{fig:entropy} the ratio of $E_{\\rm vn}$ to its corresponding maximum value $E=E_{\\rm vn}\/E_{\\rm max}$ for various numbers $N$ and external perturbation strength $V_0$~\\cite{hines2003entanglement, xie2006quantum, byrnes2012quantum}. The values of $E$ range from $0$ to $1$. Note that the ratio $E$ only represents how much the entanglement of the state $|\\Psi_{\\rm BEC}\\rangle$ with total $N$ atoms is less than its corresponding maximum entanglement, $\\log(N+1)$. It is meaningless to compare the values of $E$ for different $N$. As a comparison, the dynamical evolution of the spin polarization $\\langle s_z \\rangle$ is also shown using the same parameter $V_0$. \n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.245\\textwidth]{sz_06V}\n\t\\includegraphics[width=0.245\\textwidth]{sz_0999V}\n\t\\includegraphics[width=0.245\\textwidth]{sz_1001V}\n\t\\includegraphics[width=0.245\\textwidth]{sz_14V}\\\\\n\t\\includegraphics[width=0.245\\textwidth]{entropy_06V}\n\t\\includegraphics[width=0.245\\textwidth]{entropy_0999V}\n\t\\includegraphics[width=0.245\\textwidth]{entropy_1001V}\n\t\\includegraphics[width=0.245\\textwidth]{entropy_14V}\n\\caption{The evolution of entropic entanglement $E$ is related to the evolution of pseudospin $\\langle s_z\\rangle$. From left to right, the external perturbation strength is (a)(e) $V_0 = 0.6V_{0,{\\rm crit}}$, (b)(f) $0.999V_{0,{\\rm crit}}$, (c)(g) $1.001V_{0,{\\rm crit}}$, and (d)(h) $1.4V_{0,{\\rm crit}}$. For each situation, we plot the entropic entanglement for various numbers of atoms $N=1$ (blue), $N=10$ (green), and $N=100$ (red solid). Other parameters are used as $g_{s}n=1.0k_{0}^{2}$ and $\\Omega=0.3k_{0}^{2}$.}\n\\label{fig:entropy}\n\\end{figure*}\n\nFrom Fig.~\\ref{fig:entropy}, it is clear that the entropic entanglement evolves with time periodically, and the period is related to that of the spin oscillation. In the regime where the perturbation strength is below the critical value $V_{0}<V_{0,{\\rm crit}}$, as shown in Figs.~\\ref{fig:entropy}(a)(b)(e)(f), only less than half of the atoms can transit from one magnetized state at $k_x=k_m$ to the other at $k_x=-k_m$. Thus the spin evolves only on the upper half spherical surface of the Bloch sphere, i.e., $\\langle s_z\\rangle>0$. In such a case, the oscillatory period of the two-mode entanglement is as same as that of the spin oscillation. When all the atoms are at the magnetized state $k_x=\\pm k_m$, we have $\\langle s_z\\rangle=\\pm1$ and $E=0$, i.e., the two-mode entanglement doesn't exist. When atoms distribute equally in the two modes, i.e., $\\langle s_{z}\\rangle$ approaches zero, the two-mode entanglement reaches a maximal value. On the other hand, in the regime where the external perturbation strength exceeds the critical value, as shown in Figs.~\\ref{fig:entropy}(c)(d)(g)(h), the spin evolves on the entire spherical surface of the Bloch sphere for one oscillation period $T_R$. A complete transition of atoms from one magnetized phase to another occurs, i.e., $\\langle s_z\\rangle=-1$, at $T_R\/2$, where $E=0$ ends one complete cycle of entanglement motion. Therefore, for this case the period of entanglement oscillation is half of that of the spin evolution. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{entropy_mean_N}\n\t\\caption{The time-averaged entropic entanglement measure $\\bar{E}$ calculated as a function of perturbation strength $V_0\/V_{0,{\\rm crit}}$ (left axis). Different lines represent different total numbers of atoms $N=1$ (blue), $N=10$ (green), and $N=100$ (red). The entanglement displays a sharp peak at the critical value $V_{0,{\\rm crit}}$ where the DPT occurs as characterized by the order parameter $\\bar{M}$ (black dashed, right axis). This observation suggests that the critical behavior of entropic entanglement measure can be used to identify the DPT. Other parameters are used as those in Fig.~\\ref{fig:entropy}.}\n\t\\label{fig:time-average_entropy}\n\\end{figure}\n\nThe spin DPT across $V_{0,{\\rm crit}}$ can be characterized by the order parameter $\\bar{M}$ defined in Eq.~(\\ref{order-parameter}), which is depicted by the dashed black line in Fig.~\\ref{fig:time-average_entropy}. To reveal the connection between DPT and entanglement, we define the time-averaged entropic entanglement measure as\n\\begin{equation}\n\t\\bar{E}(V_{0})=\\frac{1}{T_{R}}\\int_{0}^{T_{R}}E(t)dt,\n\t\\label{time-average_entropy}\n\\end{equation}\nwhich is plotted as a function of perturbation strength for different particle numbers by solid lines in Fig.~\\ref{fig:time-average_entropy}. Apparently, the entanglement measure ${\\bar E}$ features a sharp peak at the critical point where the order parameter ${\\bar M}$ distinctly changed from $\\bar{M}>0$ (dynamical magnetized phase) to $\\bar{M}=0$ (dynamical unmagnetized phase), indicating that one can achieve maximal entanglement in the vicinity of the critical point of DPT. We also notice that the entanglement peak at DPT is less sharp with increasing particle number $N$. This is because when we define the entanglement measure $E$, the maximal entanglement $E_{\\rm max}$ in the denominator increases logarithmically with $N$. Thus, when $N$ goes larger, the normalization leads to a flatter peak of $E$, making the usage of entropic entanglement measure as an indicator of DPT rather ambiguous. \n\n\\section{Correlation-based entanglement measure for two-mode system}\\label{Sec:HZ}\n\nAlthough the entropy of entanglement is a useful measure to characterize the DPT, it is an entanglement measure only for pure states and therefore cannot account for the effects of finite temperatures. In addition, measuring entropic entanglement requires reconstruction of the quantum states via tomography, which demands the state-of-art technique in experiments even for special systems~\\cite{islam2015measuring} and remains challenging for general quantum systems. To circumvent these difficulties, in the following we use an experimentally feasible correlation-based entanglement criterion to detect entanglement undergoing a DPT and discuss the influence of thermal excitations. \n\nA sufficient entanglement criterion for a two-mode system is the operator product measure $|\\langle ab^{\\dagger}\\rangle|^2>\\langle a^\\dagger ab^\\dagger b\\rangle$ given by Hillery and Zubairy (HZ)~\\cite{hillery2006entanglement}, where $a$ and $b$ denote the annihilation operators of the two modes. \nFor double-well BEC systems, a spin version of HZ criterion has also been developed~\\cite{he2011einstein, he2012einstein}. Specifically, the state is entangled if\n\\begin{equation}\n\tE_{\\rm HZ}=\\frac{\\Delta^2 J_x+\\Delta^2 J_y}{\\langle \\hat{N} \\rangle\/2}<1,\n\t\\label{HZ_J}\n\\end{equation}\nwhere the spin operators $J_{x}=(\\psi_{R}^{\\dagger}\\psi_{L}+\\psi_{R}\\psi_{L}^{\\dagger})\/2$, $J_{y}=(\\psi_{R}^{\\dagger}\\psi_{L}-\\psi_{R}\\psi_{L}^{\\dagger})\/(2i)$, $J_{z}=(\\psi_{R}^{\\dagger}\\psi_{R}-\\psi_{L}^{\\dagger}\\psi_{L})\/2$ with canonical commutation relations $[J_{x},J_{y}]=iJ_{z}$ (and cyclic permutations). The variance of measurements of $J_{x,y}$ is defined as $\\Delta^2 J_{x,y}\\equiv \\langle J^2_{x,y}\\rangle-\\langle J_{x,y}\\rangle^2$, and the expectation values for $\\hat{N}=\\psi_{R}^{\\dagger}\\psi_{R}+\\psi_{L}^{\\dagger}\\psi_{L}$ are fixed at $N$ for state~(\\ref{two-mode_wave_function}). It is convenient to quantify entanglement using spin-operator methods, and this type of spin-operator variance has been measured experimentally by expanding the two condensates and measuring the absorption imaging average fringe visibility~\\cite{esteve2008squeezing, ji2014experimental}. \nWe emphasize that the variances $\\Delta^2 J_{x,y}$ for the state~(\\ref{two-mode_wave_function}) are proportional to the number of atoms, so that the value of $E_{\\rm HZ}$ is independent of $N$ as shown in spatial double-well BEC systems~\\cite{he2012einstein}.\n\nTo get the evolution of the entanglement parameter $E_{\\rm HZ}$, we consider the Heisenberg equation for $\\langle \\psi_{R}^{\\dagger}\\psi_{R} \\rangle$, $\\langle \\psi_{R}^{\\dagger}\\psi_{L} \\rangle$, $\\langle\\psi_{L}^{\\dagger}\\psi_{L}\\rangle$, $\\langle \\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}^{\\dagger}\\psi_{L} \\rangle$, etc. Due to the nonlinear interaction terms in the Hamiltonian, the sets of equations can't be closed. Thus we truncate the set of equations at the fourth order of operators with mean field approximation, leading to\n\\begin{widetext}\n\\allowdisplaybreaks\n\\begin{eqnarray}\n\t\\frac{d}{dt}\\langle\\psi_{R}^{\\dagger}\\psi_{R}\\rangle &=& -iV_{p}\\left[ \\langle\\psi_{R}^{\\dagger}\\psi_{L}\\rangle-\\langle\\psi_{L}^{\\dagger}\\psi_{R}\\rangle \\right] , \\nonumber\\\\\n\t\\frac{d}{dt}\\langle\\psi_{L}^{\\dagger}\\psi_{L}\\rangle &=& -iV_{p}\\left[ \\langle\\psi_{L}^{\\dagger}\\psi_{R}\\rangle-\\langle\\psi_{R}^{\\dagger}\\psi_{L}\\rangle \\right] , \\nonumber\\\\\n\t\\frac{d}{dt}\\langle\\psi_{R}^{\\dagger}\\psi_{L}\\rangle &=& -i\\left[ V_{p}( \\langle\\psi_{R}^{\\dagger}\\psi_{R}\\rangle-\\langle\\psi_{L}^{\\dagger}\\psi_{L}\\rangle )+2(E_{s}-E_{m})( \\langle \\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L} \\rangle - \\langle \\psi_{R}^{\\dagger}\\psi_{L}^{\\dagger}\\psi_{L}\\psi_{L}\\rangle ) \\right] , \\nonumber\\\\\n\t\\frac{d}{dt}\\langle\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}^{\\dagger}\\psi_{L}\\rangle &\\approx& -iV_{p}\\left[ \\langle \\psi_{R}^{\\dagger}\\psi_{R}\\psi_{R}\\psi_{L}^{\\dagger}\\rangle+\\langle\\psi_{R}^{\\dagger}\\psi_{L}^{\\dagger}\\psi_{L}\\psi_{L}\\rangle-\\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}\\rangle-\\langle\\psi_{R}\\psi_{L}^{\\dagger}\\psi_{L}^{\\dagger}\\psi_{L}\\rangle \\right] , \\nonumber\\\\\n\t\\frac{d}{dt}\\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}\\rangle &\\approx& -i\\left[ V_{p}( \\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{R}\\rangle+\\langle \\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{L}\\psi_{L}\\rangle -2\\langle\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}^{\\dagger}\\psi_{L}\\rangle)\\right.