{"text":"\\section{Introduction}\n\\label{sec:introduction}\nQuivers with relations play a fundamental role in the representation theory of finite dimensional algebras.\n\nEvery basic finite dimensional algebra $\\Lambda$ over an algebraically closed field $\\Bbbk$ is isomorphic to a path algebra $\\Bbbk Q\/R$ of a finite quiver $Q$ with admissible relations $R$. Moreover, the modules of $\\Lambda$ are in bijection with the representations of $(Q,I)$ over $\\Bbbk$. Without relations, we get a correspondence only to hereditary algebras \\cite{Gabriel}; see also \\cite[Ch.~III.1]{ARS}. \n\nNon-hereditary algebras are central in most fields of modern representation theory of algebras. For one, higher homological algebra requires algebras of global dimension at least 2 \\cite{Jasso, Kvamme}. There is a rich tradition of studying classes of non-hereditary algebras, such as gentle \\cite{AH, AS}, clannish \\cite{C-B}, Schur \\cite{Erdmann}, preprojective \\cite{Ringel}, and self--injective \\cite{SY} algebras.\n\nContinuous quivers and their representations were first explicitly studied in \\cite{IRT}. They are a natural generalisation of quivers, replacing finite sets of vertices with uncountably infinite sets. In the process, one gains intuition about what characteristics of representation theory come from innate properties of algebraic structures, and what comes from the discrete examples that are usually studied.\n\nOne parameter persistence modules are often defined over the real line so that persistence modules coincide with pointwise finite-dimensional representations of a continuous quiver of type $\\mathbb{A}$ (see, for example, \\cite{CdSGO}).\nIn \\cite{BBOS} the authors consider $m\\times n$ rectangular grid quivers which have the commutativity relation on each square.\nThe authors of \\cite{BBH} study homological approximations in order to obtain new invariants of these representations (persistence modules).\n\n\nGiven the important role of quiver relations in the representation theory of finite-dimensional algebra, it is natural to ask if relations can be extended to the continuous setting. This has already been done in a restricted sense by the second author and Zhu in \\cite{RZ}. We give a more general definition that works with any underlying quiver. To capture the full generality we actually go beyond quivers and consider categories instead.\n\nIn starting this work, we were motivated by two areas of study that we intend to lift to the continuous setting: gentle algebras and $d$-cluster-tilting subcategories. In gentle algebras, the relations appear in the definition, and are always generated by compositions of two arrows. This type of relations are generalized as \\emph{point relations} in \\Cref{subsec:point relations}. An important class of $d$-cluster-tilting subcategories appear in the module category of type A algebras, with relations consisting of all paths above a certain length \\cite{Vaso}. This type of relations is generalized as \\emph{length relations} in \\Cref{subsec:length}.\n\n\n\\subsection{Contributions}\nIn \\Cref{sec:definition and general}, we give essential background, before stating our main definition.\n\n\\begin{definition}[\\Cref{def:admissible}]\nLet $\\mathcal{C}$ be a category and $\\mathcal{I}$ an ideal in $\\mathcal{C}$.\n    We say $\\mathcal{I}$ is \\emph{admissible} if the following are satisfied.\n    \\begin{enumerate}\n        \\item For each $f$ in $\\mathcal{I}$, there exists a finite collection of morphisms $g_1,\\ldots,g_n$ not in $\\mathcal{I}$ such that $f = g_n\\circ\\cdots\\circ g_1$.\n        \\item For each nonzero endomorphism $f$, if $f$ is not an isomorphism then there exists $n\\geq 2\\in\\mathbb{N}$ such that $f^n\\in\\mathcal{I}(X,X)$.\n        \\end{enumerate}\n\\end{definition}\n\n\\Cref{sec:general results} gives some general results on the quotient category $\\mathcal{C} \/ \\mathcal{I}$, summarized here.\n\n\n\\begin{theorem}[\\Cref{prop:connected,prop:basic,prop:radical commutes with admissible}]\n    Let $\\mathcal{C}$ be a  category. Let $\\mathcal{I}$ be an admissible ideal of morphisms.\n    \n    \\begin{enumerate}\n        \\item If $\\mathcal{C}$ is connected, then $\\mathcal{C} \/ \\mathcal{I}$ is also connected.\n        \\item If $\\mathcal{C}$ is Krull--Remak--Schmidt--Azumaya, then $\\mathcal{C} \/ \\mathcal I$ is also Krull--Remak--Schmidt--Azumaya.\n        \\item If all endomorphism rings of $\\mathcal{C}$ are artinian, then \\[\\mathsf{Rad}(\\mathcal{C} \/ \\mathcal{I})=\\mathsf{Rad}(\\mathcal{C}) \/ \\mathcal{I}.\\]\n    \\end{enumerate}\n\\end{theorem}\n\nIn \\Cref{sec:relations} we give two important classes of relations that can generate admissible ideals. The first is \\emph{point relations}, which generalizes relations of length two. The idea is that certain paths through a vertex in the quiver are excluded, but others not. For an illustration see \\Cref{fig:point intro}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n        \\node (center) at (0,0) {$\\bullet$};\n        \\node at (0,.3) {$P$};\n        \\node (left) at (-1,0) {$\\circ$};\n        \\node (right) at (1,0) {$\\circ$};\n        \\draw[->] (left)--(center);\n        \\draw[->] (center) -- (right);\n    \\end{tikzpicture}\n    \\caption{Discrete case}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n    \\centering\n    \\begin{tikzpicture}[decoration={markings, mark=at position 0.5 with {\\arrow{>}}}]\n        \\node (center) at (0,0) {$\\bullet$};\n        \\node (left) at (-1,0) {};\n        \\node (right) at (1,0) {};\n        \\node at (0,.3) {$P$};\n        \\draw[very thick, postaction=decorate] (left.center) to[bend left] (center.center);\n        \\draw[very thick, postaction=decorate] (center.center) to[bend right] (right.center);\n    \\end{tikzpicture}\n    \\caption{Continuous case}\n    \\end{subfigure}\n    \\caption{An illustration of point relations in the discrete and continuous case. In both cases, the relations contain all paths passing through the point $P$.\n    See \\Cref{fig:arrow and line} on page~\\pageref{fig:arrow and line} for an explanation of our drawing conventions.}\n    \\label{fig:point intro}\n\\end{figure}\n\n\n\\begin{theorem}[\\Cref{thm:point relations are admissible}]\n    Let $\\{\\mathcal P_\\alpha\\}$ be an admissible collection of point relations in $\\mathcal{C}$, such that any cycles are either isomorphisms (and hence trivial) or contained in  at least one $\\mathcal P_\\alpha$.\n    Then $\\mathcal{I}=\\langle \\bigcup_\\alpha \\mathcal P_\\alpha \\rangle$ is an admissible ideal in $\\mathcal{C}$.\n\\end{theorem}\n\nThe other class of relations we define are \\emph{length relations}. This is a generalisation of relations generated by paths containing at least $n$ arrows, where $n$ is a natural number.\n\n\\begin{theorem}[\\Cref{thm:length relations are admissible}]\n    A length relation generates an admissible ideal.\n\\end{theorem}\n\\Cref{sec:examples} contains multiple examples of how relations work, including a sketch of their Auslander--Reiten theory.\n\n\\subsection{Future Work}\nThe present paper is a precursor to future work on generalizations of non-hereditary structures.\nOf note, the authors will consider point relations, such as \\Cref{ex:crossing real lines}, that generalize gentle algebras. They will also study the modding out by length relations, such as \\Cref{ex:length relations}(\\ref{ex:length relations:continuous A}), to generate higher cluster tilting subcategories.\n\n\\subsection{Acknowledgements}\nThe idea for this project was conceived at the Hausdorff Research Institute of Mathematics, KMJ visited JDR at Ghent University during this project, and JDR visited KMJ at Aarhus University during this project.\nThe authors thank each of these institutions for their hospitality.\nKMJ is supported by the Norwegian Research Council via the project Higher Homological Algebra and Tilting Theory (301046).\nJDR is supported at Ghent University by BOF grant 01P12621.\nThe authors would like to thank Jenny August, Raphael Bennett-Tennenhaus, Charles Paquette, Amit Shah, Emine Y{\\i}ld{\\i}r{\\i}m, and Shijie Zhu for helpful discussions.\n\n\\subsection{Conventions}\nWe work over $\\Bbbk=\\overline{\\Bbbk}$ be a field of characteristic 0.\nBy $\\mathsf{Vec}(\\Bbbk)$ and $\\mathsf{vec}(\\Bbbk)$ we denote the categories of $\\Bbbk$-vector spaces and finite-dimensional $\\Bbbk$-vector spaces, respectively.\nFor a $\\Bbbk$-algebra $\\Lambda$, denote by $\\mathsf{J}(\\Lambda)$ the Jacobson radical of $\\Lambda$.\nWe assume $\\mathcal{C}$ is a $\\Bbbk$-linear category.\n\nRecall that a category $\\mathcal{C}$ is called \\emph{Krull--Remak--Schmidt--Azumaya} if any object $X$ is isomorphic to a arbitrary sum $\\bigoplus X_i$, where each $\\End_\\mathcal{C} X_i$ is a local ring, which itself is unique up to isomorphism.\nIf every object is instead isomorphic to a finite sum $\\bigoplus_{i=1}^n X_i$ as above, we say $\\mathcal{C}$ is \\emph{Krull--Remak--Schmidt}.\n\nFinally, we consider (discrete) quivers, continuous generalizations of such quivers, and combinations of the two.\nWhen we draw an arrow, we  use a thin line with an arrow head at the end to indicate the direction.\nWhen we draw a continuous line segment, we use a bold line with the arrow head in the middle to indicate the direction;\nsee \\Cref{fig:arrow and line}.\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{tikzpicture}[decoration={markings, mark=at position 0.5 with {\\arrow{>}}}]\n        \\node (left-start) at (0,0) {$\\bullet$};\n        \\node (left-end) at (2,0) {$\\bullet$};\n        \\node (right-start) at (3,0) {$\\bullet$};\n        \\node (right-end) at (5,0) {$\\bullet$};\n        \\draw[->] (left-start)--(left-end);\n        \n        \\draw[very thick, postaction=decorate] (right-start.center)--(right-end.center);\n        \n    \\end{tikzpicture}\n    \\caption{On the left, how we draw arrows. On the right, how we draw line segments.}\n    \\label{fig:arrow and line}\n\\end{figure}\n\n\n\\section{Definition and General Results}\\label{sec:definition and general}\n\n\\subsection{$\\Bbbk$-linear categorization}\n\n\\begin{definition}\\label{def:categorification}\n    Let $Q$ be a (finite) quiver and $\\Bbbk Q$ its path algebra.\n    Let $\\mathcal{Q}$ be the category whose indecomposable objects are the vertices of $Q$ and morphisms between indecomposables $i$ and $j$ are given by\n    \\begin{displaymath}\n        \\Hom_{\\mathcal{Q}}(i,j) = e_j \\Bbbk Q e_i.\n    \\end{displaymath}\n    The objects in $\\mathcal{Q}$ are finite direct sums of the indecomposables (and 0).\n    The morphisms in $\\mathcal{Q}$ are given by extending bilinearly.\n    We call $\\mathcal{Q}$ the \\emph{$\\Bbbk$-linear categorification} of $Q$.\n\\end{definition}\n\n\\begin{example}\\label{ex:discrete categorification}\nLet $Q$ be the following quiver:\n\n\\begin{center}\n\\begin{tikzpicture}[xscale=1.5, yscale=.5]\n    \\node (1) at (0,1) {1};\n    \\node (2) at (1,2) {2};\n    \\node (3) at (1,0) {3};\n    \\node (4) at (2,1) {4};\n    \n    \\draw[->] (1) -- node[pos=0.6, above]{$\\alpha_1$} (2);\n    \\draw[->] (1) -- node[pos=0.6, below]{$\\beta_1$} (3);\n    \\draw[->] (2) -- node[pos=0.4, above]{$\\alpha_2$} (4);\n    \\draw[->] (3) -- node[pos=0.4, below]{$\\beta_2$} (4);\n\\end{tikzpicture}\n\\end{center}\nThen the $\\Bbbk$-linear categorification $\\mathcal{Q}$ is a category with indecomposable objects $1, 2, 3$ and $4$. The morphisms in $\\mathcal{Q}$ are given by paths in $Q$, so for example we have $\\Hom(1,4) \\cong\\Bbbk^2$, while  $\\Hom(4,1) = 0$\n\\end{example}\n\n\\begin{proposition}\\label{lem:correspondence between paths and morphisms}\n    There is a bijection between nonzero elements in $\\Bbbk Q$ and nonzero morphisms in $\\mathcal{Q}$.\n\\end{proposition}\n\\begin{proof}\n    A non-zero element in $\\Bbbk Q$ is a finite sum of paths in $Q$. We can therefore define a map $F$ from the elements in $\\Bbbk Q$ to morphisms in $\\mathcal{Q}$ by specifying the action of $F$ on paths in $Q$. We let this mapping be determined by $\\Hom_{\\mathcal{Q}}(i,j) = e_j \\Bbbk Q e_i$. This map is a bijection by bilinearity of $\\mathcal{Q}$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{prop:same representations}\n    Let $\\mathsf{Mod}(\\mathcal{Q})$ be the category of functors $\\mathcal{Q}\\to \\mathsf{Vec}(\\Bbbk)$.\n    Then there exists an isomorphism of categories $\\Phi:\\mathsf{Mod}(\\mathcal{Q}) \\to \\mathsf{Rep}(Q)$.\n\\end{lemma}\n\n\\begin{proof}\n    Let $F$ be a functor in $\\mathsf{Mod}(\\mathcal{Q})$.\n    We now define the corresponding representation $V=\\Phi(F)$.\n    Let $M$ be the representation of $Q$ over $\\Bbbk$ where $V(i)=F(i)$ for each $i\\in Q_0$.\n    For a path $\\rho$ in $Q$, let $V(\\rho)$ be the $\\Bbbk$-linear map $F(\\rho)$.\n    \n    Let $f:F\\to G$ be a morphism in $\\mathsf{Mod}(\\mathcal{Q})$.\n    Then $\\Phi(f):\\Phi(F)\\to \\Phi(G)$ is defined by the $f_i:F(i)\\to G(i)$ for each $i\\in Q_0$.\n    Straightforward computations show that $\\Phi$ respects composition and so it is a functor.\n    \n    Define $\\Phi^{-1}:\\mathsf{Rep}(Q)\\to \\mathsf{Mod}(\\mathcal{Q})$ in the following way.\n    For a representation $V$ of $Q$, let $F=\\Phi^{-1}(V)$ be determined by $F(i)=V(i)$, for each $i\\in Q_0$, and $F(\\rho)=V(\\rho)$ for each path in $Q$.\n    Morphisms are defined similarly.\n    One may check $\\Phi^{-1}\\Phi$ and $\\Phi\\Phi^{-1}$ are the identity functors on $\\mathsf{Mod}(\\mathcal{Q})$ and $\\mathsf{Rep}(Q)$, respectively.\n\\end{proof}\n\n\\subsection{The Jacobson Radical}\\label{sec:jacobson radical}\n\n\\begin{definition}\\label{def:radical of C}\n    Let $\\mathcal{C}$ be a category.\n    The \\emph{radical} $\\mathsf{Rad}(\\mathcal{C})$ of $\\mathcal{C}$ is the ideal consisting of\n    \\begin{displaymath}\n        \\mathsf{Rad}_{\\mathcal{C}}(X,Y) := \\left\\{ f\\in\\Hom_{\\mathcal{C}}(X,Y) \\mid \\forall g\\in\\Hom_{\\mathcal{C}}(Y,X),\\, f\\circ g\\in\\mathsf{J}(\\End_{\\mathcal{C}}(Y))  \\right\\},\n    \\end{displaymath}\n    for each pair of objects $X,Y$ in $\\mathcal{C}$.\n\\end{definition}\n\n\\begin{proposition}[\\cite{Krause}]\\label{prop:Krause}\n    Let $X$ and $Y$ be objects in $\\mathcal{C}$.\n    Then $\\mathsf{Rad}_{\\mathcal{C}}(X,Y)=\\mathsf{J}(\\Hom_{\\mathcal{C}}(X,Y))$.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:easy radical}\n    Let $f:X\\to Y$ be an morphism for indecomposable objects $X,Y$ in $\\mathcal{C}$.\n    Then $f\\in\\mathsf{Rad}(\\mathcal{C})$ if and only if $f$ is not an isomorphism.\n\\end{proposition}\n\\begin{proof}\n    Suppose $f$ is not an isomorphism.\n    If $\\mathcal{C}$ does not have cycles we are done.\n    If $\\mathcal{C}$ has cycles, let $g:Y\\to X$ be a nonzero morphism.\n    Then $f\\circ g\\in\\mathsf{J}(\\End_{\\mathcal{C}}(Y))$ and so $f\\in \\mathsf{Rad}_{\\mathcal{C}}(X,Y)$.\n    Reversing the argument shows that if $f\\in\\mathsf{Rad}_{\\mathcal{C}}(X,Y)$ then $f$ is not an isomorphism.\n\\end{proof}\n\nRecall that $\\mathcal{C}$ is semi-simple if every object in $\\mathcal{C}$ is a finite direct sum of simple objects and all such direct sums exist.\n\n\\begin{proposition}\\label{prop:C is basic}\n    If $\\mathcal{C}$ is Krull--Remak--Schmidt, then $\\mathcal{C}\/ \\mathsf{Rad}(\\mathcal{C})$ is semi-simple.\n\\end{proposition}\n\\begin{proof}\n    Let $\\mathcal{C}$ be a Krull--Remak--Schmidt category.\n    Let $X$ and $Y$ be indecomposables in $\\mathcal{C}$ such that $X\\not\\cong Y$.\n    Then $\\Hom_{\\mathcal{C}}(X,Y)=\\mathsf{Rad}_{\\mathcal{C}}(X,Y)$ and so $\\Hom_{\\mathcal{C} \/ \\mathsf{Rad}(\\mathcal{C})}(X,Y)=0$.\n    Extending bilinearly we see $\\mathcal{C} \/ \\mathsf{Rad}(\\mathcal{C})$ is semi-simple.\n\\end{proof}\n\n\\begin{remark}\\label{rmk:categorification is basic}\n    It follows immediately from \\Cref{prop:C is basic} that if $Q$ is a finite acyclic quiver then $\\mathcal{Q} \/\\mathsf{Rad}(\\mathcal{Q})$ is semi-simple.\n\\end{remark}\n\n\\subsection{Admissible Ideals}\\label{sec:admissible ideals}\n\n\\begin{definition}[\\cite{Krause}]\n    Let $\\{\\mathcal{C}_i\\}_{i\\in I}$ be a family of full additive subcategories of $\\mathcal{C}$. We have an \\emph{orthogonal decomposition} $\\coprod_{i\\in I} \\mathcal{C}_i$ of $\\mathcal{C}$ if every object $X$ in $\\mathcal{C}$ is isomorphic to a direct sum $\\bigoplus_{i\\in I} X_i$, where $X_i$ is an object of $\\mathcal{C}_i$, and for $X_i\\in \\mathcal{C}_i,X_j\\in C_j$ we have $\\Hom_{\\mathcal{C}}(X_i,X_j)=0$ when $i\\neq j$.\n    \n    We say $\\mathcal{C}$ is \\emph{connected} if the only orthogonal decomposition of $\\mathcal{C}$ is the trivial one. \n\\end{definition}\n\nAn \\emph{ideal} $\\mathcal{I}$ of a category $C$ is a collection of morphisms is $\\mathcal{C}$ such that for any $f\\in \\mathcal{I}$ and for any $g$ and $h$ such that the composition $gfh $ is defined, the composition $gfh\\in \\mathcal{I}$.\nFor an ideal $\\mathcal{I}$ of $\\mathcal{C}$, we denote by $\\mathcal{I}(X,Y)$ the morphisms in $\\Hom_{\\mathcal{C}}(X,Y)\\cap \\mathcal{I}$.\n\n\\begin{remark}\n    For an ideal $\\mathcal{I}$ of $\\mathcal{C}$, the category $\\mathcal{C} \/\\mathcal{I}$ has the same objects as $\\mathcal{C}$.\n    The morphisms of $\\mathcal{C} \/ \\mathcal{I}$ are given by $\\Hom_{\\mathcal{C}}(X,Y) \/ \\mathcal{I}(X,Y)$.\n    A representation $V: \\mathcal{C} \/ \\mathcal{I} \\to \\Bbbk\\text{vec}$ is also a representation of $\\mathcal{C}$ by precomposition with the quotient functor $\\pi$. Thus we obtain a representation $\\widetilde{V}:\\mathcal{C} \\stackrel{\\pi}{\\to} \\mathcal{C} \/ \\mathcal{I} \\stackrel{V}{\\to} \\Bbbk\\text{vec}$.\n    Hence\n    we may consider the representations of $\\mathcal{C}\/\\mathcal{I}$ as a subcategory of the representations of $\\mathcal{C}$.\n    In particular, representations of $\\mathcal{C}\/\\mathcal{I}$ are those representations $V$ of $\\mathcal{C}$ such that if $f\\in\\mathcal{I}$ then $V(f)=0$.\n\\end{remark}\n\n\\begin{definition}\\label{def:admissible}\n    Let $\\mathcal{C}$ be a $\\Bbbk$-linear, Krull--Remak--Schmidt--Azumaya category and $\\mathcal{I}$ an ideal in $\\mathcal{C}$.\n    We say $\\mathcal{I}$ is \\emph{admissible} if the following are satisfied.\n    \\begin{enumerate}\n        \\item For each $f$ in $\\mathcal{I}$, there exists a finite collection of morphisms $g_1,\\ldots,g_n$ not in $\\mathcal{I}$ such that $f = g_n\\circ\\cdots\\circ g_1$.\n        \\item For each indecomposable $X$ in $\\mathcal{C}$, the endomorphism ring $\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)$ is finite-dimensional.\n    \\end{enumerate}\n\\end{definition}\n\nWe remark that, in \\Cref{def:admissible}(2), we do not want to require that there is some $n$ that works for all nonisomorphism endomorphisms $f$.\nSee \\Cref{ex: cycles length} for an explicit example why.\n\n\n\\begin{lemma}\\label{lem: admissible in radical}\n\tLet $\\mathcal{C}$ be a Krull--Remak--Schmidt category and $\\mathcal{I}$ an ideal in $\\mathcal{C}$.\n    If $\\mathcal{I}$ is admissible, then $\\mathcal{I}$ is contained in the radical of $\\mathcal{C}$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $f:X\\rightarrow Y$ be a morphism in $\\mathcal{I}$ between indecomposable objects. Then we know by \\Cref{prop:easy radical} that if $f$ is not contained in the radical, it is an isomorphism. However, if $f$ is an isomorphism, we have $1_X\\in\\mathcal{I}$.\n\tThen \\emph{every} morphism to \/ from $X$ is in $\\mathcal{I}$.\n\tThus, if $1_x=g_n\\circ\\cdots\\circ g_1$ for any composition, both $g_n,g_1\\in \\mathcal{I}$, which contradicts condition \\Cref{def:admissible}(1).\n\tHence $\\mathcal{I}(X,Y)\\subseteq \\mathsf{Rad}(X,Y)$ for indecomposable $X,Y$. \n\t\n\tNow let $X=\\bigoplus_{i=0}^m X_i$ and $Y=\\bigoplus_{j=0}^n Y_j$, where each $X_i$ and $Y_j$ is indecomposable. Consider $f\\in \\mathcal{I}(X,Y)$. We can rewrite $f$ as $f=(f_{ij})$, where $f_{ij}:X_i\\rightarrow Y_j$. By composition with the canonical injections and projections, we see that $f_{ij}\\in \\mathcal{I}(X_i,Y_j)$, so by the above, $f_{ij}\\in \\mathsf{Rad}(X_i,Y_j)$.\n\tThen by linearity, $f\\in \\mathsf{Rad}(X,Y)$.\n\\end{proof}\n\n\nLet $Q$ be a finite quiver and let $\\mathcal{Q}$ be its $\\Bbbk$-linear categorification. Suppose $I$ is an ideal of the path algebra $\\Bbbk Q$. We show how to build an ideal $\\mathcal{I}$ in $\\mathcal{Q}$ from $I$. \n\nFrom the definition of the $\\Bbbk$-linear categorification, we know that each path in $I$ corresponds to a non-zero morphism in $\\mathcal{Q}$, see \\Cref{lem:correspondence between paths and morphisms}. We (na\\\"ively) let $\\mathcal{I}$ be the set of morphisms obtained by mapping $I$ to $\\mathcal{Q}$. We now show that $\\mathcal{I}$ is an ideal of the category $\\mathcal{Q}$. \n\nBy $\\Bbbk$-linearity of $Q$, it is enough to consider morphisms between indecomposable objects.\nSuppose $f\\in\\mathcal{I}(i,j)$ for some $i,j\\in Q_0=\\operatorname{Ind}\\mathcal{Q}$, and let $g:j\\rightarrow k$ and $h:l\\rightarrow i$ be two nonzero morphisms in $\\mathcal{Q}$.\nBy \\Cref{lem:correspondence between paths and morphisms} we know that $f$ corresponds to an element $\\rho$ in $e_j\\Bbbk Qe_i$.\nFurther, $g$ corresponds to an element $\\psi$ in $\\Bbbk Qe_j$ and $h$ corresponds to an element $\\phi$ in $e_i\\Bbbk Q$.\nEach of $\\rho$, $\\phi$, and $\\psi$ are are sums of paths in $Q$ from the respective source and to the respective target.\nWithout loss of generality, due to $\\Bbbk$-linearity, suppose each of $\\rho$, $\\phi$, and $\\psi$ is a path in $Q$.\nWe see $\\psi\\rho\\phi$ is an element of $I$ since $I$ is a two-sided ideal containing $\\rho$.\nThe image of the composition $\\psi\\rho\\phi$ is the composition $gfh$, which must therefore be in $\\mathcal{I}$.\n\n\\begin{proposition}\\label{prop:admissible is correct}\n    Let $Q$ be a finite quiver and $I$ an ideal of $\\Bbbk Q$ as a path algebra.\n    Let $\\mathcal{Q}$ be the $\\Bbbk$-linear category induced by $Q$ and let $\\mathcal{I}$ be the ideal induced by $I$ in $\\mathcal{Q}$.\n    Then $\\mathcal{I}$ is an admissible ideal of $\\mathcal{Q}$ as in \\Cref{def:admissible} if and only if $I$ is an admissible ideal of $\\Bbbk Q$.\n\\end{proposition}\n\n\\begin{proof}\n    Let $I$ be an admissible ideal of $\\Bbbk Q$ and $\\rho\\in I$.\n    We first prove $\\mathcal I$ satisfies property (1) of \\Cref{def:admissible}.\n    Without loss of generality, assume $\\rho$ is a path in $Q$.\n    Then $\\rho = \\alpha_n \\alpha_{n-1}\\cdots \\alpha_2\\alpha_1$, where each $\\alpha_i$ is an arrow in $Q$.\n    Now, let $f$ be the morphism in $\\mathcal{Q}$ corresponding to $\\rho$ and $g_i$ the morphism in $\\mathcal{Q}$ corresponding to $\\alpha_i$, for each $i$.\n    Then we know each $g_i\\notin \\mathcal I$ and have satisfied property (1) of \\Cref{def:admissible}.\n    Reversing the argument proves the converse.\n    \n    If $Q$ has no cycles then the proposition immediately holds for property (2) of \\Cref{def:admissible}.\n    So, suppose $Q$ has at least one oriented cycle.\n    Since $I \\subset \\mathsf{Rad}^n(\\Bbbk Q)$, for some $n\\geq 2$, we see that $\\mathcal I$ must immediately satisfy property (2) of \\Cref{def:admissible}.\n    \n    Now suppose $\\mathcal{I}$ satisfies \\Cref{def:admissible}(2).\n    Since $Q$ is finite, there are finitely many cycles.\n    For each cycle $\\rho$ at each vertex $i$, let \n    \\[ m_\\rho= \\min_m \\{ m \\mid \\rho^m \\in \\mathcal{I}(i,i)\\}.\\]\n    We know such an $m_\\rho$ exists since $\\End_{\\mathcal{Q}}(i) \/ \\mathcal{I}(i,i)$ is finite-dimensional.\n    Let $n_\\rho$ be $m_\\rho$ time the length of $\\rho$.\n    Then let \\[ N = \\max\\left(\\max_\\rho \\{n_\\rho\\}\\cup \\{\\text{length of longest path without cycles in }Q\\}\\right).\\]\n    Thus, $\\mathsf{Rad}^N(\\Bbbk Q)\\supset I$.\n    This concludes the proof.\n\\end{proof}\n\n\\begin{remark}\nThe second half of the proof above can be extended to more general quivers.\nSuppose $\\mathcal{Q}$ is the $\\Bbbk$-linear categorification of a (not necessarily finite) quiver $Q$ with finitely many cycles. Suppose that for each cycle $\\rho$ in the quiver with corresponding morphism $f_\\rho:X\\rightarrow X$, there is some $n\\geq 2$ such that $f_\\rho\\in \\mathcal{I}(X,X)$.\nThen $\\mathcal{I}$ satisfies criterion (2) in \\Cref{def:admissible}. \n\nFor the majority of our examples, this will be the criterion we actually use.\n\\end{remark}\n\n\\begin{example}\\label{ex:discrete ideal}\n    Consider the quiver from \\Cref{ex:discrete categorification}. Let $I$ be the commutative ideal generated by $\\{\\alpha_2\\alpha_1-\\beta_2\\beta_1\\}$.\n    \n    In the $\\Bbbk$-linear categorification, the relation $\\alpha_2\\alpha_1-\\beta_2\\beta_1$ can be written as $\\left [ \\begin{smallmatrix}\\alpha_2 & \\beta_2\\end{smallmatrix}\\right]\\left [ \\begin{smallmatrix}\\alpha_1 \\\\ -\\beta_1\\end{smallmatrix}\\right]$. The ideal generated by this morphism fulfills the criteria for being an admissible ideal. \n\\end{example}\n\n\\subsection{General Results}\\label{sec:general results}\n\n\\begin{proposition}\\label{prop:connected}\n    Let $\\mathcal{C}$ be connected, and let $\\mathcal{I}$ be an admissible ideal of morphisms. Then $\\mathcal{C} \/ \\mathcal{I}$ is connected.\n\\end{proposition}\n\n\\begin{proof}\n Assume towards a contradiction that $\\mathcal{C} \/ \\mathcal{I}$ is not connected; then there exists a decomposition of $\\mathcal{C} \/ \\mathcal{I}$ into mutually orthogonal subcategories $\\mathcal{C}'_1, \\cdots, \\mathcal{C}'_n$. We can lift these subcategories to subcategories $\\mathcal{C}_1, \\cdots, \\mathcal{C}_n$ of $\\mathcal{C}$. As $\\mathcal{C}$ is connected, these subcategories cannot all be mutually orthogonal, so assume that there exists some morphism $f:X_i\\rightarrow X_j$, with $X_i\\in \\mathcal{C}_i, X_j\\in \\mathcal{C}_j$ and $i\\neq j$. To preserve mutual orthogonality in $\\mathcal{C}\/\\mathcal{I}$, we must have $f\\in \\mathcal{I}$. Then since $I$ is an admissible ideal, we can write $f=g_m\\circ \\cdots \\circ g_1$, where each of the $g_1, \\cdots, g_m$ are not in $\\mathcal{I}$.\n \n Consider $g_1: X_i\\rightarrow Y$ and denote its image in $\\mathcal{C} \/ \\mathcal{I}$ by $\\overline{g_1}$. As $\\mathcal{C} \/ \\mathcal{I}$ is not connected, we can write $Y=\\bigoplus_{i=1}^n Y_i$, with $Y_i\\in \\mathcal{C}'_i$ and $\\overline{g_1}=(g^1_1,\\cdots g_1^n)$. Now, as $\\overline{g_1}$ is nonzero, $g_1^k$ is nonzero for some $k$. If $k\\neq i$, we have reached a contradiction. If $k=i$, we can repeat the argument with $\\overline{g_2}|_{Y_i}$, eventually reaching a contradiction. \n\\end{proof}\n\n\\begin{proposition}\\label{prop:basic}\n    Let $\\mathcal{C}$ be Krull--Remak--Schmidt--Azumaya and $\\mathcal{I}$ an admissible ideal.\n    Then $\\mathcal{C} \/ \\mathcal I$ is Krull--Remak--Schmidt--Azumaya.\n\\end{proposition}\n\\begin{proof}\nLet $X$ be an object in $\\mathcal{C} \/ \\mathcal I$. Then $X$ is an object in $\\mathcal{C}$, which we assume to be Krull--Remak--Schmidt--Azumaya. It follows that $X=\\bigoplus_{\\alpha} X_\\alpha$, where $\\End_\\mathcal{C}(X_\\alpha)$ is local. If we can show that $\\End_{\\mathcal{C}\/\\mathcal{I}}(X_\\alpha)$ is local, we are done.\n\nWe know that $\\End_{\\mathcal{C}\/\\mathcal{I}}(X_{\\alpha})=\\End_\\mathcal{C}(X_{\\alpha})\/\\mathcal{I}(X_{\\alpha},X_{\\alpha})$. By property 1 of \\Cref{def:admissible}, the identity on $X_{\\alpha}$ cannot be an element of $\\mathcal{I}(X_{\\alpha},X_{\\alpha})$, so $\\End_{\\mathcal{C}\/\\mathcal{I}}(X_{\\alpha})$ is a nonzero quotient ring of a local ring, which is local by the ideal correspondence theorem for quotient rings.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:radical commutes with admissible}\n    Let $\\mathcal{C}$ be a category such that all endomorphism rings are artinian and $\\mathcal{I}$ an admissible ideal.\n    Then \\[\\mathsf{Rad}(\\mathcal{C} \/ \\mathcal{I})=\\mathsf{Rad}(\\mathcal{C}) \/ \\mathcal{I}.\\] \n\\end{proposition}\n\\begin{proof}\nThe equation $\\mathsf{Rad}(\\mathcal{C} \/ \\mathcal{I}) = \\mathsf{Rad}(\\mathcal{C}) \/ \\mathcal{I}$ holds if and only if $\\mathsf{Rad}_{\\mathcal{C} \/ \\mathcal{I}}(X,Y) = \\mathsf{Rad}_\\mathcal{C}(X,Y) \/ \\mathcal{I}$ holds for all pairs of objects $X,Y\\in \\mathcal{C}$.\n\nFirst note that if $\\End_{\\mathcal{C} }(Y)$ is artinian, then\n\\begin{align*}\n\t \\mathsf{Rad}_\\mathcal{C}(Y,Y) \/ \\mathcal{I} &= \\mathsf{J}(\\End_{\\mathcal{C} }(Y))\/ \\mathcal{I}=  \\mathsf{J}(\\End_{\\mathcal{C} }(Y)+\\mathcal{I})\/\\mathcal{I} \\\\ \n\t &= \\mathsf{J}(\\End_{\\mathcal{C} \/ \\mathcal{I}}(Y)) =\\mathsf{Rad}_{\\mathcal{C} \/ \\mathcal{I}}(Y,Y).\n\\end{align*}\nNow we can see that \n\\begin{align*}\n\tf \\in \\mathsf{Rad}_\\mathcal{C}(X,Y) \/ \\mathcal{I} &\\Leftrightarrow f = f'+\\mathcal{I}(X,Y)\\text{, with } f'\\in \\mathsf{Rad}_\\mathcal{C}(X,Y) \\\\\n\t& \\Leftrightarrow  f = f'+\\mathcal{I}(X,Y)\\text{, s.t.\\ } f'\\circ g'\\in \\mathsf{J}(\\End_{\\mathcal{C} }(Y))\\, \\forall\\, g'\\in \\Hom_\\mathcal{C}  (Y,X)\\\\\n\t& \\Leftrightarrow f\\circ (g'+\\mathcal{I}(Y,X))\\in \\mathsf{J}(\\End_{\\mathcal{C} }(Y)  )\/\\mathcal{I}\\, \\forall \\, g'\\in \\Hom_\\mathcal{C}  (Y,X)\\\\\n\t& \\Leftrightarrow f\\circ g \\in \\mathsf{J}(\\End_{\\mathcal{C} \/\\mathcal{I}}(Y)  )\\, \\forall\\, g\\in \\Hom_{\\mathcal{C}\/\\mathcal{I}}  (Y,X)\\\\\n\t& \\Leftrightarrow f \\in \\mathsf{Rad}_{\\mathcal{C}\/\\mathcal{I}}(X,Y). \n\\end{align*}\n\\end{proof}\nIn particular, if $\\mathcal{C}$ is $\\Bbbk$-linear and Krull--Remak--Schmidt, then all endomorphism rings are artinian and the above Proposition holds.\n\nThe following lemma is useful for our examples in \\Cref{sec:examples}.\n\\begin{lemma}\\label{lem:stack admissible}\n    Let $\\mathcal{C}$ be a small, $\\Bbbk$-linear, Krull--Remak--Schmidt--Azumaya category and $\\mathcal{I}$ an admissible ideal in $\\mathcal{C}$.\n    Let $\\mathcal{J}$ be an admissible ideal in $\\mathcal{C}\/\\mathcal{I}$ and $\\widetilde{\\Jay}$ the set of morphisms $f$ in $\\mathcal{C}$ such that $f+\\mathcal{I}\\in\\mathcal{J}$ in $\\mathcal{C}\/\\mathcal{I}$. Then the following hold:\n    \\begin{enumerate}\n        \\item $\\widetilde{\\Jay}$ contains $\\mathcal{I}$.\n        \\item $\\widetilde{\\Jay}$ is an ideal.\n        \\item $\\widetilde{\\Jay}$ is admissible in $\\mathcal{C}$.\n        \\item $(\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J} \\simeq \\mathcal{C}\/\\widetilde{\\Jay}$.\n    \\end{enumerate} \n\\end{lemma}\n\\begin{proof}\n    \\textbf{1.} Let $f\\in\\mathcal{I}$.\n    Then $f\\mapsto 0$ in $\\mathcal{C}\/\\mathcal{I}$.\n    Since all zero morphisms in $\\mathcal{C}\/\\mathcal{I}$ are in $\\mathcal{J}$, we see $f\\in \\widetilde{\\Jay}$.\n    \n    \\textbf{2.} Let $f:X\\to Y$ be nonzero in $\\widetilde{\\Jay}$ and let $g:Y\\to Z$ be nonzero in $\\mathcal{C}$.\n    Then $f+\\mathcal{I}$ and $g+\\mathcal{I}$ are in $\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})$.\n    So, $(g+\\mathcal{I})\\circ(f+\\mathcal{I})$ is in $\\mathcal{J}$ and is equal to $gf + \\mathcal{I}$.\n    Then $gf \\in \\widetilde{\\Jay}$.\n    \n    \\textbf{3.} Let $f\\in\\widetilde{\\Jay}$.\n    Then $f+\\mathcal{I}\\in\\mathcal{J}$ and, by assumption, there exists $(g_1+\\mathcal{I}),\\ldots,(g_n+\\mathcal{I})$ in $\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})\\setminus\\mathcal{J}$ such that \\[ f + \\mathcal{I} = (g_n+\\mathcal{I})\\circ (g_{n-1}+\\mathcal{I})\\circ\\cdots\\circ (g_2+\\mathcal{I})\\circ(g_1+\\mathcal{I}).\\]\n    Then for each $g_i$ there is $h_i\\in\\mathcal{I}$ such that $g_i+h_i\\mapsto g_i+\\mathcal{I}$ and \\[f = (g_n+h_n)\\circ(g_{n-1}+h_{n-1})\\circ\\cdots\\circ(g_2+h_2)\\circ (g_1+h_1). \\]\n    Since each $g_i\\notin \\mathcal{J}$, we know each $g_i+h_i\\notin\\widetilde{\\Jay}$ and so $f$ is a finite composition of morphisms not in $\\widetilde{\\Jay}$.\n    Additionally, for any nonzero, nonisomorphism endomorphism $f$, we have $f^n\\in\\mathcal{I}$ for some $n\\in\\mathbb{N}$.\n    Then $f^n\\in\\widetilde{\\Jay}$ by statement 1.\n    Therefore, $\\widetilde{\\Jay}$ is admissible.\n    \n    \\textbf{4.} Recall $\\mathsf{Ob}((\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J}) = \\mathsf{Ob}(\\mathcal{C}\/\\widetilde{\\Jay})$.\n    We now produce a bijection between $\\mathsf{Mor}((\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J})$ and  $\\mathsf{Mor}(\\mathcal{C}\/\\widetilde{\\Jay})$ by producing bijections \\[ \\phi_{X,Y}:\\Hom_{\\mathcal{C}\/\\widetilde{\\Jay}}(X,Y) \\to \\Hom_{(\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J}}(X,Y) \\] for each ordered pair $X,Y$ of objects.\n    \n    Let $f+\\widetilde{\\Jay}\\in\\Hom_{\\mathcal{C}\/\\widetilde{\\Jay}}(X,Y)$.\n    Then there exists $g\\in\\widetilde{\\Jay}\\subset \\mathsf{Mor}(\\mathcal{C})$ such that $f+g\\mapsto f+\\widetilde{\\Jay}\\in\\mathsf{Mor}(\\mathcal{C}\/\\widetilde{\\Jay})$.\n    If $g\\in\\mathcal{I}$ then $f+g\\mapsto f+\\mathcal{I}\\in\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})$; otherewise $f+g\\mapsto f+g+\\mathcal{I}\\in\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})$.\n    In either case, $f+g\\mapsto f+\\mathcal{J}$ in $\\Hom_{(\\mathcal{C}\/\\mathcal{I})\/\\mathsf{J}}(X,Y)$.\n    We define $\\phi_{X,Y}(f+\\widetilde{\\Jay}):= f+\\mathcal{J}$.\n    \n    It is immediate that $\\phi_{X,Y}$ is injective.\n    Suppose $f+\\mathcal{J}\\in\\Hom_{(\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J}}$.\n    Then there exists $g+\\mathcal{I}$ in $\\Hom_{\\mathcal{C}\/\\mathcal{I}}(X,Y)$ such that $f+g+\\mathcal{I}\\mapsto f+\\mathcal{J}$.\n    Then there exists $h\\in\\Hom_{\\mathcal{C}}(X,Y)$ such that $f+g+h\\mapsto f+g+\\mathcal{I}$.\n    But this means $g+h\\in\\widetilde{\\Jay}$ and so $f+(g+h)\\mapsto f+\\widetilde{\\Jay}$ in $\\Hom_{\\mathcal{C}\/\\widetilde{\\Jay}}(X,Y)$.\n    Thus, $\\phi_{X,Y}$ is surjective.\n\\end{proof}\n\n\\section{Relations}\\label{sec:relations}\n\nIn this section we look at two types of admissible ideals: those generated by point relations (\\Cref{subsec:point relations}) and those generated by length relations (\\Cref{subsec:length}).\nThese generalize a relation generated by a single path of length two and relations generated by all paths of a particular length, respectively.\n\n\\subsection{Point Relations}\\label{subsec:point relations}\nHere we generalize relations generated by a single path of length 2 to a point relation (\\Cref{def:point relation}) and prove this generates an admissible ideal (\\Cref{thm:point relations are admissible}).\nWe then give examples that point to a continuous version of a gentle algebra (\\Cref{ex:monomial points,ex:crossing real lines}).\n\n\\begin{definition}\\label{def:decomposition point}\n    Let $f:X\\to Y$ be a nonisomorphism between indecomposables in $\\mathcal{C}$.\n    A \\emph{decomposition point} $Z$ in $\\mathcal{C}$ is an indecomposable object such that there exists nonisomorphisms $g:X\\to Z$ and $h:Z\\to Y$ where $f=h\\circ g$.\n\\end{definition}\n\n\\begin{definition}\\label{def:acyclic morphism}\n    Let $f:X\\to Y$ be a nonisomorphism of indecomposables in $\\mathcal{C}$, $Z$ a decomposition point of $f$, and $f=g\\circ h$ such a decomposition.\n    We call $f$ an \\emph{acyclic morphism} if for all pairs $g':X\\to Z$ and $h':Z\\to Y$ such that $f=h'\\circ g'$ then $h'$ and $g'$ are scalar multiples of $h$ and $g$, respectively.\n\\end{definition}\n\nNote that an acyclic morphism cannot be irreducible. (I.e., it must be a path of length at least 2 in a quiver.)\n\n\\begin{definition}\\label{def:point relation}\n    Let $f$ be an acyclic morphism and $Z$ a decomposition point of $f$.\n    Let $P$ be the set of all nonisomorphisms $g$ between indecomposables satisfying the following.\n    \\begin{itemize}\n        \\item There exists $h_1$ and $h_2$ morphisms of indecomposables such that $f=h_2\\circ g\\circ h_1$.