\\nonumber\\\\\n\t&&\\ \\ \\ \\ \\ \\  \\left.+2(E_{s}-E_{m})\\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}\\rangle( \\langle\\psi_{R}^{\\dagger}\\psi_{R}\\rangle-\\langle\\psi_{L}^{\\dagger}\\psi_{L}\\rangle+1) \\right] , \\nonumber\\\\\n\t\\frac{d}{dt}\\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{L}\\psi_{L}\\rangle &\\approx& -i\\left[ 2V_{p}( \\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}\\rangle-\\langle\\psi_{R}^{\\dagger}\\psi_{L}^{\\dagger}\\psi_{L}\\psi_{L}\\rangle )+4(E_{s}-E_{m})\\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{L}\\psi_{L}\\rangle( \\langle\\psi_{R}^{\\dagger}\\psi_{R}\\rangle-\\langle\\psi_{L}^{\\dagger}\\psi_{L}\\rangle ) \\right] ,\\nonumber\\\\\n      \\frac{d}{dt}\\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{R}\\rangle&=&-2iV_{p}\\left[ \\langle\\psi_{R}^{\\dagger}\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{L}\\rangle-\\langle\\psi_{R}^{\\dagger}\\psi_{R}\\psi_{R}\\psi_{L}^{\\dagger}\\rangle \\right] .\n\\end{eqnarray}\n\\allowdisplaybreaks[0]\n\\end{widetext}\nOther terms can be derived directly by considering symmetry and conjugate properties of the equations.\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.245\\textwidth]{EHZ_compare_06V}\n\t\\includegraphics[width=0.245\\textwidth]{EHZ_compare_0999V}\n\t\\includegraphics[width=0.245\\textwidth]{EHZ_compare_1001V}\n\t\\includegraphics[width=0.245\\textwidth]{EHZ_compare_14V}\n\t\\caption{A comparison between the evolution of HZ entanglement parameter $E_{\\rm HZ}$ (blue dashed) and the entropic entanglement $1-E$ with total number of atoms $N=10$ (green solid) and $N=100$ (red solid). The strength of external perturbation is (a) $V_{0}=0.6V_{0,{\\rm crit}}$, (b) $0.999V_{0,{\\rm crit}}$, (c) $1.001V_{0,{\\rm crit}}$, and (d) $1.4V_{0,{\\rm crit}}$, respectively. The HZ entanglement parameters behaves similarly to the entropic entanglement for different $N$. Other parameters are used as those in Fig.~\\ref{fig:entropy}.}\n\t\\label{fig:HZ-compare}\n\\end{figure*}\n\nBy solving the equations above numerically, we show in Fig.~\\ref{fig:HZ-compare} the evolution of entropic and HZ entanglement signature. To make a direct comparison, the entropic entanglement measure is plotted as $1-E < 1$.\nWe find that the evolution of HZ entanglement parameter exhibits the very same qualitative behavior as that of the entropic entanglement measure. Specifically, if the system is in a magnetized state where $|\\alpha|=0,\\, 1$ and $\\langle s_z\\rangle\\rightarrow \\pm 1$, the HZ entanglement parameter $E_{\\rm HZ}\\rightarrow 1$ showing zero entanglement. On the other hand, the best entanglement would be obtained when the system is in a stripe state where atoms are distributed equally in the two modes with $|\\alpha|^2=0.5$ and $\\langle s_z\\rangle\\rightarrow 0$.  \n\nIn order to characterize the DPT, we introduce the time-averaged HZ entanglement parameter\n\\begin{equation}\n\t\\bar{E}_{\\rm HZ}=\\frac{1}{T_{R}}\\int_{0}^{T_R} E_{\\rm HZ}(t)dt .\n\t\\label{HZ-t}\n\\end{equation}\nOur results for the entropic entanglement measure $1-\\bar{E}$ and the correlation-based HZ entanglement parameter $\\bar{E}_{\\rm HZ}$ are depicted in Fig.~\\ref{fig:HJ-mean}, showing that the HZ measure is an excellent proxy for entropic entanglement measure. The time-averaged HZ entanglement parameter presents a sharp dip in the vicinity of the critical point, which can be used as an indicator for the DPT. Comparing to the entropic measure discussed in Sec.~\\ref{Sec:entropy}, the HZ parameter benefits not only from the experimental feasibility, but also from the fact that the dip is not affected by the total numbers of atoms, and thus can identify the DPT in systems with large particle numbers.\n\n\n\\section{Thermal Effects}\\label{Sec:thermal}\n\nSo far we have studied how the two-mode entanglement can characterize the DPT at zero temperature. In the practical experiments, there are inevitable thermal excitations due to the finite temperature. In general, thermal excitations reduce entanglement because they will cause decoherence and degrade the purity. In the present system, an increasing finite temperature reduces the condensate density $n_{C}$ and consequently changes the interaction energies of the condensate $g_sn_{C}$, as well as the critical perturbation strength $V_{0,{\\rm crit}}$. This may change the dynamics of pseudospin and induce new critical points for DPT. We explore how these new critical points at finite temperatures connect with the two-mode entanglement examined by the HZ entanglement criterion. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{EHZ_compare_mean}\n\t\\caption{The time-averaged HZ entanglement parameter $\\bar{E}_{\\rm HZ}$ (blue dashed) calculated as a function of perturbation strength $V_{0}\/V_{0,{\\rm crit}}$, in comparison with the time-averaged entropic entanglement measure $1-\\bar{E}$ with $N=10$ (green solid) and $N=100$ (red solid). In the vicinity of the critical point $V_{0,{\\rm crit}}$, $\\bar{E}_{\\rm HZ}$ shows a discontinuity in the first-order derivative, which signifies the occurrence of DPT. Other parameters are used as those in Fig.~\\ref{fig:entropy}.}\n\t\\label{fig:HJ-mean}\n\\end{figure}\n\nIn order to study the effects of thermal excitations, we solve the quasiparticle spectra by using Hartree-Fock-Bogoliubov theory with Popov approximation~\\cite{pethick2002bose,liao2014spin,chen2015collective,PhysRevA.96.013625}. The general wave function $\\Psi_{\\pm k_{m},\\sigma}$ with $\\sigma=\\uparrow, \\downarrow$ is given by\n\\begin{equation}\n\t\\Psi_{\\pm k_{m},\\sigma}(x,t)=e^{-i\\mu t\\pm ik_{m}x}\\left[ \\Phi_{\\pm k_{m},\\sigma}+\\delta\\Phi_{\\pm k_{m},\\sigma}(x,t) \\right] ,\n\t\\label{general_wave_function}\n\\end{equation}\nwhere $\\mu$ is the chemical potential, $\\Phi_{\\pm k_{m}, \\sigma}$ are the condensate wave functions, and $\\delta\\Phi_{\\pm k_{m},\\sigma}(x,t)$ are the fluctuations with the following form\n\\begin{eqnarray}\n\t\\delta\\Phi_{\\pm k_{m},\\sigma}&=&\\psi_{\\pm k_{m}+q,\\sigma}e^{-i\\omega t}+\\phi_{\\pm k_{m}-q,\\sigma}^{\\dagger}e^{i\\omega t}.\n\\end{eqnarray}\nHere, $q$ is the quasi momentum and $\\omega$ is the frequency. It is convenient to solve the Bogoliubov spectrum by expanding $\\psi_{\\pm k_{m}+q,\\sigma}$ and $\\phi_{\\pm k_{m}-q,\\sigma}$ in the Bloch form~\\cite{li2013superstripes,poon2016quantum} with basis $\\psi_{\\tilde{q}+2 \\ell k_{m},\\sigma}$ and $\\phi_{-\\tilde{q}+2 \\ell k_{m},\\sigma}$, \n\\begin{eqnarray}\n\t\\psi_{\\pm k_{m}+q,\\sigma}&=&\\sum_{\\ell}\\psi_{\\tilde{q}+2 \\ell k_{m},\\sigma} ,\\nonumber\\\\\n\t\\phi_{\\pm k_{m}-q,\\sigma}&=&\\sum_{\\ell}\\phi_{-\\tilde{q}+2 \\ell k_{m},\\sigma} ,\n\\end{eqnarray}\nwhere $\\ell$ is an integer and $|\\tilde{q}|<k_{m}$.\n\nSolving the Heisenberg equations $i\\partial_{t}\\Psi_{\\pm k_{m},\\sigma}=[\\Psi_{\\pm k_{m},\\sigma}, H]$ by substituting the general wave function of Eq.~(\\ref{general_wave_function}) with the mean-field decoupling of interaction and ignoring the anomalous densities $n_{a}=\\langle \\delta\\Phi_{\\pm k_m,\\sigma}\\delta\\Phi_{\\pm k_m, \\sigma'} \\rangle$ (i.e. the Hartee-Fock-Bogliubov-Popov approximation)~\\cite{pethick2002bose,liao2014spin,chen2015collective,PhysRevA.96.013625}, we can obtain the structure of the energy bands and the density of states $g(\\alpha,E)$ of the Bogoliubov quasiparticles at energy $E$ as functions of the superposition coefficient $\\alpha$. Please note that we have applied the approximation to the condensate wave functions\n\\begin{eqnarray}\n\t\\Phi_{k_{m},\\uparrow}&=&\\sqrt{n_C}\\alpha\\cos\\theta_{k_{m}} , \\Phi_{k_{m},\\downarrow}=-\\sqrt{n_C}\\alpha\\sin\\theta_{k_{m}} , \\nonumber\\\\\n\t\\Phi_{-k_{m},\\uparrow}&=&\\sqrt{n_C}\\beta\\sin\\theta_{k_{m}} , \\Phi_{-k_{m},\\downarrow}=-\\sqrt{n_C}\\beta\\cos\\theta_{k_{m}} .  \\nonumber\\\\\n\\end{eqnarray}\nThe density of excited atoms $n_{\\rm ex}$ can be extracted as\n\\begin{equation}\n\tn_{\\rm ex}=\\sum_{s=\\pm}\\sum_{\\sigma=\\uparrow,\\downarrow}\\langle \\delta\\Psi^{\\dagger}_{sk_m,\\sigma}\\delta\\Psi_{sk_m,\\sigma} \\rangle .\n\t\\label{nex}\n\\end{equation}\nThe population of the excited states can then be directly derived as $\\Gamma(\\alpha,T)={n_{\\rm ex}}\/{n}$ with $n$ the total density. Using realistic experimental parameters, we estimate that the ratio $\\Gamma$ is usually below $0.20$ with $T<70$nK, such that the thermal fluctuations are less important when the system evolves between the magnetized states and stripe state~\\cite{poon2016quantum}. \n\nBecause the excitations only take a very small fraction, we expect that the thermal effects on DPT and two-mode entanglement are also limited. Considering the fact that the system becomes a mixed state due to the presence of thermal excitations, the entropic entanglement measure cannot be used to characterize the DPT, while the HZ entanglement parameter $E_{\\rm HZ}$ is still suitable within two-mode approximation.\n\nIn the low excitation region, we assume that the mixed state of the system takes the following form\n\\begin{equation}\n\t\\rho=[1-\\Gamma(\\alpha,T)]\\rho_{g}+\\Gamma(\\alpha,T)\\delta\\rho ,\n\t\\label{mixed_state}\n\\end{equation}\nwhere $\\rho_{g}$ and $\\delta\\rho$ are the density operators of ground state and excited states of the system, respectively. Note that the effect induced by all excited states are summed over and denoted by $\\delta\\rho$, which can be taken as a perturbation when the thermal excitations are only factional at low temperature~\\cite{poon2016quantum}. The expectation value of an arbitrary operator $O$ can be calculated by $\\langle O\\rangle={\\rm Tr}(O\\rho)$. Thus we can calculate the evolution of the order parameter $\\bar{M}$ of DPT and the time-averaged entanglement parameter $\\bar{E}_{\\rm HZ}$ with thermal excitations at finite temperatures. \n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{order_full_mean_HFBP}\n\t\\includegraphics[width=0.4\\textwidth]{EHZT_full_mean_HFBP}\n\t\\caption{(a) The order parameter $\\bar{M}$ and (b) the time-averaged HZ entanglement parameter $\\bar{E}_{\\rm HZ}$ at temperatures of $T=0$nK (blue), $30$nK (green), $50$nK (red), and $70$nK (cyan) by using Hartree-Fock-Bogoliubov-Popov approximation. The critical point of the DPT shifts towards a smaller value of perturbation strength with elevated temperature, where the HZ entanglement parameter features a sharp dip as an unambiguous signature. Other parameters are used as those in Fig.~\\ref{fig:entropy}.}\n\t\\label{fig:EHZ_mean}\n\\end{figure}\n\nThe numerical results of the order parameter $\\bar{M}$ are given in Fig.~\\ref{fig:EHZ_mean}(a) for different finite temperatures. It can be seen that the transition point shifts towards lower value of $V_0$ with increasing temperature. This is because the thermal excitations reduce the condensate density, which leads to smaller interaction energy and thus induces smaller value of critical perturbation strength defined in Eq.~(\\ref{V0c}). The transition points can also be identified from the results of entanglement parameter $\\bar{E}_{\\rm HZ}$ as shown in Fig.~\\ref{fig:EHZ_mean}(b), where the maximal value of entanglement (i.e., minimum of parameter $\\bar{E}_{\\rm HZ}$ represented by the sharp dip) appear exactly at the transition points where the order parameter $\\bar{M}>0$ switching to $\\bar{M}=0$ at different temperatures. This observation suggests that the DPT can be characterized by the time-average two-mode entanglement even under finite temperatures. \n\nWe also notice from Fig.~\\ref{fig:EHZ_mean}(b) that at the sharp dip, the lowest value of the parameter $\\bar{E}_{\\rm HZ}$ increases with elevated temperature, indicating that thermal effect is in general detrimental to entanglement. However, since the transition point also shifts with temperature, for a fixed perturbation strength $V_0 < V^{T=0}_{0,{\\rm crit}}$ one may find that a lower value of $\\bar{E}_{\\rm HZ}$ can be obtained with increasing temperature. Physically, this corresponds to the fact that although a finite temperature will inevitably introduce thermal fluctuations and hence degrade entanglement, in certain circumstances it can also bring the system closer to a phase transition point, which features maximal entanglement. \n\n\n\\section{Conclusion}\\label{Sec:conclusion}\n\nIn summary, we have studied the possibility that using two-mode entanglement in the synthetic (i.e., spin) space to characterize the dynamical phase transitions in a BEC with spin-orbit coupling. By adding an additional lattice potential in the system Hamiltonian as perturbation, we show that the time-averaged entropic entanglement reaches a maximal value at the critical values of perturbation strength where the system is driven from dynamic magnetized phase to a dynamic nonmagnetized phase. The sharp peak of entropic entanglement measure can be used to identify the existence of the DPT. On the other hand, this provides inspiration for generating the maximal entanglement between two modes. Then, considering the difficulty of measuring entropic entanglement in experiments, we have also examined another correlation-based entanglement criterion which is more feasible for experimental test. Our results shows that the time-averaged HZ entanglement parameter is an excellent substitute for entropic entanglement measure, which can not only determine the existence, but also qualitatively characterize the extent of entanglement. Furthermore, it can also be used to account for the effects of thermal excitations induced by finite temperatures. We find that the thermal effects will change the critical point of the DPT, which can be revealed by the shift of the sharp dips of the time-averaged HZ entanglement parameter. \nThis work may broaden the understanding of the connection between quantum correlations and dynamical phase transitions in the SOC systems with interactions. \n\nIn the end, we would briefly comment about the experimental feasibility of the study. The synthetic spin-orbit coupling in ultracold atomic gases for pseudospin-1\/2 systems has been realized experimentally~\\cite{lin2011spin, wang2012spin, cheuk2012spin, ji2014experimental}. The periodic perturbation $V_{\\rm ex}$ is simply a lattice potential which can be generated by standing-wave lasers with wave vector $k_{m}$. The time-averaged two-mode entanglement parameter $\\bar{E}_{\\rm HZ}$ can be detected by measuring the evolution of pseudospin variance of the BEC which has been realized in many experiments for quantum squeezing and metrology~\\cite{esteve2008squeezing,riedel2010atom,luo2017deterministic}. Thus, we expect that the system can be readily prepared and investigated with present experimental technique. \n\n\n\\begin{acknowledgements}\n\nWe acknowledge illuminating discussions with X.-J. Liu, T. F. Jeffrey Poon, and H. Zhai. This work is supported by the Ministry of Science and Technology of China (Grant No. 2016YFA0301302), National Natural Science Foundation of China (Grants No. 11622428, No. 11274025, No. 61475006, No. 11434011, No. 11522436, and No. 11774425), the Research Funds of Renmin University of China (Grants No. 10XNL016 and No. 16XNLQ03), and the National 973 Program (Grant No. 2014CB921403).\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nIn recent years, deep neural networks have been shown to be highly effective in solving many real world problems, creating an increased demand for their deployment in a variety of applications. However, such deep neural networks often require a large amount of computational resources for both training and inference purposes, which limits the adoption and spread of this technology in scenarios where large computational resources are not available. This issue is likely to persist in the near future, since recent trends have demonstrated that increased model size is often correlated with improved performance \\cite{GPT3, tan2019efficientnet, BiT}. To mitigate this issue, recent efforts have been focused on developing specialized hardware to support the computational demands \\cite{hardware_survey} as well as model compression methods in order to reduce them \\cite{compression_survey}. These include broad families of techniques such as pruning \\cite{pruning_survey}, knowledge distillation \\cite{distillation_survey}, neural architecture search (NAS)\\cite{nas_survey} and as is the focus of this paper, quantization \\cite{hubara2017quantized}.\n\nQuantization methods enable the computations performed by neural networks to be carried out with fixed point operations, rather than floating point arithmetic.\nThis improves the computational efficiency of neural networks and reduces their memory requirements. However, as with other compression methods, this typically comes at the cost of reduced performance \\cite{compression_survey}. Recent efforts in the field have focused on improving the trade-offs between model compression and performance, by proposing a plethora of quantization schemes tailored for different scenarios.\n\n\\subsection{Quantization Methods}\nQuantization schemes can first be divided into post training and quantization aware training schemes. Post training schemes decouple the tasks of model training and the quantization of its weights and\/or activations. These methods are most suitable when the training data is no longer available when compressing the network, and only the trained network is available \\cite{soudry1, postq1}. On the other hand, quantization aware training schemes perform both optimization tasks together, and do require training data, which tends to provide better performance.