\n        \\item We have $Z$ as a decomposition point of $g$.\n    \\end{itemize}\n    Let $\\mathcal P_{f,Z}$ be the ideal in $\\mathcal{C}$ generated by $P$.\n    We call $\\mathcal P_{f,Z}$ the \\emph{point relation through $Z$ by $f$}.\n\\end{definition}\n\n\\begin{definition}\\label{def:admissible point relations}\n    Let $\\{\\mathcal P_\\alpha\\}$ be a collection of point relations in $\\mathcal{C}$.\n    We say $\\{\\mathcal P_\\alpha\\}$ is \\emph{admissible} if each morphism of indecomposables appears in at most finitely-many $\\mathcal P_\\alpha$.\n\\end{definition}\n\n\\begin{theorem}\\label{thm:point relations are admissible}\n    Let $\\{\\mathcal P_\\alpha\\}$ be an admissible collection of point relations in $\\mathcal{C}$ and let $\\mathcal{I}=\\langle \\bigcup_\\alpha P_\\alpha\\rangle$.\n    Suppose also that for each indecomposable $X$ in $\\mathcal{C}$, we have $\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)$ is finite dimensional.\n    Then $\\mathcal{I}$ is an admissible ideal.\n\\end{theorem}\n\\begin{proof}\nWe satisfy \\Cref{def:admissible}(2) by assumption.\n\nNow suppose $f\\in \\mathcal{I}$; we show that $f$ can be written as a finite composition of morphisms not in $\\mathcal{I}$.\nSince $f$ is a finite sum of morphisms of indecomposables, we assume without loss of generality $f$ is a morphism of indecomposables.\nThen, by assumption there are at most finitely-many $\\mathcal P_\\alpha$ such that $f\\in \\mathcal P_\\alpha$.\n\nWe proceed by induction beginning with $f$ is only in $\\mathcal P_1$.\nLet $f_1$, $P_1$, and $Z_1$ be as in \\Cref{def:point relation}.\nThen there exists morphisms $h_1$ and $h_2$ in $Mor(\\mathcal{C})$ and $g\\in P_1$ such that $f = h_2\\circ g \\circ h_1$.\nFurther, $g=h'_2\\circ h'_1$ where the target of $h'_1$ is $Z_\\alpha$ and the source of $h'_2$ is $Z_\\alpha$.\nSo, let $g_1 = h'_1\\circ h_1$ and let $g_2=h_2\\circ h'_2$.\nNote that neither $g_1$ nor $g_2$ is in $\\mathcal P_1$.\nFurther, neither $g_1$ nor $g_2$ is in $\\mathcal{I}$ or else $f$ would be in another $\\mathcal P_\\alpha$ as well.\nThus, we have our desired decomposition.\n\nNow assume that if $f$ is in $n$ of the $\\mathcal P_\\alpha$, then $f$ is a finite composition of morphisms not in $\\mathcal{I}$.\nSuppose $f$ is in $n+1$ of the $\\mathcal P_\\alpha$ and denote one of them by $\\mathcal P_1$.\nLet $f_1$, $P_1$, and $Z_1$ be as before for $\\mathcal P_1$.\nWe find $g_1$ and $g_2$ as before, but they may be in $\\mathcal{I}$.\nHowever, each $g_1$ and $g_2$ may only be in $n$ or fewer $\\mathcal P_\\alpha$ and so are a finite composition of morphisms not in $\\mathcal{I}$.\nTherefore, $f$ is a finite composition of morphisms not in $\\mathcal{I}$.\n\\end{proof}\n\n\\begin{example}[Discrete quiver]\\label{ex:monomial points}\nLet $Q$ be a discrete quiver. Then any quadratic monomial relation in $Q$ corresponds to a point relation in the $\\Bbbk$-linear categorification of $Q$.\n\nIn particular, any gentle algebra can be obtained by considering a quiver with point relations. \n\\end{example}\n\n\\begin{example}[Continuous ``gentle'', crossing real lines]\\label{ex:crossing real lines}\nConsider two copies of the real line, labeled $\\mathbb{R}$ and $\\mathbb{R}'$. We label the numbers in $\\mathbb{R}$ by $x$ and the numbers in $\\mathbb{R}'$ by $x'$. Identify $0$ and $0'$ and label the category of $\\Bbbk$-representations  of the resulting partially ordered set by $\\mathcal{C}$. \n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[inner sep = 0cm, outer sep = 0cm, xscale = 1, decoration={\n    markings,\n    mark=at position 0.5 with {\\arrow{>}}}, very thick\n    ]\n    \\coordinate (topleft) at (0,1);\n    \\coordinate (bottomleft) at (0,0);\n    \\coordinate (topright) at (10,1);\n    \\coordinate (bottomright) at (10,0);\n    \\coordinate (mid) at (5,0.5);\n    \\draw[blue, postaction=decorate] (topleft) ..controls +(4,0).. (mid); \n    \\draw[red, postaction=decorate] (mid) ..controls +(1,-.5).. (bottomright);\n    \\draw[red, postaction=decorate] (bottomleft)  ..controls +(4,0).. (mid); \n    \\draw[blue, postaction=decorate] (mid) ..controls +(1,.5).. (topright);\n    \\draw[fill=white] (mid) circle[radius=1mm];\n    \\end{tikzpicture}\n    \\caption{The category considered in \\Cref{ex:crossing real lines}. The two copies of the real line have been drawn in different colours}\n    \\label{fig:my_label}\n\\end{figure}\n\nLet $\\mathcal P$ be the point relation at $0$ generated morphisms starting in $\\mathbb{R}_{<0}$ and ending in  $\\mathbb{R}'_{>0}$. Dually, let $\\mathcal P'$ be the point relation at $0$ generated morphisms starting in $\\mathbb{R}'_{<0}$ and ending in  $\\mathbb{R}_{>0}$. The collection $\\{\\mathcal P, \\mathcal P'\\}$ generates an admissible ideal.\nIn later work, we will argue that this $\\mathcal{C}$ with this ideal yields a continuous generalization of a gentle algebra. \n\n\\begin{remark}\nIf we do not assume that $\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)$ is finite dimensional in our hypothesis of \\Cref{thm:point relations are admissible}, it is possible that we do not have an admissible ideal.\nSee \\Cref{ex:big wedge}.\n\\end{remark}\n\n\\end{example}\n\\subsection{Length Relations}\\label{subsec:length}\nWe now generalize relations generated by all paths of a certain length to length relations.\nTo do this we define a way of measuring length in our category (\\Cref{def:weakly archimedean monoid,def:category with length in Lambda}) and provide examples (\\Cref{ex:weakly Archimedian monoids,ex:categories with length in Lambda}).\nThen we define the length relations (\\Cref{def:length relation}) and provide examples (\\Cref{ex:length relations}) and prove that length relations generate admissible ideals (\\Cref{thm:length relations are admissible}).\nIn \\Cref{apx:length} we discuss the proof of \\Cref{thm:length relations are admissible} (\\Cref{subsec:need weakly archimedean}), why we require the specific setup that we have (\\Cref{sec:more on length}), and compare our notion of length to the notion of a metric on a category, introduced by Lawvere \\cite{L73} (\\Cref{subsec:length vs metric}).\n\nRecall a commutative monoid $\\Lambda$ is a set with an associative, commutative, binary operation $+_{\\Lambda}:\\Lambda\\times\\Lambda \\to \\Lambda$ and an identity 0.\n\\begin{definition}\\label{def:weakly archimedean monoid}\n    Let $\\Lambda$ be a commutative monoid.\n    We say $\\Lambda$ is \\emph{weakly Archimedian} if it satisfies the following.\n    \\begin{itemize}\n        \\item There is a total order $\\leq$ on $\\Lambda$.\n        \\item If $\\lambda\\neq 0$ then $\\lambda>0$.\n        \\item If $\\lambda_1>\\lambda_2$ then, for any $\\lambda_3$, we have $\\lambda_1+\\lambda_3> \\lambda_2+\\lambda_3$ or $\\lambda_1+\\lambda_3 = \\lambda_2+\\lambda_3=\\max \\Lambda$.\n        \\item For all $0<\\lambda_1<\\lambda_2$ in $\\Lambda$, there exists $n\\in\\mathbb{N}$ such that\n        \\begin{displaymath}\n            n\\lambda_1 := \\underbrace{\\lambda_1 +_{\\Lambda} \\lambda_1 +_{\\Lambda} \\cdots +_{\\Lambda} \\lambda_1}_{n} \\geq \\lambda_2.\n        \\end{displaymath}\n    \\end{itemize}\n\\end{definition}\n\n\\begin{example}\\label{ex:weakly Archimedian monoids}\n    We give three examples, two of which the reader might expect.\n    \\begin{enumerate}\n        \\item The set $\\mathbb{N}$ with the usual total order and $+_{\\mathbb{N}}$ given in the usual way is weakly Archimedian.\n        \\item The set $\\mathbb{R}_{\\geq 0}$ with the usual total order and $+_{\\mathbb{R}}$ given in the usual way is weakly Archimedian.\n        \\item\\label{ex:weakly Archimedian monoids:with max} Let $\\Lambda = \\{0,1,2,\\ldots,n-1,n,\\infty\\}$. Let $+_{\\Lambda}$ be given by\n        \\begin{displaymath}\n            \\lambda_1+_{\\Lambda}\\lambda_2 = \\begin{cases}\n                \\lambda_1+_{\\mathbb{N}}\\lambda_2 & (\\lambda_1+_{\\mathbb{N}}\\lambda_2)\\leq n \\\\\n                \\infty & \\text{otherwise}.\n            \\end{cases}\n        \\end{displaymath}\n        For the total order, we say $0<1<2<\\cdots<n-1<n<\\infty$.\n        Then $\\Lambda$ is weakly Archimedian.\n    \\end{enumerate}\n\\end{example}\nWhen the weakly Archimedian monoid \\textcolor{karin}{structure} is clear, we write $+$ instead of $+_{\\Lambda}$.\n\n\\begin{definition}\\label{def:stem category}\n    Let $\\widehat{\\mathcal{C}}$ be a small category and ignore any additional structure.\n    The \\emph{additive $\\Bbbk$-linearization} $\\mathcal{C}$ of $\\widehat{\\mathcal{C}}$ is the category such that\n    \\begin{itemize}\n        \\item $\\mathsf{Ob}(\\mathcal{C})$ contains finite direct sums of objects in $\\widehat{\\mathcal{C}}$ and\n        \\item $\\Hom_{\\mathcal{C}}(X,Y) = \\langle \\Hom_{\\widehat{\\mathcal{C}}}(X,Y) \\rangle_\\Bbbk$ for indecomposables $X,Y\\in\\mathsf{Ob}(\\mathcal{C})$.\n    \\end{itemize}\n    The other Hom spaces and composition are given by extending bilinearly and using composition in $\\widehat{\\mathcal{C}}$.\n    \n    Let $\\mathcal{C}$ be a small $\\Bbbk$-linear category.\n    We say subcategory $\\widehat{\\mathcal{C}}$ of $\\mathcal{C}$ is a \\emph{stem category} of $\\mathcal{C}$ if $\\mathcal{C}$ is isomorphic to the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{C}}$.\n\\end{definition}\nThis is similar to the use of ``stem category'' in \\cite{Bongartz}.\nWe note that Bongartz considered stem categories of locally bounded categories.\nWe say $\\mathcal{C}$ is \\emph{locally bounded} if, for each object $X\\in\\mathsf{Ob}(\\mathcal{C})$, we have \\[\\dim_{\\Bbbk}\\left( \\bigoplus_{Y\\in\\mathsf{Ob}(\\mathcal{C})} \\Hom_{\\mathcal{C}}(X,Y)\\oplus\\Hom_{\\mathcal{C}}(Y,X) \\right) <\\infty. \\]\nImmediately, we note that when we consider the $\\Bbbk$-linearization of $\\mathbb{R}$ as a category, this condition fails.\nIn fact, many categories in the present paper are not locally bounded, especially those considered in \\Cref{sec:examples}.\n\n\\begin{remark}\\label{rmk:quiver additive linearization}\n    If we consider a quiver $Q$ as a category $\\widehat{\\mathcal{Q}}$, then the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{Q}}$ is precisely the $\\Bbbk$-linear categorification $\\mathcal{Q}$ of $Q$.\n\\end{remark}\n\n\\begin{definition}\\label{def:category with length in Lambda}\n    Let $\\mathcal{C}$ be a $\\Bbbk$-linear category, $\\widehat{\\mathcal{C}}$ be a stem category of $\\mathcal{C}$, and $\\Lambda$ a weakly archimedian monoid.\n    \n    We say \\emph{$\\mathcal{C}$ has length in $\\Lambda$} if there is a function $\\ell:\\mathsf{Mor}(\\widehat{\\mathcal{C}})\\to\\Lambda$ satisfying the following.\n    \\begin{enumerate}\n        \\item If $f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ is an isomorphism then $\\ell(f)=0$.\n        \\item For each $f,g\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ such that $g\\circ f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$, we have $\\ell(g\\circ f)=\\ell(g)+\\ell(f)$.\n        \\item For each $f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ with $\\ell(f)>0$ and for each $\\lambda<\\ell(f)$ there are $g,h\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ such that $f=h\\circ g$ and either $\\ell(g)=\\lambda$ or $\\ell(h)=\\lambda$.\n    \\end{enumerate}\n\\end{definition}\n\n\\begin{example}\\label{ex:categories with length in Lambda}\n    We give some existing examples of \\Cref{def:category with length in Lambda}.\n    \\begin{enumerate}\n        \\item\\label{ex:categories with length in Lambda:classic} Let $Q$ be a quiver, $\\mathcal{Q}$ the $\\Bbbk$-linear categorification of $Q$, and $\\widehat{\\mathcal{Q}}$ a stem category of $\\mathcal{Q}$ whose morphisms are generated by arrows.\n        Let $\\Lambda=\\mathbb{N}$ and set $\\ell(\\alpha)=1$ for each morphism in $\\widehat{\\mathcal{Q}}$ from an arrow $\\alpha$ in $Q$.\n        Then $\\mathcal{Q}$ has length in $\\mathbb{N}$.\n        \\item\\label{ex:categories with length in Lambda:AR} Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of a continuous quiver $\\widehat{\\mathcal{Q}}$ of type $A$ as in \\cite{IRT}.\n        Then $\\widehat{\\mathcal{Q}}$ is a stem category of $\\mathcal{Q}$.\n        \n        Define $\\ell: \\mathsf{Mor}(\\widehat{\\mathcal{Q}})\\to\\mathbb{R}_{\\geq 0}$ by $\\ell(g_{x,y}) = |x-y|$ where $g_{x,y}$ is the unique nonzero morphism in $\\widehat{\\mathcal{Q}}$ from $x$ to $y$.\n        Then $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n        Note $\\mathcal{Q}$ is also a category with a metric.\n        See \\Cref{subsec:length vs metric}.\n    \\end{enumerate}\n\\end{example}\n\nWe now define a length relation.\n\\begin{definition}\\label{def:length relation}\n    Let $\\Lambda$ be a weakly Archimedian monoid and $\\mathcal{C}$ a category with length in $\\Lambda$ with stem category $\\widehat{\\mathcal{C}}$.\n    Consider $\\Lambda_1,\\Lambda_2$ subsets of $\\Lambda$ such that $\\Lambda_1\\amalg\\Lambda_2=\\Lambda$, $|\\Lambda_1|\\geq 2$, and for all $\\lambda_1\\in\\Lambda_1, \\lambda_2\\in\\Lambda_2$ we have $\\lambda_1<\\lambda_2$.\n    Then the set $\\ell^{-1}(\\Lambda_2)$ in $\\widehat{\\mathcal{C}}$ generates an ideal $\\mathcal{I}$ in $\\mathcal{C}$.\n    We call $\\mathcal{I}$ a \\emph{length relation}.\n\\end{definition}\n\n\\begin{remark}\\label{rmk:length is not a number}\n    It is possible that $\\Lambda_1$ has no maximum element and $\\Lambda_2$ has no minimum element.\n    (Consider, for example, $\\Lambda=\\mathbb{Q}_{\\geq 0}$.)\n    Thus, we may not always be able to say that we are taking ``paths longer than $\\lambda$'' for some $\\lambda\\in\\Lambda$.\n\\end{remark}\n\n\\begin{example}\\label{ex:length relations}\n    We give three examples of length relations.\n    \\begin{enumerate}\n        \\item Let $Q$ be a quiver and $\\mathcal{Q}$ its $\\Bbbk$-linear categorification, which has length in $\\mathbb{N}$ (\\Cref{ex:categories with length in Lambda}(\\ref{ex:categories with length in Lambda:classic})).\n        Let $\\widehat{\\mathcal{Q}}$ be the stem category of $\\mathcal{Q}$ seen, effectively, as $Q$ embedded in $\\mathcal{Q}$.\n        Let $\\Lambda_1 = \\{0,1,2\\}$ and $\\Lambda_2=\\{3,4,5,\\cdots\\}$.\n        Then $\\mathcal{I}=\\langle \\ell^{-1}(\\Lambda_2)\\rangle $ is the set of morphisms in $\\mathcal{Q}$ generated by paths with length $\\geq 3$ in $Q$.\n        \\item Any Nakayama algebra where the relations have constant length $l$ can be realized as the $\\Bbbk$-linear categorification of its underlying quiver with length relations of length $l$ in $\\mathbb{N}$.\n        \\item\\label{ex:length relations:continuous A} Let $\\mathcal{Q}$ and $\\widehat{\\mathcal{Q}}$ be as in \\Cref{ex:categories with length in Lambda}(\\ref{ex:categories with length in Lambda:AR}).\n        Recall $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n        Let $\\Lambda_1 =[0,4]$ and $\\Lambda_2=(4,+\\infty)$.\n        Then $\\langle\\ell^{-1}(\\Lambda_2)\\rangle$ is the set of morphisms in $\\mathcal{Q}$ of length \\emph{strictly greater than $4$}.\n    \\end{enumerate}\n\\end{example}\n\n\\begin{theorem}\\label{thm:length relations are admissible}\n    Let $\\Lambda$ be a weakly Archimedian monoid, $\\mathcal{C}$ a category with length in $\\Lambda$ with stem category $\\widehat{\\mathcal{C}}$, and $\\mathcal{I}$ a length relation.\n    If $\\End_{\\widehat{\\mathcal{C}}}(X)$ is a finitely-generated monoid, for each $X\\in\\mathsf{Ob}(\\widehat{\\mathcal{C}})$, then $\\mathcal{I}$ is an admissible ideal.\n\\end{theorem}\n\\begin{proof}\n    If $\\mathcal{I}=\\emptyset$ then condition (1) is vacously satisfied.\n    Assume $\\mathcal{I}\\neq\\emptyset$, let $f\\in \\mathcal{I}$ such that $f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$, and let $\\lambda\\in\\Lambda_1$ such that $\\lambda>0$.\n    Then there is $n\\in\\mathbb{N}$ such that $n\\lambda \\geq \\ell(f)$.\n    Thus, there is some decomposition $f=g_n\\circ\\cdots\\circ g_1$ where $\\ell(g_i)\\in\\Lambda_1$ for each $g_i$.\n    Thus, each $g_i$ is not in $\\mathcal{I}$.\n    \n    Since $\\End_{\\widehat{\\mathcal{C}}}(X)$ is a finitely-generated monoid, let $m$ be the number of generators and let $\\{f_i\\}_{i=1}^m$ be the set of generators.\n    Let\n    \\[ N = \\max_i \\{ \\min_n \\{ n\\ell(f_i) \\mid n\\ell(f_i)\\in\\Lambda_2\\}\\}.\\]\n    Then\n    \\[ \\dim_{\\Bbbk}(\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)) \\leq m\\cdot N + 1,\\]\n    where we need the ``$+1$'' to account for the identity in $\\End_{\\widehat{\\mathcal{C}}}(X)$.\n    Therefore, $\\mathcal{I}$ is an admissible ideal.\n\\end{proof}\n\n\n\n\\section{Examples}\\label{sec:examples}\n\\begin{figure}[b]\n    \\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \n\\begin{tikzpicture}[xscale=2]\n\\node (1) at (0,0) {1};\n\\node (2) at (1,1) {2};\n\\node (3) at (1,0) {3};\n\\node (4) at (1,-1) {4};\n\\node (5) at (2,0) {5};\n\n\\draw[->] (1) -- node[pos=0.6, above]{$\\alpha_1$} (2);\n\\draw[->] (1) -- node[pos=0.7, above]{$\\beta_1$} (3);\n\\draw[->] (1) -- node[pos=0.6, above]{$\\gamma_1$} (4);\n\\draw[->] (2) -- node[pos=0.4, above]{$\\alpha_2$} (5);\n\\draw[->] (3) -- node[pos=0.3, above]{$\\beta_2$} (5);\n\\draw[->] (4) -- node[pos=0.4, above]{$\\gamma_2$} (5);\n\\end{tikzpicture}\n        \\caption{}\n        \\label{fig:three-path-discrete}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[xscale=2, inner sep = 0cm, outer sep = 0cm, very thick,decoration={\n    markings,\n    mark=at position 0.5 with {\\arrow{>}}}\n    ]\n\\node (start) at (0,0) {$\\bullet$};\n\\node (end) at (2,0) {$\\bullet$};\n\\node at (-.1,0){$0$};\n\\node at (2.1,0){$1$};\n\n\\draw[postaction={decorate}] (start) .. controls (.5,1.3) and (1.5,1.3).. node[above=2pt, pos=0.6]{$\\alpha$} (end);\n\\draw[postaction={decorate}] (start) -- node[pos=0.6, above=2pt]{$\\beta$} (end);\n\\draw[postaction={decorate}] (start) .. controls (.5,-1.3) and (1.5,-1.3).. node[pos=0.6, above=2pt]{$\\gamma$} (end);\n\\end{tikzpicture}\n        \\caption{}\n        \\label{fig:three-path-continuous}\n    \\end{subfigure}\n    \\caption{The quivers considered in \\Cref{ex: discrete 3 paths,ex: cts 3 paths}, respectively.