\n\nIn addition, quantization schemes can be further divided into methods which perform uniform or non-uniform quantization. Uniform quantization divides the real-valued domain into equally sized bins, whereas non-uniform methods do so with bins of varying sizes.  The former method tends to be more resource efficient and compatible with modern hardware accelerators, while the latter tends to provide better model performance \\cite{li2019additive, jung2019learning}.\n\nQuantization schemes can be further subdivided into per-channel methods, which utilize different quantization parameters for different channels within each layer, as opposed to per-layer methods, which use the same parameters for all channels in each layer, but different parameters for different layers. However, as with non-uniform quantization, per-channel methods can be difficult to implement efficiently on standard hardware components, making per-layer methods more feasible for deployment on edge devices.\n\nFor the reasons stated above, in this work we focus on per-layer, uniform quantization aware training with a different number of bits allocated per layer, referred to as dynamic quantization.\n\\subsection{Dynamic Quantization}\\label{dynamicmotivation}\n\\begin{figure}\n\\vskip 0.1in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{Figure_1}}\n\\caption{Our proposed training scheme, which uses iterative optimization of the model weights and bit allocation. Given a fixed allocation of bits for each layer, the weights are optimised using a quantization aware training procedure. Then, the bit allocation is optimized for those fixed weights and the process is repeated.}\n\\label{fig:OptScheme}\n\\end{center}\n\\vskip -0.1in\n\\end{figure}\nDynamic quantization aims to further improve the trade-offs between performance and computational requirements of QNNs for deployment on specialized edge devices. However, the proper allocation of bits between layers is combinatorial in nature and is hard to optimize. Recent efforts to tackle this problem include the use of reinforcement learning \\cite{elthakeb2018releq, wang2019haq}, Hessian analysis \\cite{dong2019hawqv1, dong2019hawqv2}, quantizer parametrization \\cite{mpd}, and differentiable NAS approaches \\cite{dnas_dynamic, li2020efficient, edmips}.  Among these methods, only the NAS approaches account for dependencies between bit allocations in different layers, by forming a super network that includes multiple branches for each precision at each layer. NAS approaches, however, are more expensive to train due to the multiple network branches. Here we offer an alternative approach to achieve the same goal.\n\n\\subsection{Our Contributions}\nIn this work we wish to optimize for the bit allocation of the network as a whole, and address the full interactions between different components of QNNs. To this end, we propose to utilize gradient-free optimization algorithms for the bit allocation. Such algorithms are known to perform well in difficult scenarios with complex dependencies between variables, while maintaining excellent sample efficiency during optimization \\cite{conn2009introduction, rios2013derivative}. In particular, we use the algorithm Covariance Matrix Adaptation Evolution Strategy (CMA-ES) \\cite{hansen2003reducing}, which aims to achieve the mentioned goals, though other alternatives may also be suitable for the task.\n\nWe propose a novel quantization aware training procedure for dynamic QNNs. In this procedure, CMA-ES is used for computing the optimal (per-layer) dynamic bit allocation for the weights and activations of the network. This method is interchangeably applied with gradient-based methods. That is, the network weights are updated by a gradient-based method, while the bit allocation is updated using the gradient-free method CMA-ES---see Fig. \\ref{fig:OptScheme}.\n\nThe advantages of our approach are as follows:\n\\begin{itemize}\n\\item Our training scheme for the dynamic bit allocation optimizes for the network as a whole. That is, it takes into account dependencies between layers of the neural network, as well as dependencies between weights and their bit allocation.\n\\item Our method for bit allocation is gradient-free, hence it can handle multiple (possibly non-differentiable) hardware constraints. This enables the QNN to be tailored to the resources of the specific edge devices it is to be deployed on.\n\\item The systematic combination of gradient-based and gradient-free optimization algorithms can be utilized in other applications and scenarios, e.g. systematic search of the network's other hyper-parameters.\n\\end{itemize}\n\n\\section{Preliminaries}\n\\subsection{Quantization Aware Training}\\label{qat}\nThe quantization schemes of weights or activations in neural networks that we consider here can typically be broken into three steps: clip, scale and round. In the first step, the real values of the weights or activations are clipped to be within a range of values: i.e., $[-\\alpha, \\alpha]$ for symmetric signed quantization (typically weights) and $[0, \\alpha]$ for unsigned quantization (typically activations, outputs of ReLU operations). Then, the range is mapped to the target integer range  $[-2^{b-1} + 1, 2^{b-1} - 1]$ and $[0, 2^{b} - 1]$, for signed and unsigned quantization respectively, where $b$ is the number of bits. The values $\\alpha$ are  referred to as ``scales'' or clipping parameters and take on different values for different layers. The clipping parameters are optimized throughout the training procedure (along with the weights), so that they yield the lowest possible generalization error when quantization applied to the models' weights and\/or activations.\n\nSeveral approaches have been proposed in recent years, in order to optimize for the scales. In \\cite{zhang2018lq} the scales are chosen based on quantization error minimization. \\cite{choi2018pact} proposes differentiable clipping parameters which are learned through backpropagation together with the model's weights. This approach is further improved in \\cite{li2019additive}, which also supports a computationally efficient \\emph{non-uniform} quantization scheme. In this work we base our uniform quantization aware training scheme (for the weights and activations) on \\cite{li2019additive}, while interchangeably utilizing CMA-ES for the dynamic bit allocation.\n\n\\begin{figure}\n\\vskip 0.1in\n\\begin{center}\n\\centerline{\\includegraphics[width=7cm, height=3cm]{Quantized_tensor}}\n\\caption{Example of a signed 4-bit quantized weights tensor and $\\alpha=0.16$ (original-left, quantized-right). Values which are lesser or greater than the clipping parameters $\\pm \\alpha$ are clipped to $-\\alpha$ or $+\\alpha$ respectively, then scaled and rounded to the signed integer range. The value of 0 is preserved in this process.}\n\\label{fig:quantized_weights_example}\n\\end{center}\n\\vskip -0.1in\n\\end{figure}\n\nTo formally introduce the point-wise quantization operations we first define the quantization operator:\n\\begin{equation} \\label{quantoperator}\n    Q_b(x) = \\frac{\\mbox{round}((2^b - 1) \\cdot x)}{2^b - 1},\n\\end{equation}\nwhere $x$ is a real-valued tensor in [-1, 1] or [0, 1] for signed or unsigned quantization, respectively. $b$ is the number of bits that are used to represent $x$.\nGiven this operator, we use the reparameterized clipping function \\cite{li2019additive} to define the quantized weights and activations.\n\\begin{eqnarray} \\label{quantweights}\n    W_b &=& \\alpha_W Q_{b-1}(\\mbox{clip}(\\frac{W}{\\alpha_W}, -1, 1))\\\\\n    X_b &=& \\alpha_X Q_{b}(\\mbox{clip}(\\frac{X}{\\alpha_X}, 0, 1)) \\label{quantacts}.\n\\end{eqnarray}\nHere, $W, W_b$ are the real-valued and quantized weight tensors, $X, X_b$ are the real-valued and quantized input tensors, and $\\alpha_W, \\alpha_X$ are their associated scale (or clipping) parameters, respectively.\nAn example of Eq. \\eqref{quantweights} being applied to a weights tensor, using 4-bit signed quantization (without multiplying by $\\alpha_W$), can be seen in Figure \\ref{fig:quantized_weights_example}.\n\nWe note that equations \\eqref{quantoperator}-\\eqref{quantacts} are used for training only. During inference, weights and activations are dynamically quantized, and all operations are performed using integers in mixed precision, while taking the scales into account.\n\nIn this quantization aware training scheme, both weights and activations are quantized during the forward pass, and during the backward pass, the Straight Through Estimator (STE) \\cite{STE} is used to optimize the weights. That is, we ignore the derivative of $Q_b$ in the backward pass, and iterate on the floating point values of $W$ using the derivatives w.r.t $W_b$ in the SGD optimization. Given Eq. \\eqref{quantweights}-\\eqref{quantacts} the STE can also be used to calculate the gradients with respect to $\\alpha_W, \\alpha_X$ \\cite{li2019additive}.\nThis enables the quantized network to be trained in an end-to-end manner using backpropogation.