}\n    \\label{fig:three-path-quivers}\n\\end{figure}\n\n\\begin{example}\\label{ex: discrete 3 paths}\nConsider the (discrete) quiver $Q$ shown in \\Cref{fig:three-path-discrete}.\nThe relation $\\alpha_2\\alpha_1-2\\beta_2\\beta_1+3\\gamma_2\\gamma_1$ generates an admissible ideal in the classical sense. In the $\\Bbbk$-linear categorification of $Q$, we rewrite the relation as the composition of morphisms\n\\[1\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_1 \\\\ -2\\beta_1\\\\3\\gamma_1\\end{smallmatrix}\\right]} 2\\oplus 3\\oplus 4\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_2 & \\beta_2 &\\gamma_2\\end{smallmatrix}\\right]}  5,\n    \\]\n    which generates an admissible ideal\n    \n\\end{example}\n\n\n\\begin{example} \\label{ex: cts 3 paths}\n\nConsider the continuous analogue of \\Cref{ex: discrete 3 paths}, displayed in \\Cref{fig:three-path-continuous}.\nWe consider a similar relation $\\alpha-2\\beta+3\\gamma$.\nLet $X$, $Y$, and $Z$ be points on the interior of the paths $\\alpha$, $\\beta$, and $\\gamma$, respectively.\nThen $\\alpha=\\alpha_2\\alpha_1$ where $\\alpha_1:0\\to X$ and $\\alpha_2:X\\to 1$.\nWe similarly write $\\beta=\\beta_2\\beta_1$ and $\\gamma=\\gamma_2\\gamma_1$.\nThen the relation $\\alpha-2\\beta+3\\gamma$ can be written as the composition\n\\[0\n\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_1 \\\\ -2\\beta_1\\\\3\\gamma_1\\end{smallmatrix}\\right]}\nX\\oplus Y\\oplus Z\n\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_2 & \\beta_2 &\\gamma_2\\end{smallmatrix}\\right]}\n1,\n    \\]\nand it generates an admissible ideal.\n\\end{example}\n\n\\begin{example}[Real line with point relations on integers]\\label{ex:real line integers}\nLet $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\mathbb{R}$ as a category where paths move upwards.\nFor a point $c\\in \\mathbb{R}$, let the (unique) point relation at $c$ be $\\mathcal P_c$. The collection of point relations on the integers, $\\{\\mathcal P_z\\}_{z\\in \\mathbb{Z}}$ generates an admissible ideal by \\Cref{thm:point relations are admissible}.\n\nThe ``Auslander--Reiten space'' of the representations of this quiver is shaped like a mountain range; it is a set of triangles joined at their bottom vertices, see \\Cref{fig:AR mountain range}\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n        \\draw[dotted, thick] (-1.5,0) -- (0,0)--(-1.5,1);\n        \\draw[dotted,thick] (10.5,1) -- (9,0) -- (10.5,0);\n        \\draw[] (0,0) -- (1.5,1)--(3,0)--(4.5,1)--(6,0)--(7.5,1)--(9,0)--(0,0);\n    \\end{tikzpicture}\n    \\caption{The ``Auslander--Reiten space'' of representations of the quiver from \\Cref{ex:real line integers}}\n    \\label{fig:AR mountain range}\n    \\end{subfigure}\n    \n    \\begin{subfigure}{\\textwidth}\n    \\begin{tikzpicture}\n    \\draw[color=white] (0,0)--(0,1.3);\n        \\draw[dotted, thick] (-1.5,0) -- (0,0)--(-1,1)--(-1.5,1);\n        \\draw[dotted,thick] (10.5,1) -- (10,1) -- (9,0) -- (10.5,0);\n        \\draw[] (0,0) -- (1,1)--(2,1)--(3,0)--(4,1)--(5,1)--(6,0)--(7,1)--(8,1)--(9,0)--(0,0);\n    \\end{tikzpicture}\n    \n    \\caption{The ``Auslander--Reiten space'' for representations of $\\mathbb{R}$ modulo paths longer than $s$ and modulo paths that go through any $x\\in r\\mathbb{Z}$, where $r>s\\in\\mathbb{R}_{>0}$.}\\label{fig:AR chopped mountains}\n    \\end{subfigure}\n    \\caption{The ``Auslander--Reiten spaces'' of representations of the quivers in \\Cref{ex:real line integers,ex:real line points}}\n    \\label{fig:AR spaces}\n\\end{figure}\n\\end{example}\n\n\\begin{example}[Circle with length\/Kupisch relations]\\label{ex:circle length}\n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of a continuous quiver $\\widehat{\\mathcal{Q}}$ of type $\\widetilde{A}$ as in \\cite{HR}.\n    Define $\\ell: \\mathsf{Mor}(\\widehat{\\mathcal{Q}})\\to\\mathbb{R}_{\\geq 0}$ by $\\ell(f)=\\phi-\\theta + 2n\\pi$ where $f:e^{i\\theta}\\to e^{i\\phi}$, and $0\\leq \\phi-\\theta<2\\pi$, and $n$ is the number of full rotations around the circle at $e^{i\\theta}$ before moving to $e^{i\\phi}$.\n    Then $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n    If $\\mathcal{Q}$ is acyclic, we may replace $\\mathbb{R}_{\\geq 0}$ with $\\Lambda=[0,2\\pi)\\cup\\{\\infty\\}$ and define $+_\\Lambda$ similarly to \\Cref{ex:weakly Archimedian monoids}(\\ref{ex:weakly Archimedian monoids:with max}).\n    \n    \\begin{figure}\n        \\centering\n        \\begin{tikzpicture}[very thick, decoration={markings,mark=at position 0.5 with {\\arrow{>}}}]\n            \\draw[radius=3cm] (0,0) circle;\n            \\draw[radius=2.8cm, color=red,postaction=decorate] (180:2.8cm) arc[start angle =180, delta angle= 180] node[above left, pos=0.666, red]{\\XSolidBrush};\n            \\draw[radius=3.2cm, color=blue, postaction=decorate] (0:3.2cm) arc[start angle = 0, delta angle= 90] node[above right, pos=0.666, blue]{\\CheckmarkBold};\n            \\foreach \\x in {0, 30,...,360}\n            \t\\draw[thin] (\\x:2.9cm) -- (\\x:3.1cm);  \n        \\end{tikzpicture}\n        \\caption{The circle with length relations as described in \\Cref{ex:circle length}. In this figure, the relations have length $\\frac{2\\pi}{3}$.}\n        \n        \\label{fig:circle length}\n    \\end{figure}\n    \n    Now assume $\\widehat{\\mathcal{Q}}$ has cyclic orientation.\n    Let $\\kappa$ be a Kupisch function as in \\cite[Definition~3.9]{RZ}.\n    That is, $\\kappa:\\mathbb{R}\\to\\mathbb{R}_{> 0}$ is a function such that $\\kappa(t)+t>t$ and $\\kappa(t+1)=\\kappa(t)$, for all $t\\in\\mathbb{R}$.\n    This yields a map $\\mathbb{S}^1\\to \\mathbb{R}_{>0}$ where $\\mathbb{S}^1=[0,1]\/\\{0\\sim 1\\}$.\n    If $\\kappa$ is constant with value $a$, then this yields a length relation where $\\Lambda_1=[0,a]$ and $\\Lambda_2=(a,+\\infty)$.\n    If $\\kappa$ is not constant, then we do not have a length relation.\n    However, if $\\kappa$ does not have any separation points \\cite[Definition~4.2]{RZ}, then $\\kappa$ still induces an admissible ideal.\n\\end{example}\n\n\\begin{example}[Real line with length and point relations]\\label{ex:real line points}\n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\mathbb{R}$ as a category where paths move upwards.\n    Let $r,s$ be positive real numbers and for each $x\\in r\\mathbb{Z}\\subset \\mathbb{R}$, let $\\mathcal P_x$ be the (unique) point relation in $\\mathcal{Q}$ through $x$ and $\\mathcal{I}$ the admissible ideal generated by $\\bigcup_{x\\in r\\mathbb{Z}} \\mathcal{P}_x$.\n    Let $\\mathcal{J}$ be the the length relation in $\\mathcal{Q}\/\\mathcal{I}$ obtained by modding out by paths of length greater than $s$.\n    By \\Cref{thm:length relations are admissible,thm:point relations are admissible} with \\Cref{lem:stack admissible} we obtain an admissible ideal $\\widetilde{\\Jay}$ given by the point relations at each $x\\in r\\mathbb{Z}$ and paths of length greater than $s$.\n    \n    If $r\\leq s$ then $\\mathcal{C}\/\\mathcal{I} = \\mathcal{C}\/\\widetilde{\\Jay}$ since we cannot have a morphism of length greater than $r$ in $\\mathcal{C}\/\\mathcal{I}$ anyway.\n    If $r>s$ then we obtain paths of of length less than or equal to $s$ that do not pass through any $x\\in r\\mathbb{Z}$.\n    The ``Auslander--Reiten space'' for the case $r>s$ is in \\Cref{fig:AR chopped mountains}; notice the similarity with \\Cref{fig:AR mountain range}.\n\\end{example}\n\n\\begin{example}[Complications with Cycles]\\label{ex: cycles length}\n    For each $n\\in\\mathbb{N}$, let $\\mathcal{C}_n$ be circle whose radius is $\\frac{1}{2}e^{-n}$.\n    Let $\\mathcal{C}$ be the additive $\\Bbbk$-linearization of $\\mathbb{R}_{\\leq 0}\\amalg(\\coprod_{n\\in\\mathbb{N}} \\mathcal{C}_n)\/ \\sim$, where $\\mathbb{R}\\ni -n\\sim 0\\in\\mathcal{C}_n$.\n    See \\Cref{fig:cycles with decreasing length} for a visual depiction.\n    \n    We see that $\\mathcal{C}$ has length in $\\mathbb{R}_{\\geq 0}$.\n    Let $\\mathcal{I}$ be a length relation.\n    Since our length is in $\\mathbb{R}_{\\geq 0}$ we can say we are modding out by length $>L$ or $\\geq L$ for some $L>0\\in\\mathbb{R}$.\n    \n    Notice that for each $N\\in\\mathbb{N}$, there exists some $\\mathcal{C}_n$ with radius $r$ such that $Nr < L$.\n    Therefore, there is no natural number $n$ such that for all nonisomorphism endomorphisms $f$ we have $f^n\\in\\mathcal{I}$.\n    \n    \\begin{figure}\n        \\centering\n        \\begin{tikzpicture}[scale=2, decoration={markings, mark=at position 0.6 with {\\arrow{>}}}]\n    \\foreach \\x in {0, -1,...,-3}{\n    \t\\node[outer sep=4+\\x] (\\x) at (\\x,0){$\\bullet$};\n    \t\\node at (\\x.south){\\x};}\n    \\node (-4) at (-4,0){$\\cdots$};\n    \\foreach \\x [remember=\\x as \\y (initially 0)]in {-1, -2,...,-4}\n    \t\\draw[very thick, postaction=decorate] (\\x.center)--(\\y.center);\n    \\foreach \\x [evaluate=\\x as \\r using .5*e^(\\x\/2)] in {0, -1,...,-3}\n         \\draw[radius=-\\r, very thick, postaction=decorate](\\x)+(0,\\r ) circle;\n    \\end{tikzpicture}\n        \\caption{Illustration of $\\mathcal{C}$ in \\Cref{ex: cycles length}, where we have glued smaller and smaller circles to each non-positive integer $n$ in $\\mathbb{R}_{\\leq 0}$.}\n        \\label{fig:cycles with decreasing length}\n    \\end{figure}\n\\end{example}\n\n\\begin{example}[Big wedge]\\label{ex:big wedge}\n    Let $\\mathcal{C}$ be a cyclic continuous quiver of type $\\widetilde{A}$ as in \\cite{HR}.\n    Let $\\widehat{\\mathcal{Q}} = (\\coprod_{\\mathbb{N}} \\mathcal{C} )\/\\sim$ where we join all the copies of the $\\mathcal{C}$ together at one point.\n    Denote the wedge point by $X$.\n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{Q}}$.\n    Let us discuss the contruction of an admissible ideal out of point relations and a length relation.\n\n    Notice $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n    However, since the endomorphism ring of $X$ is an infinitely-generated monoid in $\\widehat{\\mathcal{Q}}$, we see that $\\End_{\\mathcal{C}}(X)\/ \\mathcal{I}(X,X)$ is not finite-dimensional.\n    Instead, we must add a point relation on all but finitely-many different copies of $\\mathcal{C}$.\n    If we do not have a point relation on all the cycles, we also need a length relation.\n    Without such a combination, it is not possible to build an admissible ideal out of point relations and a length relation.\n\\end{example}\n\n\\subsection{The Real Plane}\\label{subsec:real plane}\n    We now consider a continuous version of the grid quiver with commutativity relations as examined in \\cite{BBOS}.\n    Let $\\widehat{\\mathcal{Q}}$ be the category whose objects are points in $\\mathbb{R}^2$.\n    \n    We now define the $\\Hom$ set between an arbitrary pair of points $(x,y)$ and $(z,w)$.\n    Hom sets are given by considering paths made of up of horizontal and vertical line segments.\n    For a pair $(x,y)$ and $(z,w)$, consider the set $P_{x,y}^{z,w}$ of all finite sequences $\\{(x_i,y_i)\\}_{i=i}^n$ such that\n    \\begin{itemize}\n        \\item $(x_1,y_1)=(x,y)$ and $(x_n,y_n)=(z,w)$,\n        \\item $x_1\\leq x_2\\leq\\cdots \\leq x_n$ and $y_1\\leq y_2\\leq\\cdots \\leq y_n$,\n        \\item $(x_i,y_i)\\neq (x_{i+1},y_{i+1})$ for all $1\\leq i<n$,\n        \\item $x_1=x_2$ or $y_1=y_2$,\n        \\item for all $1\\leq i<n-1$, if $x_i=x_{i+1}$ then $y_{i+1}=y_{i+2}$, and\n        \\item for all $1\\leq i<n-1$, if $y_i=y_{i+1}$ then $x_{i+1}=x_{i+2}$\n    \\end{itemize}\n    We define \\[\\Hom_{\\widehat{\\mathcal{Q}}}((x,y),(z,w))=P_{x,y}^{z,w}.\\]\n    \n    Note that $P_{x,y}^{z,w}$ may be empty.\n    If either $x>z$ or $y>w$, then\n    \\[ \\Hom_{\\widehat{\\mathcal{Q}}}((x,y), (z,w)) = \\emptyset.\\]\n    If (i) $x=z$ and $y< w$ or (ii) $x< z$ and $y=w$, then\n    \\[ \\Hom_{\\widehat{\\mathcal{Q}}}((x,y), (z,w)) = \\left\\{ \\{ (x,y), (z,w) \\}\\right\\}. \\]\n    If $(x,y)=(z,w)$ then\n    \\[ \\Hom_{\\widehat{\\mathcal{Q}}}((x,y), (x,y)) = \\left\\{ \\{ (x,y) \\}\\right\\}. \\]\n    Composition is given by concatenating sequences and, if necessary, deleting a repeated term.\n    \n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{Q}}$ and let $(x,y),(z,w)\\in \\mathbb{R}^2$ such that $x<z$ and $y<w$.\n    Then, we define\n    \\[\n    \\mathcal{I}((x,y),(z,w)) = \\left\\langle \\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n \\mid \\{(x_i,y_i)\\}_{i=1}^m \\neq \\{(x'_j,y'_j)\\}_{j=1}^n \\right\\rangle.\n    \\]\n    If $x=z$ or $y=w$, then $\\mathcal{I}((x,y),(z,w))=0$.\n    \n    Consider\n    \\begin{align*}\n        \\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n &\\in \\mathcal{I}((x,y),(z,w)) \\\\\n        \\{(z_k,y_k)\\}_{k=1}^p - \\{(z'_l,z'_l)\\}_{l=1}^q &\\in \\mathcal{I}((z,w),(u,v)).\n    \\end{align*}\n    We show the composition is in $\\mathcal{I}((x,y),(u,v))$.\n    \\begin{align*}\n       & (\\{(z_k,y_k)\\}_{k=1}^p - \\{(z'_l,z'_l)\\}_{l=1}^q)\\circ (\\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n) \\\\\n       =& \\{(z_k,y_k)\\}\\circ \\{(x_i,y_i)\\} - \\{(z_k,y_k)\\}\\circ \\{(x'_j,y'_j)\\} \\\\& - \\{(z'_l,z'_l)\\}\\circ \\{(x_i,y_i)\\} + \\{(z'_l,z'_l)\\}\\circ \\{(x'_j,y'_j)\\}.\n    \\end{align*}\n    We know \n    \\begin{align*}\n        \\{(z_k,y_k)\\}\\circ \\{(x_i,y_i)\\} & \\neq \\{(z_k,y_k)\\}\\circ \\{(x'_j,y'_j)\\} \\\\\n        &\\text{and} \\\\\n        \\{(z'_l,z'_l)\\}\\circ \\{(x_i,y_i)\\} & \\neq \\{(z'_l,z'_l)\\}\\circ \\{(x'_j,y'_j)\\}.\n    \\end{align*}\n    Thus, the composition is in $\\mathcal{I}((x,y),(u,v))$.\n    \n    Consider $\\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n\\in\\mathcal{I}((x,y),(z,w))$.\n    By assumption there is $1\\leq k \\leq m$ and $1\\leq l \\leq n$ such that $(x_k,y_k)\\neq (x'_l,y'_l)$.\n    Let\n    \\begin{align*}\n        f:(x,y)\\to (x_k,y_k)\\oplus(x'_l,y'_l) &= \\left[\\begin{matrix} \\{(x_i,y_i)\\}_{i=1}^k \\\\ - \\{(x'_j,y'_j)\\}_{j=1}^l \\end{matrix} \\right] \\\\\n        g:(x_k,y_k)\\oplus(x'_l,y'_l) \\to (z,w) &= \\left[\\begin{matrix} \\{(x_i,y_i)\\}_{i=k}^m & \\{(x'_j,y'_j)\\}_{j=l}^n \\end{matrix} \\right].\n    \\end{align*}\n    Notice $f\\notin \\mathcal{I}((x,y),(x_k,y_k)\\oplus(x'_l,y'_l))$, $g\\notin \\mathcal{I}((x_k,y_k)\\oplus(x'_l,y'_l), (z,w))$, and\n    \\[ \\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n = g\\circ f.\\]\n    Thus, every morphism in $\\mathcal{I}$ is given by a finite composition of morphisms not in $\\mathcal{I}$.\n    Furthermore, there are no cycles in $\\mathcal{Q}$.\n    Thus, $\\mathcal{I}$ is an admissible ideal.\n    \n    The resulting $\\mathcal{Q}\/\\mathcal{I}$ is the category where the objects are fnite direct sums of points in $\\mathbb{R}^2$ and Hom spaces between points is given by\n    \\[\n        \\Hom_{\\mathcal{Q}\/\\mathcal{I}}((x,y),(z,w)) = \\begin{cases}\n            \\Bbbk & x\\leq z \\text{ and }y\\leq w \\\\\n            0 & \\text{otherwise}.\n        \\end{cases}\n    \\]\n    This means that $\\mathcal{Q}\/\\mathcal{I}$ is the continuous generalization of (finite) discrete commutative grid quivers as examined in \\cite{BBOS}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \n\nTwo structures $R$ and $S$ are called {\\em equimorphic}, denoted by $R\\approx S$, when each embeds in the other; we may also say that one is a {\\em sibling} of the other. Equimorphic finite structures are necessarily isomorphic, but this is no longer the case for infinite structures. For instance, the rational numbers, considered as a linear order, has continuum many siblings, up to isomorphism. The number of isomorphism classes of siblings of a structure $S$ is called the {\\em sibling number} of $S$ and we denote it by $Sib(S)$. \nA {\\em relation} $R$ is a pair $(V,E)$ where $V$ is a non-empty set, called the {\\em domain} of $R$, and $E\\subseteq V^n$ for some positive integer $n$; the integer $n$ is called the arity of $E$ and $E$ is called an $n$-ary relational symbol. A {\\em binary relation} is a relation $(V,E)$ where $E$ is of arity 2. The cardinality of a relation is the cardinality of its domain. \nWhen studying infinite relations, Thomass\\'e \\cite{TH2} made the following conjecture:\n\n\\begin{conjecture} [Thomass\\'e's Conjecture, \\cite{TH2}]\nLet R be a countable relation. Then $Sib(R)=1$, $\\aleph_0$ or $2^{\\aleph_0}$. \n\\end{conjecture}\n\nThere is also an alternate version of the conjecture as follows:\n\n\\begin{conjecture}[The Alternate Thomass\\'e Conjecture]\nLet R be a relation of any cardinality. Then $Sib(R)=1$ or $\\infty$.  \n\\end{conjecture}\n\nThe alternate Thomass\\'{e} conjecture is connected to a conjecture of Bonato-Tardif \\cite{BT} ({\\em The Tree Alternative Conjecture}) stating that the sibling number of a tree of any cardinality is one or infinite in the category of trees. The connection between these two conjectures is through this fact that each sibling of a tree $T$ is a tree if and only if $T\\oplus 1$, the graph obtained by adding an isolated vertex to $T$, does not embed in $T$. Therefore, for a tree $T$ not equimorphic to $T\\oplus 1$, the tree alternative conjecture and the alternate Thomass\\'e conjecture are equivalent. The tree alternative conjecture has been verified for rayless trees by Bonato and Tardif \\cite{BT}, for rooted trees by Tyomkyn \\cite{TY} and for scattered trees by Laflamme, Pouzet and Sauer \\cite{LPS}. Besides, the alternate Thomass\\'e conjecture was proved for rayless graphs by Bonato, Bruhn, Diestel and Spr\\\"ussel \\cite{BBDS}, for countable $\\aleph_0$-categorical structures by Laflamme, Pouzet, Sauer and Woodrow \\cite{LPSW}, for countable cographs by Hahn, Pouzet and Woodrow \\cite{HPW} and for countable universal theories by Braunfeld and Laskowski \\cite{BL}. Furthermore, Thomass\\'e's conjecture for countable chains and the alternate Thomass\\'e conjecture for all chains have been proved by Laflamme, Pouzet and Woodrow \\cite{LPW}. Tateno \\cite{TAT} claimed a counterexample to the tree alternative conjecture in his thesis.