\nTo improve stability, \\cite{li2019additive} also use weight normalization before quantization:\n\\begin{equation} \\label{eq:normalize}\n    \\hat W = \\frac{W - \\mu}{\\sigma  + \\epsilon}.\n\\end{equation}\nHere, $\\mu$ and $\\sigma$ are the mean and standard deviation of the weight tensor, respectively, and $\\epsilon=10^{-6}$.\n\n\\subsection{CMA-ES}\\label{prelim_cma}\nCovariance Matrix Adaptation Evolution Strategies (CMA-ES) \\cite{hansen2003reducing}, is a population based gradient-free optimization algorithm. It is known to be highly versatile and has been applied to a large variety of settings, such as RL \\cite{heidrich2008evolution}, placement of wave energy converters \\cite{neshat2019ahybrid}, hyper-parameter optimization \\cite{loshchilov2016cma}, evolving game levels in Mario \\cite{volz2018evolving}, adversarial attacks \\cite{8917642} and more. It is designed to work in $d$-dimensional continuous spaces and optimize discontinuous, ill-conditioned and non-separable objective functions, in a black box optimization setting \\cite{cma_bbob}.\n\nAt a high level, the optimization process of CMA-ES is as follows. At the $g$-th generation, a set of $\\lambda$ $d$-dimensional samples are drawn from a multivariate normal distribution $\\mathcal{N} (m^{(g)}, \\mathcal{C}^{(g)})$:\n\\begin{equation}\\label{cmaes}\n    x_k^{(g+1)} \\sim  m^{(g)} + \\sigma^{(g)} \\mathcal{N} (0, \\mathcal{C}^{(g)}), \\mbox{  for }k = 1,...,\\lambda\n\\end{equation}\nWhere $m^{(g)}$, $\\mathcal{C}^{(g)}$ are the mean and covariance matrrix of the population at the previous generation, respectively. $\\lambda$ is the population size and $\\sigma^{(g)}$ is the step-size.\n\nOnce the samples are drawn from this distribution, they are evaluated and ranked based on the objective function.\nThen, keeping only the top $\\mu$ samples with the best objective values,  $m^{(g+1)}$, $\\mathcal{C}^{(g+1)}$ and $\\sigma^{(g+1)}$ are calculated using a set of update rules. Please refer to \\cite{hansen2016cma} for more details.\nThese are then used to draw samples for the next generation using Eq. \\eqref{cmaes}.\nThis process is repeated until one of several convergence criteria are full-filled, or until the algorithm exceeds its predefined budget of allowed samples.\n\nThough CMA-ES is an effective gradient-free optimization algorithm, it has several downsides. First, its computational complexity is $O(d^2)$ in space and time \\cite{hansen2003reducing}, where $d$ is the dimension of the parameter vector to be optimized.\nSecond, its convergence rate is often also $O(d^2)$ \\cite{cma_bbob}, making it inefficient for solving high dimensional problems where $d$ is larger than a few thousands.\n\n\\section{The GradFreeBits Method}\\label{method}\nIn this section we describe the components of our optimization procedure for dynamic QNNs.\n\n\\subsection{Motivation: CMA-ES for Dynamic Precision.}\nWe argue that CMA-ES is highly compatible with the problem of bit-allocation in QNNs. Here we assume the objective function is the differentiable loss function used during training, with additional possibly non-differentiable constraints related to computational requirements (exact details are provided below).\n\nAs recent evidence suggests \\cite{wang2019haq, dong2019hawqv1}, the optimization landscape of the bit-allocation problem is likely to be discontinuous, ill-conditioned and amenable for optimization using gradient-free optimizers. Since the constraints may be non-differentiable, they can be sampled in a black-box setting, as is done in gradient-free optimization (CMA-ES) and reinforcement learning \\cite{wang2019haq}.\nAdditionally, as shown in \\cite{dong2019hawqv1, dong2019hawqv2}, the Hessian eigenvalues show large variations for different layers in QNNs, meaning that certain layers are typically more sensitive to changes in bit-allocation than others. This is in part what motivated us to choose CMA-ES for this work, as it is capable of adapting to high variations in the Hessian eigenvalues, and is therefore considered to be one of the best and widely used gradient-free methods.\nLastly, since neural networks typically have an order of tens to hundreds of layers, the problem of bit optimization falls well within the range of dimensionallity where CMA-ES is known to perform well, as discussed in the previous section.\n\n\n\\subsection{Setting the Stage for CMA-ES}\\label{bitaloc}\nTo enable the gradient-free optimization algorithm to efficiently optimize the bit-allocation of a QNN, two items must be defined: its search space and the objective function.\n\n\\paragraph{Search Space}\nWe define the search space as a vector containing the bit allocation of the weights and activations in all the layers of the network, asides the first and the last layer, as these are quantized using a fixed allocation of 8 bits.\nWe found it beneficial to optimize the logarithm (base 2) of this vector, rather than the vector itself.\nThus, the vector to be optimized by CMA-ES is the log-precision vector:\n\\begin{equation}\\label{cma_search_space}\n    \\mathbf{v} = [\\mathbf{v_W}, \\mathbf{v_X}] = \\log_2 ([\\mathbf{r_W}, \\mathbf{r_X}]),\n\\end{equation}\nwhere $\\mathbf{r_W}, \\mathbf{r_X}$ are the bit allocations of the network's weights and activations respectively, and $[\\cdot, \\cdot]$ is the concatenation operation.\n\n\\paragraph{Objective Function}\nAs mentioned in Section \\ref{prelim_cma}, the objective function to minimize is a combination of the network's performance measure, the differentiable loss function, subject to a number of non-differrentiable computational constraints. More formally:\n\\begin{equation}\n    \\min_\\mathbf{v} \\textit{ } \\mathcal{L}(\\mathbf{v}; \\mathbf{\\theta}), \\\\\n    \\hspace{4pt} \\mbox{ s.t. }  h_j(\\mathbf{v}) \\leq C_j \\hspace{4pt} \\mbox{ for } j=1,...,M\n\\end{equation}\nWhere $\\mathcal{L}(\\mathbf{v}; \\mathbf{\\theta})$ is the loss function over the training set, parameterized by network parameters $\\mathbf{\\theta}$, which are assumed to be fixed during the bit-allocation optimization stage.\nFurthermore, $h_j(\\mathbf{v})$ are the computational requirements for a given precision vector $\\mathbf{v}$ (e.g., model size, inference time etc.), $C_j$'s are the target requirements that we wish to achieve, and $M$ is the number of constraints.\nIn our framework, we combine the constraints into the objective function using the penalty method \\cite{penalty}:\n\\begin{equation}\\label{objective}\n    \\min_\\mathbf{v} \\textit{ } \\mathcal{L}(\\mathbf{v}; \\mathbf{\\theta})\n    + \\sum_{j=1}^M \\rho_j \\max(0, h_j(\\mathbf{v}) - C_j)^2,\n\\end{equation}\nWhere $\\rho_j$ are balancing constraint parameters. This is similar to the approach taken in \\cite{mpd}, but here it is applied to gradient-free optimization.\n\nWe define the computational constraints by comparing the requirements between the dynamic and static precision networks. For example, we wish that the model size will be equal to that of a static 4 bits allocation.\nSpecifically, we use constraints on the model size for the weights entries $\\mathbf{v_W}$ and the mean bit allocation for the activation entries $\\mathbf{v_X}$:\n\\begin{equation} \\label{constraint1}\n\t    h_1({\\mathbf{v}}) = MB({\\mathbf{v_W}}), \\hspace{4pt} C_1 = \\beta_1 MB({\\mathbf{v}^S_{\\textbf{W}}})\n\\end{equation}\n\\begin{equation} \\label{constraint2}\n\t    h_2({\\mathbf{v}}) =\n\t    \\frac{1}{L} \\sum_{i=1}^L v_{X_i}, \\hspace{4pt} C_2 =  \\frac{\\beta_2}{L} \\sum_{i=1}^L v^S_{X_i}\n\\end{equation}\nWhere $MB(\\cdot)$ calculates the model size given weight entries $\\mathbf{v_W}$, $\\mathbf{v}^S$ is the log-precision vector of the target static precision we wish to achieve, $L$ is the number of relevant layers, and $\\beta_1, \\beta_2>0$ control the target compression rates of the weights and activations, respectively.\n\n\nThese constraints are designed to limit the computational requirements, while  allowing the gradient-free optimization algorithm to explore non-trivial solutions which satisfy them.\nIt is important to note that though these constraints are only related to memory requirements, other constraints can easily be incorporated into our framework - such as power usage or inference time measurements, chip area, etc.\n\n\\subsection{Variance Reduction}\\label{variance_reduction}\nVariance reduction has been shown to improve the convergence rate of optimization algorithms \\cite{variance_reduction}. The main source of variance in our objective function (Eq. \\eqref{objective}) is in the first term, related to the performance of the model for different bit-allocations. There are two main causes of variance in this term: sub-sampling noise, caused by using small mini-batches of randomly selected samples, and sensitivity to quantization errors, which networks are typically not robust to.\nTo mitigate these issues and improve the convergence rate of CMA-ES we adopt two strategies for reducing the variance in the objective function, utilizing moving super-batches and quantization aware pretraining, which will be described promptly.