\nAbdi, Laflamme, Tateno and Woodrow \\cite{ALTW} revisited and verified Tateno's claim and provided locally finite trees having an arbitrary finite number of siblings. Moreover, using an adaptation, Abdi, Laflamme, Tateno and Woodrow \\cite{ALTW} provided countable partial orders with a similar conclusion which disproves Thomass\\'e's conjecture. It turns out that both conjectures of Bonato-Tardif and Thomass\\'e  are false. \nThis is a major development in the program of understanding siblings of a given mathematical structure. We now know that the simple dichotomy one vs infinite for the number of siblings is no longer valid, and the situation is much more intricate. While counting the number of siblings provides a good first insight into the siblings of a structure, in particular understanding which structures exactly do satisfy the dichotomy, the equimorphy programme is now ready to move on and focus on the actual structure of those siblings. In this paper, we show that a countable $NE$-free poset fits in the programme.  \n \n\\section{Overview}\n\nIn order-theoretic mathematics, the class of series-parallel posets is the smallest class of finite posets obtained from the one element poset by iteration of two operations: direct sum and linear sum. This class of posets was introduced by Lawler and studied by Valdes and Tarjan.  \nGrillet and Heuchenne introduced $NE$-free posets and later Habib and Jegou \\cite{HJ} showed that $NE$-free posets are generalisations of series-parallel posets by giving a recursive construction of $NE$-free posets.  \nFinite and some infinite $NE$-free posets can be obtained by direct sum and linear sum. However, there are infinite $NE$-free posets which need another operation, namely sum over a labelled chain in which the chain has no least element. In order to count the siblings of countable $NE$-free posets, we first determine their structure. For this purpose, we use modular decomposition of $NE$-free posets in Section \\ref{DT}. Section \\ref{StructureNE} determines the structure of $NE$-free posets by classifying them. \nThen Section \\ref{SibNE} will provide a proof of the main result of this article: a countable $NE$-free poset has one or infinitely many siblings. In Section \\ref{Finiteposetsub}, we will show that a countable $NE$-free poset $P$ whose comparability graph has one sibling is a finite poset substitution of chains or antichains and that Thomass\\'{e}'s conjecture holds for $P$. This is a generalisation of a result of \\cite{LPW} for countable chains whose comparability graph is a clique. More, the result of Section \\ref{Finiteposetsub} leaves Thomass\\'e's conjecture open for a countable $NE$-free poset whose comparability graph has infinitely many siblings. \n\n\n\n\\section{Partial Orders and $NE$-Free Posets} \n\n\nA \\textit{poset (partially ordered set)} is a binary relation $P=(V,\\leq)$ where $\\leq$ is reflexive, antisymmetric and transitive. If a poset $P$ is given, by $\\leq_P$, we mean the order of $P$. Let $P=(V,\\leq)$ be a poset. By an element $x$ of $P$ we mean $x\\in V$. By $\\leq^*$ we mean the partial order obtained by reversing the order $\\leq$, that is $x\\leq^* y$ if and only if $y\\leq x$ where $x, y\\in P$. We denote the poset $(V,\\leq^*)$ by $P^*$. Any substructure, resp induced substructure, of $P$ is called a \\textit{subposet}, resp an {\\em induced subposet}, of $P$. For two subposets $P_1$ and $P_2$ of $P$ we write $P_1 \\leq P_2$ if for every $x\\in P_1$ and every $y\\in P_2$ we have $x \\leq y$. Note that $P_1$ or $P_2$ might be empty. The empty set is both greater than and less than any set. Therefore, $\\leq$ on set is not a partial order.  Two elements $x, y \\in P$ are called \\textit{comparable} if $x \\leq y$ or $y \\leq x$, otherwise they are \\textit{incomparable}, denoted by $x \\perp y$. A {\\em chain}, resp an {\\em antichain}, is a poset whose elements are pairwise comparable, resp incomparable. Two  subposets $P_1, P_2$ of $P$ are {\\em comparable} if $P_1 \\leq P_2$ or $P_2 \\leq P_1$, and {\\em incomparable}, denoted by $P_1\\perp P_2$, if $x\\perp y$ for every $x\\in P_1$ and every $y\\in P_2$. Two elements $x, y \\in P$ are \\textit{compatible} if the pair $\\{x, y\\}$ has a lower bound, that is, there is $z\\in P$ such that $z\\leq x$ and $z\\leq y$.  The graph $CG(P)=(V,E)$ such that for two distinct $x, y \\in P$, $xy \\in E$ if $x < y$ or $y < x$, is called the \\textit{comparability graph} of $P$. Note that a partial order is reflexive meaning that each element of a poset is comparable to itself, however, in order to get simple graphs, our definition of a comparability graph prevents having loops in the graphs. A \\textit{connected}, resp \\textit{disconnected}, poset is a poset whose comparability graph is connected, resp disconnected. By a \\textit{component} of a poset $P=(V,\\leq)$, we mean an induced subposet $Q=(W,\\leq)$ of $P$ where $W$ is the vertex set of a component of $CG(P)$. We say two elements $x, y$ of $P$ are connected if they belong to the same component of $P$; equivalently, they are connected in $CG(P)$ by a path and we say $x, y$ are disconnected when they belong to different components of $P$. \n\nLet us denote by an `$N$' the following poset on four elements $\\{a, b, c, d\\}$ such that $a < b$, $c < b$, $c < d$, $a\\perp c$, $b \\perp d$ and $a \\perp d$. An $NE$-\\textit{free poset} is a poset which does not embed an $N$. For instance, a chain is an $NE$-free poset. \n\n\n\n\n\n\\subsection{Graph Substitution and Poset Substitution} \\label{GPSubstitution}\n\nLet $G$ be a graph. By $V(G)$, resp $E(G)$, we mean the set of vertices, resp edges, of $G$. For simplicity, by $x\\in G$ we mean that $x$ is a vertex of $G$. The following definition is from \\cite{CD}. \n\n\\begin{definition} [Graph Substitution] \\label{Graphsub}\nLet $K$ be a graph and let $(H_v)_{v\\in K}$ be a pairwise disjoint family of graphs. We denote by $G:=K[H_v\/v : v\\in K]$ the graph resulting from the substitution of $H_v$ for $v\\in K$ as follows: \n\\begin{enumerate}\n    \\item $V(G)=\\bigcup_{v\\in K} V(H_v)$;\n    \\item $xy \\in E(G)$ if and only if either\n    \\begin{enumerate}\n        \\item $xy\\in E(H_v)$ for some $v\\in K$; or\n        \\item for two distinct $u, v \\in K$ we have $x \\in H_u$, $y \\in H_v$ and $uv\\in E(K)$. \n    \\end{enumerate}\n\\end{enumerate} \nThen we say $G$ is a {\\em graph substitution} of the $H_v$ for $K$. We call $K$ the {\\em context graph} and the $H_v$ the {\\em graph blocks} of $G$. If $K$ is a clique (complete graph), resp an independent set, then $G$ is called a {\\em complete}, resp {\\em direct}, {\\em sum} of graphs.\n\\end{definition} \n\nA poset substitution applies a {\\em composition operation} which is the substitution of a poset $P$ for an element $x$ in a poset $Q$. In the resulting poset, denoted by $Q[P\/x]$, the element $x$ is replaced by $P$, and all elements in $Q$ comparable to $x$ are comparable to all elements of $P$. \nA \\textit{chain} ({\\em linearly ordered set}), resp an \\textit{antichain}, is a poset whose elements are pairwise comparable, resp incomparable. Any substructure of a chain is called a {\\em subchain}. \n\n\n\\begin{definition} [Poset Substitution] \\label{PosetSub} \nLet $Q$ be a poset and $\\{P_v\\}_{v\\in Q}$ be a family of pairwise disjoint posets. For each $v\\in Q$ let $\\leq_v$ be the order of $P_v$. The {\\em poset substitution} of the $P_v$ for $Q$ is a poset, denoted by $P:=Q[P_v\/v : v \\in Q]$, defined as follows:  \n\\begin{enumerate}\n    \\item $P=\\bigcup_{v \\in Q} P_v$\n    \\item for $x, y \\in P$, $x \\leq_P y$ if and only if \n     \\begin{enumerate} \n        \\item $x, y \\in P_v$ for some $v \\in Q$ and $x \\leq_v y$; or\n        \\item for two distinct $u, v\\in Q$, we have  $x \\in P_u$, $y \\in P_v$ and $u \\leq_Q v$. \n    \\end{enumerate} \n\\end{enumerate}\nThe poset $Q$ is called the {\\em context poset} of $P$ and each $P_v$ is called a {\\em poset block} of $P$.   If $Q$ is a chain, resp an antichain, $P$ is called a {\\em linear}, resp {\\em direct, sum} of posets, denoted by $P=+_{v\\in Q} P_v$, resp $P=\\bigoplus_{v\\in Q} P_v$ and each $P_v$ is called a {\\em summand}, resp {\\em component}, of the linear, resp direct, sum.\n\\end{definition}\n\n\n\n\n\n\\section{Modular Decomposition} \\label{DT}\n\n\nCourcelle and Delhomm\\'{e} \\cite{CD} investigated the modular decomposition of infinite (mainly countable) graphs. The notion of modular decomposition is essential for establishing structural properties of graphs and related objects, in particular of partial orders and their comparability graphs. This fact is a special case of a general result about modular decomposition of binary relations. In particular,\neach $NE$-free poset can be represented by a special type of ordered and labelled tree which is called \\textit{decomposition tree}.  Due to the importance of such trees,  in this section, we give a detailed presentation of them. The materials  of this section are taken from \\cite{CD} and \\cite{HPW}. Indeed, we construct the decomposition tree of an $NE$-free poset in a way similar to the method employed by \\cite{HPW} for cographs. However, one can find more details about the decomposition tree of a countable graph in \\cite{CD}.\n\n\n\n\n\\subsection{Modules} \\label{Modules}\n\n\nLet $S$ be a set. A \\textit{binary structure over} $S$ is a pair $\\mathbb{B}:=(V,d)$ where $V$, the {\\em domain} of $\\mathbb{B}$, is a non-empty set and $d$ is a map from $V\\times V$ into $S$. A subset $M$ of $V$ is a \\textit{module} of $\\mathbb{B}$ if $d(x,y)=d(x,y')$ and $d(y,x)=d(y',x)$ for every $x \\in V\\setminus M$ and every $y, y' \\in M$. Modules are also called {\\em intervals}. The sets $\\emptyset$, $V$ and the singletons are called \\textit{trivial} modules. A binary structure is called {\\em indecomposable} if all its modules are trivial. If moreover it has more than two vertices it is called {\\em prime}. A non-empty module is called \\textit{strong} if it is comparable w.r.t inclusion to every module which it meets.\n\n\n\n\\begin{lemma} [\\cite{HPW}] \\label{Intersectionunionstrong}\nThe intersection of any set of strong modules is either empty or a strong module.\n\\end{lemma}\n\n\n\nLet $A$ be a non-empty subset of $V$. Let $S_{\\mathbb{B}}(A)$ be the intersection of strong modules of $\\mathbb{B}$  containing $A$. We write $S_{\\mathbb{B}}(x,y)$ instead of $S_{\\mathbb{B}}(\\{x,y\\})$. According to  Lemma \\ref{Intersectionunionstrong}, $S_{\\mathbb{B}}(A)$ is a strong module. A module is called \\textit{robust} if it is either a singleton or the smallest strong module containing two distinct elements. Hence, if $A$ is a non-trivial robust module, then there exist $x\\neq y \\in A$ such that $A$ is strong and every strong module containing $x$ and $y$ contains $A$, that is, $A=S_{\\mathbb{B}}(x,y)$.  A binary structure $\\mathbb{B}$ is called {\\em robust} if its domain is a robust module of $\\mathbb{B}$.  \n\nLet $A$ be a strong module and $x, y \\in A$. We set $x \\equiv_A y$ if either $x=y$ or there is a strong module containing $x$ and $y$ and properly contained in $A$. It is proven (Lemma 6.5 in \\cite{HPW}) that the relation $\\equiv_A$ is an equivalence relation on $A$ whose equivalence classes are strong modules. Also, if $A$ has more than one element, then there are at least two classes if and only if $A$ is robust. Moreover, these classes are the maximal strong modules properly contained in $A$. \nThe equivalence classes of a strong module $A$ are called \\textit{components} of $A$. We say that a module is {\\em limit} if it is a strong module such that none of its non-empty proper strong submodules is maximal (see \\cite{HPW}).\n\n\\begin{proposition} [\\cite{HPW}] \\label{Nonlimitrobust}\nLet $\\mathbb{B}=(V,d)$ be a binary structure and A a subset of V. Then A is a non-trivial robust module if and only if $A$ is not a limit module.\n\\end{proposition}\n\n\n\nBy Proposition \\ref{Nonlimitrobust}, a module is limit if and only if it is not robust. \nIf $A$ is a robust module of a binary structure with at least two elements, then there are two distinct components $I, J$ of $A$. Let $x, x' \\in I$ and $y, y' \\in J$. Since $J$ is a module and $x \\in A\\setminus J$, we have $d(x,y)=d(x,y')$. Since $I$ is a module and $y' \\in A\\setminus I$, we have $d(x,y')=d(x',y')$. So, $d(x,y)=d(x',y')$ meaning that the values $d(x,y)$ for $x \\in I$ and $y \\in J$ depend only upon $I$ and $J$. Therefore, the binary structure on $A$ induces a binary structure on the set $A\/\\equiv_A$ of components of $A$, called the \\textit{Gallai quotient} of $A$.  The following proposition is given in \\cite{HPW} as an observation without proof, however, we give a proof for completeness.  \n\n\\begin{proposition} \\label{Gallai}\nLet $A$ be a robust module of a binary structure $\\mathbb{B}$. The strong modules of the Gallai quotient of A are trivial: the empty set, the whole set $A\/\\equiv_A$ and the singletons.  \n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{B}$ be a strong module of $A\/\\equiv_A$ containing more than one component of $A$. Notice that $A$ has at least two components. Assume that $\\mathcal{B}$ contains two distinct components $I$ and $J$ of $A$. Let $K \\in A\/\\equiv_A \\setminus \\mathcal{B}$ and $I_1, I_2\\in \\mathcal{B}$. We have $d(K,I_1)=d(K,I_2)$ and $d(I_1,K)=d(I_2,K)$. It means that for every $x \\in K$ and every $y, y' \\in \\bigcup \\mathcal{B}$, we have $d(x,y)=d(x,y')$ and $d(y,x)=d(y',x)$. Thus, $\\bigcup\\mathcal{B}$ is a module of $A$. We prove that $\\bigcup\\mathcal{B}$ is a strong module of $A$. Let $N$ be a module of $A$ such that $N \\cap (\\bigcup\\mathcal{B}) \\neq \\emptyset$. Without loss of generality assume that $N$ meets $I$. Since $I$ is strong, $N \\subseteq I$ or $I \\subseteq N$. In the first case $N \\subseteq \\bigcup\\mathcal{B}$. In the second case if $I \\subset N$, then $N=A$ because $I$ is maximal and hence $\\bigcup\\mathcal{B}\\subseteq N$, so, $\\bigcup \\mathcal{B}$ is strong. Note that the components of $A$ are the maximal strong modules properly contained in $A$. Since $I$ is a proper subset of the strong module $\\bigcup\\mathcal{B}$, it follows that $\\bigcup\\mathcal{B}=A$. Equivalently, $\\mathcal{B}=A\/\\equiv_A$.\n\\end{proof} \n\nThe central result of the decomposition theory of binary structures describes the structure of the Gallai quotient. Let $\\mathbb{B}=(V,d)$ be a binary structure. We say that $\\mathbb{B}$ is {\\em constant} with value $\\alpha$ if $d(x,y)=\\alpha$ for all $x\\neq y\\in V$, and {\\em linear} with values $\\{\\alpha, \\beta\\}$, $\\alpha\\neq \\beta$, if $\\{(x,y)\\in V\\times V : x\\neq y, d(x,y)=\\alpha\\}$ is a linear order and $\\{(x,y)\\in V\\times V : x\\neq y, d(x,y)=\\beta\\}$ is the opposite linear order.  \n\n\n\\begin{theorem} [\\cite{HPW}] \\label{Primeconstantlinear}\nThe strong modules of a binary structure $\\mathbb{B}$ are trivial if and only if $\\mathbb{B}$ is prime,  constant or linear. \n\\end{theorem} \n\nLet $A$ be a non-trivial robust module of a binary structure $\\mathbb{B}$. The {\\em type} of $A$, denoted by $t(A)$, is {\\em prime} if $A\/\\equiv_A$ is prime, otherwise its type is $\\alpha$ if $A\/\\equiv_A$ is  constant with value $\\alpha$, and $\\{\\alpha, \\beta\\}$ if $A\/\\equiv_A$ is linear with values $\\{\\alpha, \\beta\\}$, $\\alpha\\neq \\beta$. By Proposition \\ref{Gallai},  the strong modules of the Gallai quotient $A\/\\equiv_A$ are trivial. Hence, the Gallai quotient $A\/\\equiv_A$ is either prime, constant or linear. \n\nLet $A$ be a non-trivial robust module of an $NE$-free poset $P$. Note that $P$ can be regarded as a binary structure $\\mathbb{B}=(P,d)$ where for $x, y\\in P$, $d(x,y)=0$ if $x\\perp y$, $d(x,y)=+1$ if $x <_P y$ and $d(x,y)=-1$ if $y <_P x$. By Proposition \\ref{Gallai}, the strong modules of $A\/\\equiv_A$ are trivial and by Theorem \\ref{Primeconstantlinear}, $A\/\\equiv_A$ is either prime, constant or linear. Further, a theorem due to Kelly \\cite{KE} asserts that any indecomposable partial order with at least three elements embeds `$N$'. By definition, a partial order with only two elements is not prime. Therefore, the Gallai quotient $A\/\\equiv_A$ is either constant or linear. In the first case, the quotient $A\/\\equiv_A$ is an antichain i.e. $d(I,J)=0$ for each $I\\neq J\\in A\/\\equiv_A$. So, the type of $A$ is 0 in this case. In the second case, the quotient $A\/\\equiv_A$ is a linearly ordered set, that is, for two $I\\neq J\\in A\/\\equiv_A$, we have $I <_P J$ or $J <_P I$. Thus, $\\{(I,J)\\in A\/\\equiv_A\\times A\/\\equiv_A : I\\neq J, d(I,J)=+1\\}$ is a linear order and $\\{(I,J)\\in A\/\\equiv_A\\times A\/\\equiv_A: I\\neq J, d(I,J)=-1\\}$ is the opposite linear order. Hence, the type of $A$ is $\\{-1,+1\\}$ in this case.  \n\n\\subsection{Decomposition Tree of $NE$-Free Posets} \\label{DTNE}\n\nOur terminology in this section borrows from \\cite{TH2} and \\cite{HPW}. We follow the way of construction employed by \\cite{HPW} in which a decomposition tree is a meet-tree ordered by reverse inclusion of robust modules. Let $P=(V,\\leq)$ be a poset.  We use the notations $P^x=\\{y \\in V : x \\leq y\\}$ and $P_x=\\{y \\in V : y \\leq x\\}$. A \\textit{forest} is a poset $F$ such that for every $x\\in F$, the set $F_x$ is a chain; the elements of $F$ are called \\textit{nodes}.  