\n\n\\begin{figure}\n\\vskip 0.1in\n\\begin{center}\n\\centerline{\n\\includegraphics[width=7.0cm, height=4.0cm]{Moving_superbatch}}\n\\caption{Batch replacement scheme used in moving super-batches, as compared to standard mini-batch replacement. At each iteration a single mini-batch is replaced, in a queue-like manner, creating a high overlap of samples between consecutive super-batches.}\n\\label{fig:moving_superbatch}\n\\end{center}\n\\vskip -0.1in\n\\end{figure}\n\n\\paragraph{Moving Super-batches}\\label{moving_superbatch_section}\nWe define a moving super-batch as a set of mini-batches which are replaced in a queue-like manner. That is, in each iteration of the super-batch we replace part of the mini-batches within it. During each objective evaluation of CMA-ES, the entire super-batch is run through the model in order to calculate the first term of \\eqref{objective}. The queue-like replacement scheme enables CMA-ES to encounter new data samples in each objective evaluation, but with a larger overlap of data samples as compared to standard SGD, where the entire mini-batch is re-sampled at each iteration. The process is illustrated in Figure \\ref{fig:moving_superbatch}. Several strategies for the frequency of replacement can be considered, such as replacing one or more mini-batches after each objective evaluation, or doing so every fixed number of evaluations. These different settings are explored in the ablation study in Section \\ref{ablation_study_section}.\n\n\\paragraph{Pretraining}\\label{pretraining}\nAnother way to reduce variability in performance, is by reducing sensitivity to quantization errors. This can be done by pretraining the network, using the quantization aware training scheme, as described in section \\ref{qat}. The bit allocation for this stage is static, meaning the weights and activations in all conv layers are quantized to the same target precision, except the first conv and last full-connected layers, which are always quantized to 8-bit precision.\n\n\\subsection{Gradient-free Steps}\nWe define gradient-free steps, as steps in which the CMA-ES algorithm optimizes the bit allocation (given the network's weights) according to the objective function in Eq. \\eqref{objective}. In each step, multiple generations of samples of the log-precision vector $\\mathbf{v}$ are evaluated on \\eqref{objective}. At each objective evaluation, the sample of $\\mathbf{v}$ is used to quantize the weights and activations of the model. Since CMA-ES typically operates in continuous spaces, the bit allocations of the weights and activations (which are positive integers) are extracted from $\\mathbf{v}$ using the following formula:\n\\begin{equation} \\label{extracting_precision}\n    \\mathbf{r_W} = \\ceil{2^{\\mathbf{v_W}}}, \\mathbf{r_X} = \\ceil{2^{\\mathbf{v_X}}}\n\\end{equation}\nWhere $\\ceil{x}$ is the \"ceil\" operator, which rounds its argument upwards to the nearest integer. Once the bit allocations have been inserted into the model, the loss \\eqref{objective} is calculated over all the mini-batches in the super-batch, yielding the value of the objective for each of the sampled bit-allocations. This process gradually minimizes the value of the objective function and enables non-trivial bit-allocations to be found. We define each gradient-free step to include a predefined number of  objective evaluations. (this a hyper-parater, typically set to 512 evaluations). It is important to note that gradient-free steps require significantly less computational resources than traditional epochs, even if the number of mini-batches are matched. This is because they do not require backpropogation and the gradient-free optimizer computations are negligible compared to those performed by the model.\n\n\n\\subsection{Iterative Alternating Retraining}\nThe primary innovation in this work is the combination of gradient-based training sessions of the model weights and gradient-free training sessions of the bit-allocation, which we refer to as iterative alternating retraining.\n\nAfter the pretraining stage, the model is passed to the gradient-free optimizer CMA-ES to optimize its bit allocation for a session of a predefined number of steps $N_{GF}$. This adapts the bit allocation to the model weights, which are fixed at this stage in their floating point values. This maximizes the performance of quantized networks, subject to the computational constraints (Eq. \\eqref{objective}), using the optimization process as described in section \\ref{prelim_cma}.\n\nOnce the gradient-free session is completed, the best bit allocation found by CMA-ES is passed to the gradient-based optimizer, for a session of a predefined number of epochs $N_{GB}$. This adapts the model weights to the bit allocation, which is kept fixed, using the quantization aware training scheme described in Section \\ref{qat}. Once this session is completed, the model is returned to the gradient-free optimizer to further improve the bit allocation, given the new model weights. This cycle is then repeated a number of times, or until the performance and computational requirements are satisfactory. The process is illustrated in Fig. \\ref{fig:OptScheme}.\n\nTo increase stability in the handover process between the optimization algorithms, we found it beneficial to restart CMA-ES after each gradient-based session and handover only the best model weights and bit allocation (with lowest loss\/objective values) found across all sessions, as opposed to handing over the best versions found in each session.\n\n\\section{Experiments}\n\n\\begin{figure}\n\\vskip 0.1in\n\\begin{center}\n\\centerline{\n\\includegraphics[width=\\columnwidth, height=3.8cm]{Res56_cifar10}\n}\n\\caption{(a) Convergence plot for a 3 and 4-bit dynamically quantized ResNet56 on CIFAR10, during the retraining process. Grey regions correspond to gradient-free steps, while white regions correspond to gradient-based epochs. (b) The bit allocations for the weights (blue, left) and activations (green, right) of the 4-bit ResNet56.}\n\\label{fig:res56_cifar10}\n\\end{center}\n\\vskip -0.1in\n\\end{figure}\n\nTo quantitatively compare our dynamic quantization scheme (GradFreeBits) to other related works, we apply it to several neural network architectures for image classification tasks. Throughout all the experiments, we consider similar numbers of bits (on average, in the dynamic case) for the weights and activations. Since we use the ReLU activation in all the settings, the activations are quantized using unsigned integers. The weights are quantized symmetrically using signed integers, so for example, 2-bit quantization will result in ternary weights in $\\{-1,0,1\\}$.\n\nFurthermore, in this section we also perform an ablation study, to identify the effects that different settings of the proposed system have on the performance metrics of the resulting dynamically quantized models.\n\nWe compare our approach to the following related works that use uniform quantization, with either static (S) or dynamic (D) bit-allocation schemes:\nLQ-Nets(S) \\cite{zhang2018lq}, PACT(S) \\cite{choi2018pact}, APoT(S) \\cite{li2019additive}, DSQ(S) \\cite{gong2019differentiable}, BSGD(S) \\cite{bcgd}, TQT(D) \\cite{tqt},  MPD(D) \\cite{mpd}, EBS(D)\\cite{li2020efficient}, EDMIPS(D) \\cite{edmips}, HAQ(D) \\cite{wang2019haq}, HAWQ-V1(D) \\cite{dong2019hawqv1}, HAWQ-V2(D) \\cite{dong2019hawqv2}, WNQ(S) \\cite{wnq}, RES(S) \\cite{res}, DoReFa-Net(S) \\cite{drfn}. Lastly, we denote our our proposed method GradFreeBits as GFB(D).\n\n\n\\subsection{CIFAR 10\/100} \\label{cifar10_section}\nThe CIFAR10 and CIFAR100 image classification benchmarks \\cite{cifar10} have 10 and 100 classes respectively, both containing 50k train and 10k test 32x32 RGB images. For these datasets, pretraining \\textit{from scratch} was conducted for 400 epochs, using a mini-batch size of 128, SGD optimizer with momentum of 0.9, cosine learning rate decay, with a maximum learning rate of 0.1 and warm-up of 10 epochs. Data augmentation used during the pretraining and iterative retraining stages included: random horizontal flips and crops, and mixup \\cite{mixup}. To approximately satisfy the constraints of \\eqref{objective}, we chose $\\beta_1=\\beta_2=0.7$ and $\\rho_1=\\rho_2=0.5$. This allows the penalty constraints to be slightly violated, while keeping model size and mean precision below their actual targets. We also limit the search space, such that all values of $\\textbf{v}$ are in $[0, 3]$.\n\nGiven the pretrained model, we applied 5\/3 rounds of the iterative alternating retraining algorithm for CIFAR10\/100 respectively. In each round, we applied 4 steps in the gradient-free session and 4\/16 epochs in the gradient-based session for CIFAR10\/100 respectively. At each gradient-free step of CMA-ES we applied 512 objective evaluations on super-batches, each containing 32 mini-batches of 128 data samples. After each of the 512 objective evaluations we swap only one mini-batch in the super-batch as illustrated in Fig. \\ref{moving_superbatch_section}.\n\n\n\n\n\n\\begin{table}[t]\n\\caption{Top1 accuracy of quantized network on CIFAR10\/100. (S) and (D) denote static and dynamic quantization, while \\textbf{*} denotes experiments which use 2-bit weights and 4-bit activations.