A \\textit{tree} is a forest whose elements are pairwise compatible. Let $(T,\\leq)$ be a tree. \nA {\\em branch} of a node $x\\in T$ is an induced and maximal (under inclusion) subtree of $T^x\\setminus \\{x\\}$. A {\\em limit branch} is a branch of a node $x\\in T$ with no minimal element. For two nodes $x, y$ of $T$, we say $x$ is a \\textit{child} of $y$ if $y < x$ and there is no $z\\in T$ such that $y < z < x$. A tree is a \\textit{meet-tree} if any two nodes $x, y$ have a greatest lower bound, called their \\textit{meet} and denoted by $x \\wedge y$. \nLet $T$ be a meet-tree. A \\textit{leaf} in $T$ is a maximal node, and a \\textit{root} is a minimum one. An \\textit{internal} node is one that is not a leaf. A tree has at most one root. We observe that if a node $x$ of $T$ is the meet of a finite set $X$ of the maximal nodes of $T$, denoted by $Max(T)$, then $x$ is the meet of a subset $X'$ of $X$ with at most two nodes. \nWe say that a meet-tree $T$ is \\textit{ramified} if every node of $T$ is the meet of a finite set of maximal nodes of $T$.  \n\nLet $T$ be a ramified meet-tree and $T':=T\\setminus Max(T)$. A $\\{0,\\{-1,+1\\}\\}$-\\textit{valuation} is a map $v :T'\\to \\{0,\\{-1,+1\\}\\}$. The valuation is \\textit{dense} if for every $x<y$ in $T'$, there is some $z$ with $x< z \\leq y$ such that $v(z)\\neq v(x)$. A ramified meet-tree densely valued by $\\{0,\\{-1,+1\\}\\}$  is called a \\textit{decomposition tree}. \n\nLet $P$ be an $NE$-free poset. We construct the decomposition tree $T=(R(P), \\leq,v)$ of $P$ as follows:\nThe nodes of the tree $T$ are the robust modules of $P$ denoted by $R(P)$. The order $\\leq$ on the nodes of the tree is reverse inclusion meaning that for two robust modules $A, B$ of $P$, $A \\leq B$ if and only if $B \\subseteq A$. To see that this order gives a tree, first let $A$ be a non-empty robust module of $P$. For two $C_1, C_2 \\in R(P)$ with $C_1, C_2 \\leq A$ we have $A \\subseteq C_1$ and $A \\subseteq C_2$. Since $A$ is non-empty, it follows that there exists some $x\\in A$. So, $x \\in C_1 \\cap C_2$. This means that $C_1 \\subseteq C_2$ or $C_2 \\subseteq C_1$ because both $C_1$ and $C_2$ are strong. Hence, the down-set of a node of the tree $T$ is a chain. Now, suppose that $A, B$ are two robust modules which are not comparable. Let $x \\in A\\setminus B$ and $y \\in B\\setminus A$. Then $x$ and $y$ are distinct. Set $C$ to be the robust module determined by $\\{x, y\\}$. We must have that $C$ is comparable to $A$ and $B$. Moreover, since $x\\in A$, $y \\in C$ and $y \\notin A$, we obtain $A \\subset C$. Interchanging the roles of $x$ and $y$, we get $B \\subset C$. Under reverse inclusion, $C$ is the meet of $A$ and $B$, that is $C=A \\wedge B$. The robust modules of $P$ consisting of singletons are exactly the leaves of $T$. Robust modules of $P$ which are not singletons are valued with one of two values 0 and $\\{-1, +1\\}$. If $A$ is a non-trivial robust module of $P$, then the {\\em value} $v(A)$ of $A$ is the type of $A$ that is $v(A):=\\{-1,+1\\}$, resp $v(A):=0$, if $t(A)$ is $\\{-1,+1\\}$, resp 0. It implies that if $A$ is the least strong module containing two distinct elements $x, y$, then $v(A)=\\{-1,+1\\}$, resp $v(A)=0$, if $x$ and $y$ are comparable, resp incomparable. Note that the map $v$ is defined on the set of nodes of $T$ which are not maximal, that is $v : T\\setminus Max(T) \\to \\{0,\\{-1,+1\\}\\}$.   Given a node $A\\in T$ which is a robust module of $P$, the union of the robust modules properly contained in $A$ is a maximal proper strong submodule of $A$ which might be limit itself.  \n \n\nThe following proposition implies that the valuation of $T$ is dense. \n\n\\begin{proposition} [\\cite{HPW}] \\label{Types}\nLet A, C be two non-trivial robust modules of a binary structure such that $A\\subset C$ and A and C have the same non-prime type $\\{\\alpha, \\beta\\}$. Then, there is a robust module B with $A\\subset B\\subset C$ whose type is distinct from the type of $\\{\\alpha, \\beta\\}$.  \n\\end{proposition}\n\n\\begin{corollary} \\label{Dense}\nLet P be an NE-free poset and T its decomposition tree. The valuation v of T is dense. \n\\end{corollary}\n\n\\begin{proof}\nRecall that for each non-trivial robust module $A$ of $P$, we have either $t(A):=0$, or $t(A):=\\{-1,+1\\}$. Now, let $A\\subset C$ be a non-trivial robust modules of $P$. If $v(A)\\neq v(C)$, then by setting $B:=C$, the result follows. Now assume that $v(A)=v(C)$. Then, both $\\mathbb{B}_{\\upharpoonright A\/\\equiv_A}$ and $\\mathbb{B}_{\\upharpoonright C\/\\equiv_C}$ are either constant or linear. Therefore, $A, C$ have the same non-prime type $\\{\\alpha,\\beta\\}$ ($\\alpha=\\beta=0$ or $\\alpha=-1, \\beta=+1$). By Proposition \\ref{Types}, there exists a robust module $B$ with $A\\subset B\\subset C$ whose type is distinct from the type $\\{\\alpha,\\beta\\}$, that is $t(B)\\neq t(A), t(C)$. This implies that $v(B)\\neq v(A)$. Therefore, the valuation $v$ is dense.\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\\section{Structure of $NE$-Free Posets} \\label{StructureNE}\n\nIn this section we classify $NE$-free posets and show that each $NE$-free poset is either a direct sum or a linear sum of $NE$-free posets or a sum over a labelled chain with no least element. \nIndeed, all the three operations are based on substituting smaller posets for three sorts of posets: chains, antichains and posets obtained from labelled chains. A \\textit{cograph} is a graph with no induced subgraph isomorphic to a path $P_4$ with four vertices  \\cite{J}.\n\n\n\n\n\\begin{proposition} \\label{Subcograph}\nLet $G=K[H_v\/v : v\\in K]$ be a graph substitution. $G$ is a cograph if and only if its context graph and graph blocks are cographs. \n\\end{proposition}\n\n\\begin{proof}\nLet $P$ be a copy  of $P_4$. \n\n($\\Rightarrow$) If $G$ is a cograph, then it is clear that all its graph blocks are cographs. If $K$ embeds $P$, then let $v_1, v_2, v_3, v_4$ be the vertices of $P$ in $K$. For every $1\\leq i \\leq 4$, pick an element $x_i\\in H_{v_i}$. Then, the $x_i$ form a copy of $P_4$ in $G$, a contradiction.  \n\n($\\Leftarrow$) For the sake of a contradiction, assume that $P$ is embedded in $G$. We show that no graph block $H_v$ contains more than one element of $P$. By assumption, no $H_v$ embeds $P$. Suppose $|P\\cap H_v|=3$ and let $a, b, c \\in H_v \\cap P$. Then $d \\in P$ belongs to some $H_u$ where $u \\neq v$. Without loss of generality assume that the vertex $d$ is connected to $c$. This means that $uv\\in E(K)$. Consequently, $d$ is connected to $a$ and $b$. But, then $a, b, c$ and $d$ do not form a path as $d$ has degree 3. Now suppose $|P\\cap H_v|=2$ and let $a,b \\in P\\cap H_v$. Suppose $b$ and $c$ are adjacent where $c \\in H_u$ with $u \\neq v$. It follows that $a$ and $c$ are adjacent as well. The vertex $d$ cannot be adjacent to $c$ as then $deg(c)=3$. Therefore, $d$ must be adjacent to  both $a$ and $b$. In this case the vertices $a, b, c$ and $d$ form a cycle. Thus, every vertex of $P$ belongs to a unique $H_v$.  It follows that $K$ itself embeds $P$, a contradiction.   \n\\end{proof} \n\nThe only partial order $P$ on $P_4$ with $CG(P)=P_4$ results in an `$N$'. On the other hand, the comparability graph of an `$N$' is $P_4$. So, we get the following result. \n\n\\begin{corollary} \\label{SubNEposet}\nLet $P=Q[P_v\/v : v\\in Q]$ be a poset substitution. $P$ is $NE$-free if and only if its context poset and poset blocks are $NE$-free. \n\\end{corollary}\n\n\n$NE$-free posets which are not direct or linear sums require poset labelled sums. \nLet $(I,\\leq)$ be a chain and $r$ a map from $I$ to $\\{-1, 0, +1\\}$. Let $Q^I_r=(I,\\leq')$ be defined as follows: for $i<j$, $i\\perp j$ if $r(i)=0$, $i<'j$ if $r(i)=-1$ and $j<'i$ if $r(i)=+1$. We prove in the following lemma that $Q^I_r$ is an $NE$-free poset. Let $\\{(P_i,\\leq_i)\\}_{i\\in I}$ be a disjoint family of non-empty posets. We call the poset substitution of the $P_i$ for $Q^I_r$ i.e. $P=Q^I_r[P_i\/i:i\\in I]$, a {\\em poset labelled sum} of the $P_i$ and denote it by $P=\\sum_{i\\in I} P_i$. The $P_i$ are called the {\\em summands} of $P$. We may view such a sum as the poset associated to the {\\em  labelled chain} $C:=(I,\\leq,\\ell)$ where $\\ell(i)=(P_i,r(i))$ and denote it by $P=\\sum C$. \n\n\\begin{lemma} \n$Q^I_r$ is an NE-free poset. Hence, a poset labelled sum $\\sum_{i\\in I}P_i$ is NE-free if and only if its summands are NE-free. \n\\end{lemma}\n\n\\begin{proof}\nIn order to prove that $Q^I_r=(I,\\leq')$ is a poset, it suffices to show that $\\leq'$ is transitive. Let $i, j, k\\in I$ and assume that $i <' j$ and $j <' k$. If $i<j<k$, then $r(i)=-1$ and if $k<j<i$, then $r(k)=+1$. In both cases we get $i <' k$. Now, assume that $i<j$ and $k<j$. Since $i <' j$, we have $r(i)=-1$. Further, since $(I,\\leq)$ is a chain, we have $i<k$ or $k<i$. In the first case, we get $i <' k$. In the second case, we have $k<i<j$ and regarding $j <' k$ we get $r(k)=+1$. Thus, $i <' k$. Note that the case $j<i$, $j<k$ is not possible because then we get $r(j)=+1$ and $r(j)=-1$, a contradiction. \n\nNow, suppose that a copy of $N$ embeds in $Q^I_r$. Let $i\\in N$ be such that $i\\leq j$ for each $j\\in N$. Such an element exists because $(I,\\leq)$ is a chain. Since $i$ is comparable w.r.t $\\leq'$ to at least one $j\\neq i$ in $N$, we have $r(i)=-1$ or $r(i)=+1$. But then $i$ is comparable w.r.t $\\leq'$ to each element of $N$, a contradiction. Thus, $Q^I_r$ is $NE$-free. By Corollary \\ref{SubNEposet}, it implies that $P=\\sum_{i\\in I} P_i$ is $NE$-free if and only if each $P_i$ is $NE$-free.   \n\\end{proof}\n\n\n\n\\subsection{Classification} \\label{Classification} \n\nThe classification of $NE$-free posets is based on the structure of their decomposition tree. Let $I$ be a chain and $r:I\\to\\{-1,0,+1\\}$ a map. We set $I_0:=\\{i\\in I : r(i)=0\\}$ and $I_1:=\\{i\\in I : |r(i)|= 1\\}$. In order to classify $NE$-free posets, we will need poset labelled sums indexed by infinite chains $C=(I,\\leq,r)$ with no least element where $r$ is a mapping from $I$ to $\\{-1, 0 , +1\\}$ such that both $I_0$ and $I_1$ are coinitial in $I$. We aim to show that if $T$ has no least element, then $P$ is a poset labelled sum. The crucial ingredient that we will use is the following (see \\cite{BD}, page 1747 Lemma 2.2). \n\n\n\n\\begin{lemma} [\\cite{BD}] \\label{BD}\nLet $\\mathbb{B}=(V,d)$ be a binary structure. The collection of robust modules of $\\mathbb{B}$ containing a given element is a chain covering the elements of $V$ (i.e. every element of V belongs to some member of the collection). \n\\end{lemma}\n\n\nLet $P$ be an $NE$-free poset. If $T$ is the decomposition tree of $P$, then the collection $\\mathfrak{C}_x$ of robust modules of $P$ containing a given element $x\\in P$ corresponds to a maximal chain in $T$ whose greatest element is $\\{x\\}$. Assume that $\\mathfrak{C}_x$ has no least element and take an arbitrary $y\\in P$. Let $\\mathfrak{C}_y$ be the collection of robust modules of $P$ containing $y$. Since $\\mathfrak{C}_y$ and $\\mathfrak{C}_x$ meet at $\\{x\\}\\wedge \\{y\\}$ in $T$, it follows that $\\mathfrak{C}_y$ does not have a least element as well. \n\n\n\n\\begin{lemma} \\label{Posetreduced}\nLet P be an NE-free poset such that for some (every) $x\\in P$, the chain of robust modules of $P$ containing $x$ has no least element. Then P is the sum $\\sum C$ of a labelled chain $C=(I,\\leq,\\ell)$ where $(I,\\leq)$ is an infinite chain with no least element and for each $i\\in I$, $\\ell(i)=(P_i,r(i))$ where $P_i$ is an NE-free poset and r is a mapping from I to $\\{-1,0,+1\\}$ such that $I_0=\\{i\\in I : r(i)=0\\}$ and $I_1=\\{i\\in I : |r(i)|=1\\}$ are coinitial in $I$. \n\\end{lemma} \n\n\\begin{proof} \nLet $P$ satisfy the condition of the lemma and $\\mathfrak{C}$ be the chain of robust modules of $P$ containing $x$ under reverse inclusion. Let $v$ be the valuation of the decomposition tree $T$ of $P$. Let $A\\in\\mathfrak{C}$ be given. If $v(A)=0$, then let $P_A$ be the poset obtained by restricting $P$ to $A \\setminus [x]_{\\equiv_A}$ and if $v(A)=\\{-1, +1\\}$, then let $P_{A^-}$, resp $P_{A^+}$, be the poset obtained by restricting $P$ to $\\{y \\in A : y\\not\\equiv_A x\\ \\text{and}\\ y <_P x\\}$, resp $\\{y\\in A : y\\not\\equiv_A x, \\ \\text{and}\\ x <_P y\\}$. For $A\\in \\mathfrak{C}$ with $v(A)=\\{-1, +1\\}$, replace $A$ by $A^-$ if $P_{A^+}=\\emptyset$, by $A^+$ if $P_{A^-}=\\emptyset$ and by the two-element chain $A^-<A^+$ if both $P_{A^-}$ and $P_{A^+}$ are non-empty. Let $I$ consist of $\\{x\\}$, the $A\\in\\mathfrak{C}$ with $v(A)=0$ and the replacements $A^-$ and $A^+$ for $A\\in\\mathfrak{C}$ with $v(A)=\\{-1, +1\\}$. For $A_1, A_2\\in I$, if $A_1$, resp $A_2$, is determined by some robust module $A$, resp $B$, where $A\\neq B$, then define $A_1 < A_2$ if and only if $B \\subset A$ ($A <_T B$). It follows that $(I,\\leq)$ is a chain with no least element because $\\mathfrak{C}$ has no least element. \n\nLet $I$ be the chain constructed above. Define $r:I\\to\\{-1,0,+1\\}$ by $r(A)=0$ if $v(A)=0$, $r(A^-)=-1$ and $r(A^+)=+1$. Since $\\mathfrak{C}$ has no least element and $v$ is dense on the elements of $\\mathfrak{C}$, it follows that $I_0$ and $I_1$ are coinitial in $I$. \n\nLet $I$ be the chain constructed above and for each $A\\in I$ let $P_A$ be the poset obtained above. Consider $\\{P_A\\}_{A\\in I}$ which is a family of pairwise disjoint $NE$-free posets. We have $P=\\bigcup_{A\\in I} P_A$. Let $A, B\\in I$ be given such that $A<B$. If $r(A)=0$ then $v(A)=0$ and we have $A <_T B$. So, $P_A\\perp P_B$ in this case. If $r(A)=-1$, resp $r(A)=+1$, then $A$ is determined by a robust module with value $\\{-1,+1\\}$ and we have $P_A <_P P_B$, resp $P_B <_P P_A$. This means that $P=Q^I_r[P_A\/A : A\\in I]$, that is $P$ is a poset labelled sum. In other words $P=\\sum C$ where $C:=(I,\\leq,\\ell)$ is a labelled chain with $\\ell(A)=(P_A,r(A))$. Note that $(I,\\leq)$ is an infinite chain with no least element and $r$ is a mapping from $I$ to $\\{-1,0,+1\\}$ such that $I_0$ and $I_1$ are coinitial in $I$. \n\\end{proof}\n\n\\begin{definition} \\label{CCGC}\nWe say that a poset $P$ has {\\em property CCGC} if the complement of the comparability graph of $P$ is connected, that is, $(CG(P))^c$ is connected. \n\\end{definition}\n\nIn Subsection \\ref{Linearsum}, when determining the sibling number of a linear sum $P$ of $NE$-free posets, we will see that the property CCGC of the summands of $P$ plays a crucial role. We also prove that an $NE$-free poset satisfying Lemma \\ref{Posetreduced} has property CCGC. \n\n\\begin{proposition} \\label{Plsccgc}\nLet P be an NE-free poset whose decomposition tree has no least element. Then P is connected and it has property CCGC. \n\\end{proposition}\n\n\\begin{proof}\nLet $T$ be the decomposition tree of $P$. By assumption $T$ has no least element. Set $G:=CG(P)$. Take $x, y\\in G$ and consider the robust module $A$ determined by $x$ and $y$. If $v(A)=\\{-1,+1\\}$, then $x$ and $y$ are comparable meaning that $xy\\in E(G)$. If $v(A)=0$, then since $T$ has no least element and $v$ is dense, we can find a robust module $B\\supset A$, equivalently $B <_T A$, with $v(B)=\\{-1,+1\\}$. Take some $z\\in B$. Then $x$ and $z$ are comparable as well as $y$ and $z$. Therefore, there is a path in $G$ connecting $x$ and $y$. Hence, $P$ is connected. \n\n\nNow we prove that $G^c$ is connected. Let $x, y\\in G$ be given. Let $A=S_\\mathbb{B}(x,y)$ as computed for $G$. If $v(A)=0$, then $x$ and $y$ are incomparable w.r.t $\\leq_P$. Thus, $xy\\in E(G^c)$ meaning that in this case $x$ and $y$ are connected in $G^c$. If $v(A)=\\{-1,+1\\}$, then by density of $v$ and the fact that $T$ has no least element we can find a robust module $B$ of $P$ with $B<_T A$ and $v(B)=0$. Take some $z\\in B$. We have $z\\perp x$ and $z\\perp y$. Therefore, $xz, zy\\in E(G^c)$ meaning that $x$ and $y$ are connected in $G^c$. This completes the proof. \n\\end{proof}\n\nWe have the following classification of $NE$-free posets which will be used in determining the sibling number of an $NE$-free poset.  \n\n\\begin{theorem} \\label{Pstructure}\nLet P be an NE-free poset with more than one element. Then either \n\\begin{enumerate}\n    \\item P is a direct sum of at least two non-empty connected NE-free posets, or\n    \\item P is a linear sum of at least two non-empty NE-free posets with property CCGC, or\n    \\item P is the sum $\\sum C$ of a labelled chain $C=(I,\\leq,\\ell)$ such that $(I,\\leq)$ is an infinite chain with no first element, and each label $\\ell(i)$ is the pair $(P_i,r(i))$ made of a non-empty NE-free poset $P_i$ and an element $r(i)\\in\\{-1,0 , +1\\}$ in such a way that r is a mapping on the chain $(I,\\leq)$ such that $I_0=\\{i\\in I : r(i)=0\\}$ and $I_1=\\{i\\in I : |r(i)|=1\\}$ are coinitial in I. \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $T$ be the decomposition tree of $P$. If $P$ is the least element of $T$, then $P$ is a robust module with value  0 or $\\{-1, +1\\}$. In the first, resp second, case, each maximal proper robust submodule of $P$ is connected, resp has property CCGC, the quotient $P\/\\equiv_P$ is an antichain, resp a chain, and $P$ is a direct, resp linear, sum of its components, resp summands. Finally, if $T$ has no least element, then by Lemma \\ref{Posetreduced}, $P$ satisfies (3). \n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Siblings of Countable $NE$-Free Posets} \\label{SibNE}\n\nIn this section our aim is to prove the following theorem which shows that the alternate Thomass\\'{e} conjecture holds for countable $NE$-free posets.\n\n\\begin{theorem} \\label{AltThomasseNEP}\nA countable $NE$-free poset has one or infinitely many siblings.\n\\end{theorem}\n\nA {\\em quasi-order} is a reflexive and transitive binary relation. A {\\em well-quasi-order (wqo)} is a quasi-order $\\mathcal{Q}$ such that any infinite sequence of elements of $\\mathcal{Q}$ contains an infinite increasing subsequence w.r.t $\\leq_\\mathcal{Q}$. \nWe denote the class of countable $NE$-free posets by $\\mathcal{N}_{\\leq\\omega}$. \nThomass\\'{e} \\cite{TH2} proved that  $\\mathcal{N}_{\\leq\\omega}$  is well-quasi-ordered under embeddability. Hence, embeddability is a well-founded relation on $\\mathcal{N}_{\\leq\\omega}$. Therefore, in order to prove that a countable $NE$-free poset $P$ has one or infinitely many siblings, we suppose that this property holds for all $NE$-free posets which strictly embed in $P$ and then show that the property holds for $P$. In order to tackle the problem we use the classification of $NE$-free posets. Recall by Theorem \\ref{Pstructure} that an $NE$-free poset with more that one element is either a direct sum of at least two non-empty connected $NE$-free posets or a linear sum of at least two non-empty $NE$-free posets with property CCGC, or it is the sum of a labelled chain such that the chain has no first element. We argue by cases.  