}\n\\label{tab:cifar10}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lccr}\n\\toprule\nModel & Method  & 3\/3 & 4\/4 \\\\\n\\midrule\n\nRes.20    & PACT(S)          & 91.1 & 91.7  \\\\\nCIFAR10    & LQ-Nets(S)        & 91.6       & -  \\\\\nFP 93.3  & BCGD(S)           & 91.2 * & 92.0 \\\\\n          & TQT(D)           & 90.4 * & - \\\\\n          & MPD(D)           & 91.4 * & - \\\\\n          & EBS(D)             & 92.7 & 92.9 \\\\\n          \\rowcolor{gray!20}\n          & GFB(D) (ours)        & \\textbf{93.3} & \\textbf{93.6}  \\\\\n           \\rowcolor{gray!20}\n         & GFB(D) (ours)        & \\textbf{93.4}* & -  \\\\\n\\hline\n\\hline\n\nRes.56 CIFAR10 & EBS(D)      & 94.1 & 94.3 \\\\\n\\rowcolor{gray!20}\nFP 95.1           & GFB(D) (ours)  & \\textbf{94.8} & \\textbf{94.9} \\\\\n\\hline\n\\hline\n\nRes.20    & DoReFa-Net(S)       & 68.4 & 68.9  \\\\\nCIFAR100   & Res(S)        & 68.3       & 68.7  \\\\\nFP 70.35   & LQ-Nets(S)        & 68.4       & 69.0  \\\\\n          & WNQ(S)        & 68.8       & 69.0  \\\\\n          \\rowcolor{gray!20}\n          & GFB(D) (ours)   & \\textbf{69.6} & \\textbf{70.6}  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\nThe results for the CIFAR10 dataset are presented in Table \\ref{tab:cifar10}. Our method out-performs the previous state of the art EBS(D) for both ResNet20 and ResNet56 models, at both dynamic precision settings: e.g. +0.7\\% for 4-bit ResNet20 and +0.6\\% for 4-bit ResNet56. Table \\ref{tab:cifar10} also includes the results for the  CIFAR100 dataset. For this benchmark, our method also outperforms all other related works: by 1.6 for 4-bit ResNet20 and 0.8 for fore 3-bit ResNet20.\n\n\n\n\n\\subsection{ImageNet}\\label{imagenet_section}\nThe ImageNet \\cite{imagenet} image classification benchmark has 1K classes, 1.2M train and 150K test RGB images. For all models, we used pretrained weights from TorchVision \\cite{torchvision}.\nAdditional quantization aware pretraining was then conducted for 30 epochs, using mini-batches of size 128, SGD optimizer with momentum of 0.9, and a cosine learning rate decay with a maximum of 0.001 and a warm-up of 3 epochs.\n\nData augmentations are identical to those used in the CIFAR10\/100 experiments (above), with the addition of image resize to $256 \\times 256$ and random crops to $224 \\times 224$. These are used for both pretraining and iterative alternating retraining stages. Furthermore, $\\beta_1=\\beta_2=0.7$, $\\rho_1=\\rho_2=0.5$ are used to satisfy the constraints and the search space for $\\textbf{v}$ is limited to $[0, 3]$. Three rounds of iterative alternating retraining were applied. In each round, we applied 4 steps in the gradient-free session and 4 epochs in the gradient-based session.\nIn each gradient-free step of CMA-ES we applied 512 objective evaluations on super-batches, each containing 8 mini-batches of 128 data samples, swapping one-minibatch in the super-batch after each objective evaluation.\nThe entire procedure requires objective evaluations that are comparable to 47 epochs (30 gradient-based pretraining epochs, 12 gradient-free steps, and 12 gradient-based epochs).\n\\begin{table}[t]\n\\caption{Top1 accuracy of quantized networks on ImageNet. (S) and (D) denote static and dynamic quantization, ($\\cdot$) denotes model size, measured in MB. $\\approx$ is used when only graphical results were available for comparison. \\textbf{*} denotes experiments which use 2-bit weights and 4-bit activations.}\n\\label{tab:ImageNet}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{lccr}\n\\toprule\nModel & Method & 3\/3 & 4\/4 \\\\\n\\midrule\nRes.18         & BCGD(S)           & -  & 67.4(-) \\\\\nFP 70.7    & LQ-Nets(S)         & 68.2(6.1)            & 69.3(7.4) \\\\\n(46.8)   & DSQ(S)          & 68.7(6.1)            & 69.6(7.4) \\\\\n          & PACT(S)       & 68.1(6.1)            & 69.2(7.4)  \\\\\n          & APoT(S)       & 69.4(4.6)            & - \\\\\n          & EBS(D)         & 69.5(-)     & 70.2(-) \\\\\n          & EDMIPS(D)      & $\\approx$67(-)       & $\\approx$68(-)  \\\\\n          & TQT(D)         &     -     & 69.5(5.6)  \\\\\n          & MPD(D)         & -      & 70.1(\\textbf{5.4})  \\\\\n         \\rowcolor{gray!20}\n        \n          & GFB(D) (ours)          & 69.2(\\textbf{3.6})   & \\textbf{70.3}(\\textbf{5.4})  \\\\\n\n\\hline \\hline\nRes.50    & DoReFa-Net(S)      & 69.9(16.6)            & 71.4(19.4) \\\\\nFP 76.4   & LQ-Nets(S)      & 74.2(16.6)            & 75.1(19.4) \\\\\n(97.5)    & PACT(S)       & 75.3(16.6)            & \\textbf{76.5}(19.4)  \\\\\n          & EDMIPS(D)      & $\\approx$72.5(-)     & $\\approx$74.0(-)   \\\\\n          & HAQ(D)       & 75.4(9.2)        & 76.1(\\textbf{12.1}) \\\\\n          & HAWQ-V1(D)      & 75.5(\\textbf{8.0})  \\textbf{*}           & -  \\\\\n          & HAWQ-V2(D)     & \\textbf{75.8}(\\textbf{8.0})  \\textbf{*}        & - \\\\\n         \\rowcolor{gray!20}\n          & GFB(D) (ours)     & 75.7(9.6) & 76.1(12.8)  \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\n\\noindent The results for the ImageNet dataset are presented in Table \\ref{tab:ImageNet}. For the Renet18 model, our method slightly outperforms the previous state of the art EBW(D)\\cite{li2020efficient} for 4-bits dynamic precision $+0.1\\%$, though slightly under-performs for 3-bits $-0.3\\%$. However, both dynamically quantized models achieve the smallest model size in their category.\n\n\nFor the 3-bit weights and 3-bit ResNet50, our method achieves similar Top1 accuracy to that of HAWQ2(D) \\cite{dong2019hawqv2} with 2-bit weights and 4-bit activations, though with a larger model size $+1.6MB$. However, our 4-bit ResNet50 falls behind PACT(S) \\cite{choi2018pact} in performance $-0.4\\%$, though with a significantly smaller model size $-6.6MB$. This is on par with the previous state of the art method HAQ(D) \\cite{wang2019haq}, even though it uses \\emph{non-uniform}, quantization which can achieve a higher accuracy than uniform.\n\nFor these experiments we can observe the advantage of the dynamically quantized models compared to the statically quantized ones. Even though the dynamically quantized models require less memory, they tend to obtain higher or comparable Top1 accuracy scores.\n\n\\subsection{Ablation Study}\\label{ablation_study_section}\nIn this section we examine the effects that different system settings have on the performance of our dynamically quantized models. More specifically, we examine the effects of pretraining, iterative alternating retraining, the different settings of moving super-batches and the number of mini-batches they contain. To enable efficient comparison, all experiments are conducted with 4-bit dynamically quantized ResNet20 models on CIFAR100 with the same hyperparameters used in \\ref{cifar10_section}. (learning rate, number of gradient-based\/free epochs, etc.).\n\nWe consider the following settings for replacing mini-batches in the super-batch with new batches of randomly selected training samples: \\textbf{NR} - no mini-batches are replaced in the super-batch, \\textbf{EB} - single mini-batch replacement after each gradient-free step, \\textbf{EF} - replacement of all mini-batches after each gradient-free step, \\textbf{SB} -  single mini-batch replacement after each objective evaluation, \\textbf{SF} - replacement of all mini-batches after each objective evaluation. The test without iterative alternating retraining runs all gradient-free steps first, followed by all the gradient-based epochs. The test without pretraining, uses the full-precision model as the initial weights.\n\n\n\\begin{table}[t]\n\\caption{Ablation study of a 4-bit dynamically quantized ResNet20 on CIFAR100, for various system settings. We use the shorthand: \"SS.\" for super-batch setting, \"NMB.\" for number of mini-batches in the super-batch, \"IT.-RET.\" for iterative alternating retraining, \"PRET.\" for pretraining and \"ACC.\" for accuracy.}\n\\label{ablation_system_settings}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lccccr}\n\\toprule\nvariable & ss. & nmb. & it.-ret. & pret. & Acc. \\\\\n\\midrule\nBaseline & SB      & 32 & $\\surd$ & $\\surd$ & \\textbf{70.61} \\\\\n\\hline\nComponents & SB      & 32 & $\\surd$ & $\\times$& 66.99 \\\\\n & SB      & 32 & $\\times$& $\\surd$ & 70.43 \\\\\n\\hline\nSuper-batch & NR      & 32 & $\\surd$ & $\\surd$ & 70.21  \\\\\nSetting & EB      & 32 & $\\surd$ & $\\surd$ & 70.36  \\\\\n & EF      & 32 & $\\surd$ & $\\surd$ & 70.26   \\\\\n & SF      & 32 & $\\surd$ & $\\surd$ & 70.39   \\\\\n\\hline\nSuper-batch & SB      & 4 & $\\surd$ & $\\surd$ & 69.68    \\\\\nSize & SB      & 8 & $\\surd$ & $\\surd$ & 70.25    \\\\\n & SB      & 16 & $\\surd$ & $\\surd$ & 70.18   \\\\\n& SB      & 64 & $\\surd$ & $\\surd$ & 70.26   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\n\n\n\\noindent The results of the ablations study are presented in Table \\ref{ablation_system_settings}. The $-3.6\\%$ accuracy degradation demonstrates that pretraining plays a crucial role in reducing performance degradation due to changes in bit-allocation. It also seems that iterative alternating retraining, as opposed to separating the bit-optimization and weight-optimization stages, leads to a small $+0.2\\%$ increase in performance, demonstrating the added value of this approach. Regarding the super-batch settings, it seems that the optimal setting is to use 32 mini-batches and replace a single batch after each objective evaluation (SB), also leading to a performance increase of $+0.2\\%$. For these reasons, we used these settings used in the rest of the experiments.\n\n\\section{Discussion}\nThough the proposed method achieves favorable trade-offs between performance and model size, several items need be taken into account when applying it to new problems.\n\nFirst, since CMA-ES is designed to operate in continuous search spaces and the search space considered here is discrete, rounding operations are required to ensure compatibility (Eq. \\eqref{extracting_precision}). This creates plateaus of constant values in the objective function, which may harm the convergence rate. To tackle this, we may use other gradient-free algorithms, which are more suitable for discrete optimization, and may lead to even better results using our framework.\n\nSecondly, though this work shows that the CMA-ES optimizer is highly effective in a variety of QNNs, applying it to larger networks (such as EfficientNet B7 \\cite{tan2019efficientnet}) may require more generations and objective evaluations in order to achieve the comparable improvements in the performance and compression rate. This is because the convergence rate of CMA-ES is typically $O(L^2)$ (see section (\\ref{prelim_cma})), where $L$ is the number of layers.\nThat said, if these consideration are taken into account, the combination of gradient-based and gradient-free optimization methods can be applied to achieve highly competitive results.\n\n\\section{Conclusion}\nWe proposed GradFreeBits, a novel framework for optimizing the bit-allocation in dynamically quantized neural networks, which enables neural networks to be customized to meet multiple hardware constraints. The framework is based on the combination of a gradient-based quantization aware training scheme for the weights and gradient-free optimization of the bit-allocation, based on CMA-ES. By benchmarking our method on multiple image classification tasks, we find that our method often outperforms several related dynamic and static quantization methods, in both accuracy and model size. We believe our method will help accelerate the deployment of low-precision neural network on resource constrained edge-devices, by enabling the network compression to be tailored to the specific hardware requirements. Additionally, the proposed framework for combining gradient-free and gradient-based optimization in an iterative alternating retraining scheme is quite general, making it likely easy to apply to other applications.\n\nFuture work in this direction includes utilizing additional constraints, such as FLOPS count and measurements from hardware simulators (such as BISMO \\cite{bismo}) - including inference time, power consumption and chip area. Furthermore, it is important to evaluate this method on new tasks such as object detection and semantic segmentation, as these tasks are more difficult than image classification, perhaps making the models more sensitive to changes in bit allocation.\nWe hope this framework will provide a foundation for future works related to the combination of gradient-free and gradient-based neural network training, and are looking forward to more contributions in this direction.\n\n\\small\n\n\\section{Appendix: CMA-ES}\\label{prelim_cma}\n\nIn this section we provide a very brief overview of the main derivations that are used in CMA-ES, without including any theoretical background for the structure of these update rules, or the choices of hyperparameters. For more details regarding these topics, we refer the curious reader to \\cite{hansen2016cma}---we follows the same notation as this paper here. \n\nCovariance Matrix Adaptation Evolution Strategies (CMA-ES) \\cite{hansen2003reducing}, is a population based gradient-free optimization algorithm.\nAt a high level, the optimization process of CMA-ES is as follows. At the $g$-th generation, a set of $\\lambda$ $d$-dimensional samples \n$\\boldsymbol{x}_{k}  \\in \\mathbb{R}^d$\nare drawn from a multivariate normal distribution $\\mathcal{N} (m^{(g)}, \\mathcal{C}^{(g)})$:\n\\begin{equation}\\label{cmaes2}\n    \\boldsymbol{x}_{k}^{(g+1)} \\sim \\boldsymbol{m}^{(g)}+\\sigma^{(g)} \\mathcal{N}\\left(\\mathbf{0}, \\boldsymbol{C}^{(g)}\\right), \\mbox{  for }k = 1,...,\\lambda\n\\end{equation}\nWhere $\\boldsymbol{m}^{(g)}$, $\\boldsymbol{C}^{(g)}$ are the mean and covariance matrix of the population at the previous generation, respectively. $\\lambda$ is the population size and $\\sigma^{(g)}$ is the step-size. \n\nOnce the samples are drawn from this distribution, they are evaluated and ranked based on their objective function values. These ranked samples $\\boldsymbol{x}_{i: \\lambda}^{(g+1)}$ are used to calculate $m^{(g+1)}$, $\\mathcal{C}^{(g+1)}$ and $\\sigma^{(g+1)}$ of the next generation, using a set of update rules which are provided below. \n\n\n\n\\subsection{Hyperparameters}\n\n\nCMA-ES uses several hyperparameters in order to perform optimization \\cite{hansen2016cma}. These include a dampling parameter $d_{\\sigma}$ and $c_{1}, c_{\\mu}, c_{\\mathrm{c}}, c_{\\mathrm{m}}, c_{\\sigma}$ which are \"momentum\"-like parameters, which control the amount of information retained from previous generations. Furthremore, $w_i$ are known as the \"recombination weights\", which are used in most update rules. They are typically chosen such that\n\n\\begin{equation}\\label{cmaes}\n\\sum_{j=1}^{\\lambda} w_{j} \\approx 0, \\quad \\sum_{i=1}^{\\mu} w_{i}=1, \\quad w_{1} \\geq \\cdots \\geq w_{\\mu}>0.\n\\end{equation}\n\nThese are also used to calculate the effective population size for recombination: $\\mu_{\\mathrm{eff}}=\\left(\\sum_{i=1}^{\\mu} w_{i}^{2}\\right)^{-1}$. For more details regarding the specific choices of these hyparamerameters, please refer to \\cite{hansen2016cma}.\n\n\\subsection{Mean Update Rule}\nAs mentioned above, several update rules are employed in CMA-ES. The first of those is the update of the mean $\\boldsymbol{m}^{(g+1)}$:\n\\begin{equation}\\label{cmaes}\n    \\boldsymbol{m}^{(g+1)}=\\boldsymbol{m}^{(g)}+c_{\\mathrm{m}} \\sum_{i=1}^{\\mu} w_{i}\\left(\\boldsymbol{x}_{i: \\lambda}^{(g+1)}-\\boldsymbol{m}^{(g)}\\right).\n\\end{equation}\n\n\\subsection{Covariance Matrix Update Rule}\nThe covariance matrix requires auxiliary vectors in order to construct its update rule, these are calculated using:\n\n\\begin{eqnarray} \\label{auxilary_vectors}\n    \\boldsymbol{p}_{\\mathrm{c}}^{(g+1)} &=& \\left(1-c_{\\mathrm{c}}\\right) \\boldsymbol{p}_{\\mathrm{c}}^{(g)} + \\tilde c_{\\mathrm{c}} \\frac{\\boldsymbol{m}^{(g+1)}-\\boldsymbol{m}^{(g)}}{\\sigma^{(g)}}\\\\\n    \\boldsymbol{y}_{i: \\lambda}^{(g+1)} &=& \\left(\\boldsymbol{x}_{i: \\lambda}^{(g+1)}-\\boldsymbol{m}^{(g)}\\right) \/ \\sigma^{(g)},\n\\end{eqnarray}\nwhere $\\tilde c_{\\mathrm{c}} = \\sqrt{c_{\\mathrm{c}}\\left(2-c_{\\mathrm{c}}\\right) \\mu_{\\mathrm{eff}}}$. These auxiliary vectors are then used to construct the rank-$\\mu$ and rank-$1$ update matrices:\n\\begin{eqnarray} \\label{auxilary_matricies}\n    \\boldsymbol{C}_{\\mu}^{(g+1)} &=&  \\sum_{i=1}^{\\lambda} w_{i} \\boldsymbol{y}_{i: \\lambda}^{(g+1)} \\left(\\boldsymbol{y}_{i: \\lambda}^{(g+1)}\\right)^{\\boldsymbol{\\top}}\\\\\n    \\boldsymbol{C}_{1}^{(g+1)} &=&  \\boldsymbol{p}_{\\mathrm{c}}^{(g+1)} \\boldsymbol{p}_{\\mathrm{c}}^{(g+1)^{\\top}}.\n\\end{eqnarray}\nFinally, by defining $c_{old} = \\left(1-c_{1}-c_{\\mu} \\sum_{i=1}^{\\lambda} w_{i}\\right)$, we get the update rule for the covariance matrix:\n\\begin{equation}\\label{covariance_matrix_update}\n\\boldsymbol{C}^{(g+1)} =  c_{old} \\boldsymbol{C}^{(g)} + c_1 \\boldsymbol{C}_{1}^{(g)} + c_{\\mu} \\boldsymbol{C}_{\\mu}^{(g+1)}.\n\\end{equation}\n\n\n\n\n\\subsection{Step Size Update Rule}\nThe last update rule is for the step size, which also requires use of an auxiliary vector: \n\\begin{equation}\n\\boldsymbol{p}_{\\sigma}^{(g+1)}=\\left(1-c_{\\sigma}\\right) \\boldsymbol{p}_{\\sigma}^{(g)}+\\tilde c_{\\sigma}  \\boldsymbol{C}^{(g)^{-\\frac{1}{2}}} \\frac{\\boldsymbol{m}^{(g+1)}-\\boldsymbol{m}^{(g)}}{\\sigma^{(g)}},\n\\end{equation}\nWhere $\\tilde c_{\\sigma} = \\sqrt{c_{\\sigma}\\left(2-c_{\\sigma}\\right) \\mu_{\\mathrm{eff}}}$. This is then used to construct the last update rule, which is for the step size:\n\\begin{equation}\n\\sigma^{(g+1)}=\\sigma^{(g)} \\exp \\left(\\frac{c_{\\sigma}}{d_{\\sigma}}\\left(\\frac{\\left\\|\\boldsymbol{p}_{\\sigma}^{(g+1)}\\right\\|}{E\\|\\mathcal{N}(\\mathbf{0}, \\mathbf{I})\\|}-1\\right)\\right),\n\\end{equation}\nwhere ${E\\|\\mathcal{N}(\\mathbf{0}, \\mathbf{I})\\|} \\approx \\sqrt{n}+\\mathcal{O}(1 \/ n)$ is the expectation of the $l_2$ norm of samples drawn from $\\mathcal{N}(\\mathbf{0}, \\mathbf{I})$.\n\n\n\\subsection{Next Generation}\nOnce  $\\boldsymbol{m}^{(g+1)}$, $\\boldsymbol{C}^{(g+1)}$ and $\\sigma^{(g+1)}$ have been calculated, they are inserted into \\eqref{cmaes2}, so that the next generation of samples can be drawn and the process repeated, until the convergence criteria are full-filled.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}