We start with the last case and prove that such a countable poset has continuum many siblings. Next, direct and linear sums will be considered.  \n\n\n\n\n\n\\subsection{Sum Over a Labelled Chain with no Least Element}  \n\n\nLet $P$ be a countable $NE$-free poset such that its decomposition tree has no least element. In this case by Lemma \\ref{Posetreduced}, $P$ is the sum $\\sum C$ of a  labelled chain $C=(I,\\ell)$ such that $I$ has no first element and $\\ell(i)=(P_i,r(i))$ where $r:I\\to\\{-1,0,+1\\}$ is a map such that $I_0$ and $I_1$ are coinitial in $I$. We show that $P$ has continuum many siblings in this case. The proof is based on constructing continuum many pairwise non-isomorphic labelled chains $C_\\alpha$. We quote some results in \\cite{HPW} and also translate some results there into the context of $NE$-free posets to show that the sums $\\sum C_\\alpha$ are pairwise non-isomorphic siblings of $\\sum C$. We also provide the notions needed. \n\nA {\\em labelled chain} is a pair $C=(I,\\ell)$ where $I$ is a chain and $\\ell$ is a map from $I$ to a quasi-order. Let $\\mathcal{Q}$ be a quasi-order. A {\\em $\\mathcal{Q}$-embedding}, (or an {\\em embedding} when the quasi-order $\\mathcal{Q}$ is definite), of a labelled chain  $C=(I,\\ell)$ into another labelled chain $C'=(I',\\ell')$, denoted by $C\\leq C'$, is an order embedding $f:I\\to I'$ such that $\\ell(i)\\leq_\\mathcal{Q} \\ell'(f(i))$ for all $i\\in I$. Two labelled chains $C, C'$ are {\\em equimorphic}, denoted by $C\\approx C'$, when they are mutually embeddable. They are {\\em isomorphic} when there is an isomorphism $\\phi$ from $I$ to $I'$ such that $\\ell'\\circ \\phi=\\ell$. \nA labelled chain  $C=(I,\\ell)$ is {\\em additively indecomposable}, or briefly {\\em indecomposable}, if for every partition of $I$ into an initial segment $J$ and a final segment $F$, $C\\leq C_{\\upharpoonright J}$ or $C\\leq C_{\\upharpoonright F}$. We say that $C$ is {\\em left-indecomposable}, resp {\\em strictly left-indecomposable}, if $C$ embeds in every non-empty initial segment, resp if $C$ is left-indecomposable and embeds in no proper final segment. The {\\em right-indecomposability} and the {\\em strict right-indecomposability} are defined similarly. \n\nThe {\\em sum} $\\sum_{i\\in I} C_i$ of labelled chains over a chain (not over a labelled chain) is defined as in the case of chains. If $I$ is the $n$-element chain $\\underline{n}:=\\{0,1,\\ldots, n-1\\}$ with $0<1<\\cdots<n-1$, then the sum is rather denoted by $C_0+C_1+\\cdots+C_{n-1}$. \nLet $C=(I,\\ell)$ be a labelled chain and $J, F$ be a partition of $I$ into an initial and a final segment, respectively. Let $C_0, C_1$ be obtained by restricting $C$ to $J, F$, respectively. With the notion of sum, $C$ is indecomposable if  $C=C_0+C_1$ implies that $C\\leq C_0$ or $C\\leq C_1$. \n\nLet $C=(I,\\ell)$ be a labelled chain. Recall that $I^*$ is obtained from $I$ by reversing the order of $I$. Define $C^*$ to be the labelled chain $(I^*,\\ell)$. \nFor two labelled chains $C_0$ and $C_1$, we define $C_0+^* C_1$ to be the sum $C_1^*+C_0^*$. For a chain $J$ define $\\sum^*_{j\\in J} C_j=\\left(\\sum_{j\\in J} C_j\\right)^*=\\sum_{j\\in J^*} C_j^*$. \nA sequence $(C_n)_{n<\\omega}$ of labelled chains is {\\em quasi-monotonic} if $\\{m : C_n\\leq C_m\\}$ is infinite for each $n<\\omega$, its sum $\\sum_{n<\\omega} C_n$  is right indecomposable or equivalently $\\sum_{n<\\omega}^* C_n$ is left indecomposable (\\cite{HPW}).  \n\nLet $C=(I,\\ell)$ be a labelled chain. Two elements $x, y\\in I$, $x\\leq y$, are {\\em equivalent}, denoted by $x\\equiv_C y$, if $C$ does not embed in the restriction of $C$ to the interval determined by $x$ and $y$, that is, $C \\not\\leq C_{\\upharpoonright[x,y]}$. If $C$ is indecomposable, then this relation is an equivalence relation. Equivalence classes are intervals of $I$. Further, either all the elements of $I$ are equivalent, that is, there is only one equivalence class, or the quotient $I\/\\equiv_C$ is dense and if $J$ is any equivalence class, $C$ does not embed into $C_{\\upharpoonright J}$. \n\nAccording to Laver \\cite{LAV1}, the class $\\mathcal{C}_{\\leq\\omega}$ of countable chains quasi-ordered by embeddability is wqo. Moreover, the class $\\mathcal{Q}^{\\mathcal{C}_{\\leq\\omega}}$ of countable chains labelled by a wqo $\\mathcal{Q}$ is wqo. Hence, the class of countable chains labelled by $\\mathcal{N}_{\\leq\\omega}$ is wqo.  \n\n\n\nWe list the properties we need. These properties are due to Laver \\cite{LAV1} and they are given in \\cite{HPW}. We quote the proof given in \\cite{HPW}. \n\n\\begin{lemma} [\\cite{HPW}]  \\label{Labelledchainsproperties}\nLet $C=(I,\\ell)$ be a countable chain labelled over a wqo $\\mathcal{Q}$.   \n\\begin{enumerate}\n   \\item If I has no least element, then there is some initial segment J of I such that $C_{\\upharpoonright J}$ is left indecomposable;\n    \\item If C is left indecomposable, then C is an $\\omega^*$ sum $\\sum^*_{n<\\omega} C_n$ where each $C_n$ is indecomposable and the set of m such that $C_n$ embeds into $C_m$ is infinite. \n   \n    \\end{enumerate}\n\\end{lemma}\n\nWe translate a result in \\cite{HPW} to the context of $NE$-free posets. \n\n\\begin{lemma}  \\label{Finalsegmentsiso} \nLet $P, P'$ be two isomorphic NE-free posets such that their decomposition tree has no least element. If $P=\\sum C$ and $P=\\sum C'$ where $C=(I,\\ell)$ and $C'=(I',\\ell')$ are two labelled chains, then there are two infinite initial segments J of I and $J'$ of $I'$ and an isomorphism h of the induced poset labelled chains $C_{\\upharpoonright J}$ and $C'_{\\upharpoonright J'}$. \n\\end{lemma}\n\n\\begin{proof}\nSince $P$ and $P'$ are isomorphic, their decomposition trees are isomorphic. We may assume that they are identical. Let $T$ be such a tree. Then, $(I,\\leq)$ and $(I'\\leq')$ correspond to maximal chains in $T$ and by Lemma \\ref{Posetreduced} they do not have a first element. If these chains are identical, then there is nothing to prove ($J=J'=I$). If not, then since $T$ has no least element, these chains meet in some element $x\\in I\\cap I'$ corresponding to a node in $T$. Set $J=J'=I_x$ and let $h$ be the identity on the initial segment $I_x$. \n\\end{proof}\n\nThe following lemma appears as Lemma 4 in page 39 of \\cite{LPSZ}. \n\n\\begin{lemma} [\\cite{LPSZ}] \\label{Continuummaps}\nThere is a set of $2^{\\aleph_0}$ maps  $f$ in $\\{0,1\\}^\\mathbb{N}$ such that for every pair f, $f'$ of distinct maps, and every isomorphism h from a final segment F of $\\mathbb{N}$ onto another $F'$, there is some $n\\in F$ such that $f(n)\\neq f'(h(n))$. \n\\end{lemma} \n\n\nThe following lemma gives a construction of siblings of a countable $NE$-free poset which are  sums of labelled chains with no least element. The proof is adapted with some passages quoted from \\cite{HPW}. \n\n\\begin{lemma} \\label{Nonisoplc}\nLet $C=(I,\\ell)$ be a countable labelled chain with no least element where $\\ell(i)$ is the pair $(P_i,r(i))$ made of a non-empty NE-free poset and $r(i)\\in \\{-1,0,+1\\}$ such that $I_0=\\{i\\in I : r(i)=0\\}$ and $I_1=\\{i\\in I : |r(i)|=1\\}$ are coinitial in I. Then, there are $2^{\\aleph_0}$  labelled chains $C_f$ where $f:\\mathbb{N}\\to\\{0,1\\}$, such that $\\sum C_f\\ncong \\sum C_g$ for $f\\neq g$. \n\\end{lemma}\n\n\\begin{proof}\nLet $C=(I,\\ell)$ be a countable labelled chain satisfying the conditions of the lemma. We select an initial segment $J$ of $I$ such that the restriction $C_J:=(J,\\ell_{\\upharpoonright J})$ is left indecomposable (this is due to Lemma \\ref{Labelledchainsproperties} (1)). Let $Ev(J)$ be the set of $i\\in J$ such that $P_i$ is a chain or an antichain of even size. Since $I$ has no least element and $I_0$ and $I_1$ are coinitial in $I$, there is an infinite descending sequence $(a_n)_{n<\\omega}$ of elements of $J$ coinitial in $I$ such that $|r(a_n)|=1$. We insert infinitely many elements $b_n, c_n$ such that $b_n$ covers $a_n$ and $c_n$ covers $b_n$. Set $\\Bar{J}:=J\\cup \\{b_n, c_n : n<\\omega\\}$, $F:=I\\setminus J$ and $\\Bar{I}:=\\Bar{J}\\cup F$. To each map $f:\\mathbb{N}\\to \\{0,1\\}$ we associate a labelled chain $C_f=(\\Bar{I},\\ell_f)$ with  $\\ell_f(i):=(\\Bar{P_i},r_f(i))$ as follows: If $i\\in I\\setminus Ev(J)$, then we set $\\ell_f(i):=\\ell(i)$. If $i\\in Ev(J)$, then set $r_f(i):=r(i)$ and let $\\Bar{P_i}$ be an extension of $P_i$ to an extra element in such a way that the new poset is a chain if $P_i$ is a chain and an antichain, otherwise. If $i\\in \\{b_n, c_n\\}$, recalling that $|r(a_n)|=1$, we set $r_f(b_n)=0$ and $r_f(c_n)=r(a_n)$, $\\Bar{P_i}$ being an antichain, resp a chain, of size $2f(n)+2$ if $i=b_n$, resp $i=c_n$. It is clear that $\\bar{I}_0=\\{i\\in \\bar{I} : r_f(i)=0\\}$ and $\\bar{I}_1=\\{i\\in \\bar{I} : |r_f(i)|=1\\}$ are coinitial in $\\bar{I}$. \n\n\n \n\n\n\nIf $f$ and $f'$ are two maps from $\\mathbb{N}$ to $\\{0,1\\}$ such that the sums $\\sum C_f$ of $C_f=(\\Bar{I},\\ell_f)$ and $\\sum C_{f'}$ of $C_{f'}=(\\Bar{I},\\ell_{f'})$ are isomorphic, then according to Lemma \\ref{Finalsegmentsiso}, there are two initial segments $\\Bar{J}$ and $\\Bar{J'}$ of $\\Bar{I}$ and an isomorphism $h$ from $\\Bar{J}$ onto $\\Bar{J'}$ preserving the labels, that is, the labels $\\ell_f(i)=(\\Bar{P_i},r_f(i))$ and $\\ell_{f'}(h(i))=(\\Bar{P}_{{h(i)}},r_{f'}(h(i)))$, $i\\in \\Bar{J}$, are isomorphic, meaning that $\\Bar{P_i}$ and $\\Bar{P}_{h(i)}$ are isomorphic and $r_f(i)=r_{f'}(h(i))$ for each $i\\in \\Bar{J}$. If $i=b_n$, resp $i=c_n$, then necessarily $h(b_n)=b_m$, resp $h(c_n)=c_m$, for some $m$. Indeed, $\\Bar{P}_{h(i)}$  must be an antichain, resp chain, of even size. So, $h(b_n), h(c_n)\\in Ev(\\Bar{J})$. Further, recall that for $i\\in Ev(J)$, $\\Bar{P_i}$ is a chain or antichain of odd size. Since $r_{f'}(h(b_n))=r_f(b_n)=0$, resp $r_{f'}(h(c_n))=r_f(c_n)=\\pm 1$ where $\\pm 1$ means $-1$ or $+1$, hence $h(b_n)$, resp $h(c_n)$, must be some $b_m$, resp some $c_m$. Consequently, there are two final segments $F$ and $F'$ of $\\mathbb{N}$, and an isomorphism $h$ from $F$ to $F'$ such that $f'(h(n))=f(n)$ for every $n\\in F$. By applying Lemma \\ref{Continuummaps}, we get $2^{\\aleph_0}$ pairwise non-isomorphic labelled chains $C_f$ having the same properties as does $C$.\n\\end{proof} \n\nIn Lemma \\ref{Nonisoplc}, we observe that the order on $P_f:=\\sum C_f$ extends both $\\sum C_{f\\upharpoonright\\Bar{J}}$ and $\\sum C_{f\\upharpoonright F}$ where the relation between an element $x$ of $\\sum C_{f\\upharpoonright \\Bar{J}}$ and an element $y$ of $\\sum C_{f\\upharpoonright F}$ is determined as follows: $x <_{P_f} y$ if and only if $r_f(i)=-1$ where $i\\in\\Bar{J}$ and $x\\in \\Bar{P_i}$; and $y <_{P_f} x$ if and only if $r_f(i)=+1$ where $i\\in \\Bar{J}$ and $x\\in \\Bar{P_i}$. We denote this by $\\mp$ and obtain $\\sum C_f=\\sum C_{f\\upharpoonright\\Bar{J}}\\mp\\sum C_{f\\upharpoonright F}$. \n\nLet $C=(I,\\ell)$ be a labelled chain and $n$ be a positive integer. The {\\em ordinal product} $\\underline{n}.C$ is the labelled chain $(\\underline{n}.I,\\ell_n)$ where $\\underline{n}.I$ is the ordinal product of the $n$-element chain $\\underline{n}:=\\{0, \\ldots, n-1\\}$, $0<1<\\cdots<n-1$, and the chain $I$, that is, the ordinal sum of $C$ copies of $\\underline{n}$, and $\\ell_n(m,i)=\\ell(i)$ for every $m\\in\\underline{n}$, $i\\in I$. \n\n\n\\begin{lemma} [\\cite{HPW}]  \\label{ncopies}\nLet $C=(I,\\ell)$ be a countably infinite labelled chain. If the labels belong to a b.q.o and C is indecomposable, then for every positive integer n, the ordinal product $\\underline{n}.C$ embeds in C. \n\\end{lemma} \n\n\nNow we show that the sum of labelled chains $C_f$ in Lemma \\ref{Nonisoplc} are equimorphic to the sum of the labelled chain $C$.  \n\n\\begin{lemma} \\label{ReducedequimorphP}\nLet $C=(I,\\ell)$ be a countable labelled chain such that I has no first element and the labels belong to $\\mathcal{N}_{\\leq\\omega}\\times\\{-1, 0,+1\\}$. Then, there is an initial segment J of I such that the restriction $C_J=(J,\\ell_{\\upharpoonright J})$ is left indecomposable. If J is such an initial segment, then for every map $f:\\mathbb{N}\\to \\{0,1\\}$,  $\\sum C_f\\approx \\sum C$ where $C_f$ is constructed as in Lemma \\ref{Nonisoplc}.\n\\end{lemma}\n\n\\begin{proof}\nThe existence of $J$ follows from the fact that the class of countable chains labelled by $\\mathcal{N}_{\\leq\\omega}\\times \\{-1, 0, +1\\}$ is b.q.o and Lemma \\ref{Labelledchainsproperties} (1). Let $f:\\mathbb{N}\\to\\{0,1\\}$ and $C_f=(\\Bar{I},\\ell_f)$ be the labelled  chain defined in Lemma \\ref{Nonisoplc} and set $P_f:=\\sum C_f$. We prove that $\\sum C_f\\leq \\sum C$.  Let $C_{f\\upharpoonright\\Bar{J}}$, resp $C_{f\\upharpoonright F}$, be the restriction of $C_f$ to $\\Bar{J}$, resp $F$. We have $C_f=C_{f\\upharpoonright\\Bar{J}}+C_{f\\upharpoonright F}$. By the observation given after Lemma \\ref{Nonisoplc}, we get $\\sum C_f=\\sum C_{f\\upharpoonright\\Bar{J}}\\mp\\sum C_{f\\upharpoonright F}$. \n\nTo conclude, it suffices to prove that $\\sum C_{f\\upharpoonright\\Bar{J}}$ embeds into $\\sum C_{\\upharpoonright J}$ and $\\sum C_{f\\upharpoonright F}$ embeds into $\\sum C_{\\upharpoonright F}$. Only the first statement needs a proof because $f$ does not affect $F$. In order to prove it, we define an auxiliary labelled chain $D=(M,d)$ as follows: the domain $M$ is $\\underline{2}.J\\cup X$ where $X=\\{b_n, c_n : n<\\omega\\}$, $b_n$ covers $(1,a_n)$ and $c_n$ covers $b_n$. The labelling $d$ is defined by $d(i)=\\ell(j)$ if $i=(m,j)\\in \\underline{2}.J$ and $d(i)=\\ell_f(i)$ if $i\\in X$. We prove that the following inequalities hold.\n$$\\sum C_{f\\upharpoonright\\Bar{J}}\\leq\\sum D\\leq \\sum\\underline{2}.C_{\\upharpoonright J}\\leq \\sum C_{\\upharpoonright J}.$$\n\nFor the first inequality, define $h:\\Bar{J}\\to \\underline{2}.J\\cup X$ by $h(j)=(0,j)$ for $j\\in J$ and $h(i)=i$ for $i\\in X$. Then $h$ is an embedding and due to the definition of $d$, it follows that $C_{f\\upharpoonright \\Bar{J}}\\leq D$ and consequently $\\sum C_{f\\upharpoonright \\Bar{J}}\\leq \\sum D$.\n\nFor the second inequality, recall that $C_{\\upharpoonright J}$ is left indecomposable. It can be verified that $\\underline{2}.C_{\\upharpoonright J}$ is left indecomposable.  Thus, by Lemma \\ref{Labelledchainsproperties} (2), we can write $\\underline{2}.C_{\\upharpoonright J}$ as $\\sum^*_{n<\\omega} C_n$ where each $C_n$ is indecomposable and for each $n<\\omega$, the set of $m$ such that $C_n$ embeds into $C_m$ is infinite. For each $n<\\omega$, let $J_n$ be the domain of $C_n$ and let $L_n:=J_n\\cup \\{b_m, c_m : a_m\\in J_n\\}$. Set $D_n:=(L_n, \\ell_{f\\upharpoonright L_n})$. Then, $D=\\sum^*_{n<\\omega} D_n$. We define an embedding from $\\sum D$ to $\\sum \\underline{2}.C_{\\upharpoonright J}$ by induction. Let $n<\\omega$. First we show that the poset labelled sum $\\sum D_n$ embeds in the poset labelled sum obtained by a finite sum of $C_m$, that is $\\sum D_n\\leq \\sum(C_{m+k}\\cdots+C_m)$ for some $m, k<\\omega$. The  labelled chain $D_n$ consists of $C_n$ plus finitely many elements, each labelled by a 2 or 4-element chain or antichain. Thus, $D_n$ can be written \n\\begin{multline*} \nD_{n,0}+\\alpha_{(0,0)}+\\alpha_{(1,0)}+D_{n,1}+\\alpha_{(0,1)}+\\alpha_{(1,1)}+\\cdots+D_{n,k_n-1}+ \\\\ \\alpha_{(0,k_n-1)}+  \\alpha_{(1,k_n-1)}+D_{n,k_n}\n\\end{multline*}\n\nwith $D_{n,0}+\\cdots+D_{n,k_n}=C_n$ and for each $0\\leq i\\leq k_n-1$, $\\alpha_{(0,i)}$, resp $\\alpha_{(1,i)}$, is a singleton belonging to $X:=\\{b_n, c_n : n\\in\\mathbb{N}\\}$ labelled by a 2 or 4-element antichain, resp chain.  Suppose that the poset labelled sum\n$$\\sum(D_{n,t} + \\cdots+D_{n,k_n-1}+\\alpha_{(0,k_n-1)}+\\alpha_{(1,k_n-1)}+D_{n,k_n})$$\nwhere $0< t \\leq k_n$, has been embedded in $\\sum(C_{\\varphi(t)}+\\cdots+C_m)$ for some $m$. \nWe know that $\\alpha_{(1,t-1)}$ is an $l$-element chain where $l=2$ or 4. Since $C_n$ embeds in infinitely many $C_m$, the coinitial sequence $(a_n)_{n<\\omega}$ has infinitely many elements with the same label of $c_{n,t-1}$ where $c_{n,t-1}\\in X$ is the singleton corresponding to $\\alpha_{(1,t-1)}$. Select $l$ elements $j_1, \\ldots, j_l$ of $J$ such that $j_1,\\ldots,j_l<L_{n,t}$ where $L_{n,t}$ is the domain of $D_{n,t}$ and $r(j_s)=r_f(c_{n,t-1})$. Without loss of generality assume that $j_l < \\cdots < j_1$. If $r(j_s)=-1$, resp $+1$, then, $P_{l1}:=P_{j_l}+\\cdots+P_{j_1}$, resp $P_{1l}:=P_{j_1} + \\cdots + P_{j_l}$, contains a chain of size $l$. Thus, $\\alpha_{(1,t-1)}$ embeds in $P_{l1}$, resp $P_{1l}$. Similarly, for $\\alpha_{(0,t-1)}$ which is an $l$-element antichain, choose $s_l < \\cdots < s_1\\in J$ with $s_1 < j_l$ and $r(s_k)=0$. Such elements exist since $I_0=\\{i\\in I : r(i)=0\\}$ is coinitial in $I$. It is clear that $\\alpha_{(0,t-1)}$ embeds in $P_{s_1}\\oplus \\cdots \\oplus P_{s_l}$. Now, consider $D_{n,t-1}$. We have $D_{n,t-1}\\leq C_n$. Since the set of $m$ such that $C_n$ embeds in $C_m$ is infinite, we can find $p$ such that $\\sum (D_{n,t-1}+\\cdots+D_{n,k_n})$ embeds in $\\sum(C_{m+p}+\\cdots+C_m)$. Then set $\\varphi(t-1)=m+p$. This proves the induction step. Thus, $\\sum D_n$ embeds in $\\sum(C_{m+k}+\\cdots+C_m)$ for some $k$.  \n\nNow, let $n<\\omega$ and suppose that the poset labelled sum $\\sum(D_{n-1}+\\cdots+D_0)$ has been embedded in $\\sum(C_{\\varphi(n-1)}+\\cdots+C_0)$. By the argument above, $\\sum D_n\\leq \\sum (C_{m+k}+\\cdots+C_m)$ for some $m>\\varphi(n-1)$ and $k$. Hence, $\\sum(D_n+\\cdots+D_0)\\leq \\sum(C_{m+k}+\\cdots+C_0)$. Set $\\varphi(n)=m+k$. This proves the second inequality.\n\nFor the last inequality, since $C_{\\upharpoonright J}$ is left indecomposable, by Lemma \\ref{ncopies} we have $\\underline{2}.C_{\\upharpoonright J}\\leq C_{\\upharpoonright J}$ which implies that $\\sum \\underline{2}.C_{\\upharpoonright J}\\leq \\sum C_{\\upharpoonright J}$. \n\\end{proof} \n\n\nLemmas \\ref{Nonisoplc} and \\ref{ReducedequimorphP} imply the following. \n\n\\begin{theorem}  \\label{Reduced}\nLet P be the sum of a countable labelled chain $C:=(I,\\ell)$ where $I$ has no least element and $\\ell(i)=(P_i,r(i))\\in \\mathcal{N}_{\\leq\\omega}\\times\\{-1, 0, +1\\}$ such that r takes 0 and $\\pm 1$ densely. Then  $Sib(P)=2^{\\aleph_0}$.    \n\\end{theorem}\n\n\n\n\n\n\\subsection{Direct Sum}\n \nThroughout this subsection $P$ is a countable direct sum of at least two non-empty connected $NE$-free posets. Thus, it is disconnected.  \nIn this section we prove that $P$ has one or infinitely many siblings on condition that this property holds for each component of $P$. \n\n\\begin{lemma} \\label{Connecteddisconnected}\nIf some sibling of $P$ is  connected, then $Sib(P)=\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nLet $P'\\approx P$ where $P'$ is connected. So, $P'$ embeds into some component $Q$ of $P$. Since $P$ has at least two non-empty components, $P'\\oplus 1\\hookrightarrow Q\\oplus 1\\hookrightarrow P\\hookrightarrow P'$. So, for every $n$, $P'\\oplus \\Bar{K}_n\\hookrightarrow P'$ where $\\Bar{K}_n$ is an antichain of size $n$. Since $P'$ is connected and $P'\\approx P'\\oplus \\Bar{K}_n$, $P'$ and consequently $P$ has infinitely many pairwise non-isomorphic siblings. \n\\end{proof}\n\n\n\\begin{lemma} \\label{Infinitecomponent} \nIf some component of P has infinitely many siblings, then \\\\ $Sib(P)=\\infty$. \n\\end{lemma}\n\n\\begin{proof}\nLet $P$ satisfy the conditions of the lemma. Let $Q$ be a  component of $P$ with infinitely many pairwise non-isomorphic siblings $Q_1, Q_2, Q_3, \\ldots$. For each $n$, let $P_n$ be the poset resulting from $P$ by replacing every component of $P$ which is equimorphic to $Q$ by $Q_n$. Now suppose that $P_m\\cong P_n$ for $m\\neq n$. Consider a component $Q'$ of $P_m$ equimorphic to $Q_m$. We know that $Q'$ is isomorphic to some component $Q''$ of $P_n$. We have $Q''\\cong Q'\\approx Q_m\\approx Q$ which implies that $Q''=Q_n$ because $Q''$ is a component of $P_n$. But then $Q_m\\cong Q_n$, a contradiction. \nTherefore, the resulting posets $P_n$ are pairwise non-isomorphic siblings of $P$. \n\\end{proof}\n\n\\begin{lemma} \\label{increasing}\nSuppose that P has infinitely many non-trivial components. Then $Sib(P)=\\infty$. \n\\end{lemma}\n\n\\begin{proof}\nSince $P$ has infinitely many non-trivial components and $\\mathcal{N}_{\\leq\\omega}$ is w.q.o, there is an increasing sequence $(P_n)_{n<\\omega}$  of non-trivial components of $P$. Let $\\mathcal{Q}$ be the direct sum of the non-trivial components of $P$ other than the $P_n$. We have $P=\\bigoplus_n P_n\\oplus \\mathcal{Q}\\oplus A$ where $A$ is the direct sum of the trivial components of $P$. Notice that since $P$ is countable, so is $A$. Therefore, $A$ embeds in $\\bigoplus_n P_{2n+1}$. Also $\\bigoplus_n P_n$ embeds into $\\bigoplus_n P_{2n}$. Hence, $P$ embeds into $P':=\\bigoplus_n P_n\\oplus \\mathcal{Q}$. That is, $P'\\approx P$. Similarly, $P'\\oplus \\Bar{K}_n \\approx P'\\approx P$ where $\\Bar{K}_n$ is an antichain of size $n$. Since $P'\\oplus\\Bar{K}_n$ has exactly $n$ trivial components, it follows that $P$ has infinitely many pairwise non-isomorphic siblings. \n\\end{proof}\n\nHence, by Lemmas  \\ref{Connecteddisconnected}, \\ref{Infinitecomponent} and \\ref{increasing} we have the following. \n\n\\begin{proposition} \\label{InfinitesiblingnumberofP}\nLet P be a countable direct sum of NE-free posets with at least two non-empty components. If P is a sibling of some connected NE-free poset, or some component of P has infinitely many siblings, or P has infinitely many non-trivial components, then $Sib(P)=\\infty$. \n\\end{proposition}\n\n\n\n\n\\begin{lemma} \\label{Finitenontrivial} \nIf  P has only finitely many non-trivial components and each component has only one  sibling, then $Sib(P)=1$.\n\\end{lemma}\n\n\\begin{proof}\nSet $P:=\\bigoplus_{i<l}P_i\\oplus A$ where each $P_i$ is non-trivial and $A$ is the direct sum of the trivial components of $P$. If $l=0$, then $P$ is an antichain and $Sib(P) = 1$. Assume that $l>0$ and that $P' \\subseteq P$ induces a sibling\nof $P$ via an embedding $f$. We prove that $f$ induces a bijection on the set of indices of components of $P$. First note that since the components of $P$ are connected, for each $i$, there is $j$ such that $f(P_i)\\subseteq P_j$. For each $i$, define $\\hat{f}(i)=j$ where $j$ is such that $f(P_i)\\subseteq P_j$. Suppose that for $i\\neq j$, $\\hat{f}(i)=\\hat{f}(j)=k$. It follows that $f$ embeds $P_i\\oplus P_j$ in $P_k$. We first show that $k$ cannot be $i$ or $j$. Suppose, without loss of generality, that $k=i$. Then $P_i\\oplus 1\\hookrightarrow P_i\\oplus P_j\\hookrightarrow P_i$ and by Lemma  \\ref{Connecteddisconnected}, $P_i$ has infinitely many siblings, a contradiction. So, $k\\neq i, j$. It follows that $\\hat{f}(k)\\neq k$ because otherwise $f$ embeds $P_i\\oplus P_k$ in $P_k$ which implies that $P_k$ has infinitely many siblings, a contradiction. Further, $\\hat{f}(k)\\neq i, j$ because otherwise $P_i\\oplus P_j\\approx P_k$. By a similar argument, $\\hat{f}^2(k)\\neq i, j, k, \\hat{f}(k)$. Iterating, the $P_{\\hat{f}^n(k)}$ provide infinitely many non-trivial components  of $P$, a contradiction. Thus, $\\hat{f}$ is injective on the set of indices of non-trivial components of $P$ and since there are finitely many such indices, $\\hat{f}$ is bijective. Extending $\\hat{f}$ to the set $I$ of indices of components of $P$, it follows that $\\hat{f}$ is bijective on $I$. Pick $i<l$ and consider the orbit $\\hat{f}.i$ of $i$ under $\\hat{f}$. We have $P_i\\approx P_j$ where $j\\in \\hat{f}.i$ and by assumption, $P_i\\cong P_{\\hat{f}.i}$ for each $i<l$. Hence $P'$ is isomorphic to $P$. \n\\end{proof}\n\n\nNow we are ready to conclude the following. \n\n\\begin{proposition} \\label{Directsumsiblings}\nLet $P$ be a countable direct sum of at least two non-empty connected NE-free posets. If each component of P has one or infinitely many siblings, then $Sib(P)=1$ or $\\infty$. \n\\end{proposition}\n\n\\begin{proof}\nIf $P$ has some component with infinitely many siblings or $P$ has infinitely many non-trivial components, then $Sib(P)=\\infty$ by Proposition \\ref{InfinitesiblingnumberofP}. If $P$ has only finitely many non-trivial components each of which has only one sibling, then $Sib(P)=1$ by Lemma \\ref{Finitenontrivial}. This completes the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Linear Sum} \\label{Linearsum}\n\n\nBy Theorem \\ref{Pstructure}, we know that an $NE$-free poset with more than one element which is not a direct sum or the sum of a labelled chain, is a linear sum $P$ of at least two non-empty $NE$-free posets with property CCGC. In this section we prove that such a countable linear sum $P$ of $NE$-free posets has one or infinitely many siblings on condition that this property holds for each summand of $P$. In this section, by a linear sum, we always mean a linear sum of $NE$-free posets consisting of summands with property CCGC. \n\n\\begin{lemma} \\label{Uniquesummand}\nLet $P=+_{i\\in I} P_i$ and $Q=+_{j\\in J} Q_j$ be two linear sums. Every embedding f from P to Q induces an order preserving map $\\hat{f}:I\\to J$.  Moreover, If f maps distinct summands of P into distinct summands of Q, resp f is an isomorphism, then $\\hat{f}$ is a chain embedding, resp a chain isomorphism.  \n\\end{lemma}\n\n\\begin{proof}\nWe show that for every $i\\in I$ there is $j\\in J$ such that $f(P_i)\\subseteq Q_j$. Let $i\\in I$ be given. Pick $x, y\\in P_i$. Set $G_i=CG(P_i)$. Since $G_i^c$ is connected, $x, y$ are connected in $G_i^c$ by a path $x=x_0, x_1, \\ldots, x_{n-1}=y$. This means that $x_m\\perp x_{m+1}$ in $P_i$ for every $0\\leq m\\leq n-2$. We have $f(x_m)\\perp f(x_{m+1})$ for every $0\\leq m\\leq n-2$. Now suppose that $f(x)\\in Q_j$ for some $j\\in J$. Since the summands of $Q$ are comparable to each other, $f(x_m)\\in Q_j$ for every $0\\leq m\\leq n-1$ because otherwise for two $m, m+1$, $f(x_m)$ and $f(x_{m+1})$ are comparable. This implies that $f(P_i)\\subseteq Q_j$. Now define $\\hat{f}:I\\to J$ by $\\hat{f}(i)=j$ where $j\\in J$ is such that $f(P_i)\\subseteq Q_j$. The mapping $\\hat{f}$ is well-defined by above argument. If $i \\leq_I i'$, then $P_i \\leq_P P_{i'}$ which implies that $f(P_i)\\leq_Q f(P_{i'})$ because $f$ is order-preserving. Consequently, $Q_j \\leq_Q Q_{j'}$ where $f(P_i)\\subseteq Q_j$ and $f(P_{i'})\\subseteq Q_{j'}$. Hence, $\\hat{f}(i) \\leq_J \\hat{f}(i')$. Moreover, if $f$ does not embed two distinct summands of $P$ into the same summand of $Q$, then $\\hat{f}$ is injective meaning that it is a chain embedding. In particular, if $f$ is an isomorphism, then so is $f^{-1}$ and for every $i$ there is $j$ such that $f(P_i)\\subseteq Q_j$ and $f^{-1}(Q_j)\\subseteq P_i$ which imply that $f(P_i)=Q_j$. Thus, $\\hat{f}$ defined as above is a chain isomorphism. \n\\end{proof}\n\n\n\n\\begin{lemma} \\label{InfiniteCproperty}\nLet P be a linear sum. Suppose that some summand of P has infinitely many pairwise non-isomorphic siblings with property CCGC. Then P has infinitely many siblings.\n\\end{lemma} \n\n\\begin{proof}\nSet $P=+_{i\\in I} P_i$.  Let $P_j$ be a summand  of $P$ such that  $\\{P_{jn}\\}_{n<\\omega}$ is an infinite family of pairwise non-isomorphic siblings of $P_j$ with property CCGC. Let $J=\\{i\\in I : P_i\\approx P_j\\}$. For each $n<\\omega$, set $P^n:=+_{i\\in I} Q_i^n$ where $Q_i^n=P_i$ if $i\\notin J$ and $Q_i^n=P_{jn}$ if $i\\in J$. Notice that for each $i\\in I$, $Q_i^n$ has property CCGC. Taking each summand $P_i$ to the summand $Q_i^n$ and vice versa, it is clear that $P^n\\approx P$ for every $n<\\omega$. Now suppose that $P^m\\cong P^n$ for $m\\neq n$ and let $f:P^m\\to P^n$ be an isomorphism. Since $f$ is an isomorphism and both $P^m, P^n$ are linear sums whose summands have property CCGC, by Lemma \\ref{Uniquesummand}, $f$ induces a chain isomorphism $\\hat{f}:I\\to I$ and for each $i\\in I$, $f(Q_i^m)= Q^n_{\\hat{f}(i)}$. More, we have $Q_i^m\\cong Q_{\\hat{f}(i)}^n$. Let $i\\in J$. Then $P_{jm}= Q_i^m \\cong Q_{\\hat{f}(i)}^n = P_{jn}$, a contradiction. The last equality is due to the fact that $Q^n_{\\hat{f}(i)}\\approx P_{jm}\\approx P_j$. This implies that the $P^n$ are pairwise non-isomorphic. Hence, $Sib(P)=\\infty$. \n\\end{proof}\n\nThe following proposition provides sufficient conditions to obtain infinitely many siblings for a countable linear sum. \n\n\\begin{proposition} \\label{InfiniteDproperty}\nLet P be a countable linear sum. If some summand $P_j$ of P is the sum of a labelled chain with no least element; or $P_j$ is disconnected and has infinitely many siblings with property CCGC; or $P_j$ is disconnected and has at least one connected sibling, then P has infinitely many siblings. \n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{Plsccgc} the sum of a countable labelled chain with no least element has property CCGC and by Theorem \\ref{Reduced} it has continuum many siblings with the same properties. If $P$ contains a summand $P_j$ which is the sum of a labelled chain with no least element or $P_j$ is disconnected having  infinitely many siblings with property CCGC, then the statement is true by Lemma \\ref{InfiniteCproperty}.\n\nAssume that there is some  \ndisconnected summand $P_j$ of $P$ which is equimorphic to some connected $NE$-free poset $Q$. Then $Q\\approx P_j'$ for some connected component $P_j'$ of $P_j$. Consequently, the $\\Bar{K}_n\\oplus Q$, $n<\\omega$, where $\\Bar{K}_n$ is an antichain of size $n$, are pairwise non-isomorphic siblings of $Q$ and thus of $P_j$ which have property CCGC. Hence, in this case $P$ has infinitely many siblings by Lemma \\ref{InfiniteCproperty}. \n\\end{proof}\n\nLet $P$ be a countable linear sum with more than one element such that all its summands have only one sibling. Then $P$ does not contain a summand which is the sum of a labelled chain with no least element. That is, each summand of $P$ is either a singleton or a disconnected $NE$-free poset. Obviously, $P$ does not embed in a singleton and if $P$ embeds in a disconnected summand $P_j$, then $P_j$ has infinitely many siblings by Lemma \\ref{Connecteddisconnected} which contradicts our assumption. In other words, each summand of $P$ strictly embeds into $P$. In case a linear sum has only finitely many non-trivial summands having only one sibling, we show that $P$ has one or infinitely many siblings. However, in light of the following result, we prove a stronger statement.\n\n\\begin{theorem} [\\cite{LPW},  Corollary 3.6] \\label{Dichchain}\nIf C is a countable chain, then $Sib(C)=1$, $\\aleph_0$ or $2^{\\aleph_0}$.  \n\\end{theorem}\n\n\n\\begin{lemma} \\label{Finitesummand}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum with only finitely many non-trivial summands each of which has only one sibling. Then $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$. \n\\end{lemma}\n\n\\begin{proof}\nLet $f$ be an embedding of $P$ and $\\hat{f}$ be defined as in Lemma \\ref{Uniquesummand}. Let $K$ be the chain of indices $i\\in I$ such that $P_i$ is  non-trivial. If $K=\\emptyset$, then $P=I$  and there is nothing to prove (the statement holds for a countable chain by Theorem \\ref{Dichchain}). Assume that $K\\neq\\emptyset$. We know that $\\hat{f}$ is order-preserving. We show that $\\hat{f}$ is injective. For the sake of a contradiction, assume that $\\hat{f}(i)=\\hat{f}(j)=k$ for some $i < j\\in I$. It implies that $k\\in K$. Since $P_k$ is non-trivial, $\\hat{f}(k), \\hat{f}^2(k), \\ldots\\in K$. Since $K$ is a finite chain and $\\hat{f}$ is order-preserving, we conclude that for some $l\\in K$ and some integer $m$, $\\hat{f}^m(k)=\\hat{f}^{m+1}(k)=l$. Without loss of generality, assume that $i\\leq_I l$. Let $\\Bar{I}=\\{i'\\in I : i\\leq_I i' \\leq_I l\\}$. We have $\\hat{f}^{m+1}(\\Bar{I})=l$. Now consider $P'=+_{i' \\in \\Bar{I}} P_{i'}$. From $\\hat{f}^{m+1}(\\Bar{I})=l$, we conclude that $f^{m+1}$ embeds $P'$ into $P_l$. Clearly, $P_l$ embeds into $P'$. That is $P'\\approx P_l$. Since $l\\in K$, $P_l$ is disconnected. Also, $P_l$ is equimorphic to a connected $NE$-free poset $P'$ which implies that $Sib(P_l)=\\infty$ by Lemma \\ref{Connecteddisconnected}, a contradiction. Therefore, $\\hat{f}$ is a chain embedding of $I$. Further, since $K$ is finite, $\\hat{f}$ is the identity on $K$. Consequently, $f(P_k)\\subseteq P_k$ for each $k\\in K$.\n\nSet $K:=\\underline{n}$. We may represent $P$ as $P=I_0+P_0+\\cdots+P_{n-1}+I_n$ where the $P_k$ are the non-trivial summands of $P$, $I_0=\\{x\\in P : \\{x\\} <_P P_0\\}$,  $I_k=\\{x\\in P : P_{k-1} <_P \\{x\\} <_P P_k\\}$ for every $1\\leq k\\leq n-1$ and $I_n=\\{x\\in P : P_{n-1} <_P \\{x\\}\\}$. Notice that each $I_k$ is a countable chain. By the above argument, each sibling of $P$ is of the form $Q=J_0+Q_0+\\cdots+Q_{n-1}+J_n$ where $J_k\\approx I_k$, $k\\in\\underline{n+1}$, and $Q_k\\approx P_k$, $k\\in\\underline{n}$.  Since $Sib(P_k)=1$, we have $Q_k\\cong P_k$. Moreover, each countable chain has one or countably many or continuum many siblings by Theorem \\ref{Dichchain}.  Hence, $Sib(P)=\\max\\{Sib(I_k) : k\\in\\underline{n+1}\\}$. It follows that $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$.  \n\\end{proof}\n\n\n\n\nIt remains to verify the case in which a linear sum contains infinitely many non-trivial summands, each having only one sibling. However, we will show that when a linear sum has infinitely many non-trivial summands, it has continuum many siblings, regardless of the sibling number of its summands. \nWe may consider a linear sum $P=+_{i\\in I}P_i$ as the sum $\\sum C$ of a labelled chain $C=(I,\\ell)$ where $\\ell:I\\to \\mathcal{N}_{\\leq\\omega}\\times \\{-1\\}$ is defined by $\\ell(i)=(P_i,r(i))$ and $r(i)=-1$ for each $i\\in I$. Since the map $r$ is constant, we may abuse the notation and write $\\ell(i)=P_i$. Note that the labels belong to a b.q.o. We denote by $K$ the chain of $i\\in I$ such that $P_i$ is non-trivial. Our proof contains two main ingredients: Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}. \n\nLet $n\\geq 1$ and consider the chains $\\underline{n}$ and $\\underline{n+2}$. We denote by $A_n$, resp $B_n$, the poset obtained by replacing each $m\\in \\underline{n}$, resp $0 < m < n+1$, with a two-element antichain. Note that each $A_n$ and $B_n$ is a linear sum whose summands have property CCGC with context chains $\\underline{n}$ and $\\underline{n+2}$, respectively.  \nTwo elements $i$ and $j$ of $K$ are equivalent, denoted by $i\\equiv_K j$, just in case every element of the interval in $I$ with endpoints $i$ and $j$ is in $K$. It is clear that $\\equiv_K$ is an equivalence relation. We denote by $\\mathcal{K}$ the set of equivalence classes of $K$. The equivalence classes of $K$ are maximal intervals of $I$ that are contained in $K$. \n\n\\begin{lemma} \\label{ConstructionLS}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum such that K is coinitial in I and without least element. Then, there are $2^{\\aleph_0}$ pairwise non-isomorphic linear sums $P_f=\\sum C_f$ where $f\\in \\{0,1\\}^\\mathbb{N}$. \n\\end{lemma}\n\n\\begin{proof}\nIt follows that $I$ has no least element. Set $C:=(I,\\ell)$ where $\\ell(i)=P_i$. \nRecall that in this case, by Lemma \\ref{Labelledchainsproperties} (1), there is an initial segment $J$ of $I$ such that $C_{\\upharpoonright J}$ is left indecomposable. For each $K'\\subset J$ with $K'\\in\\mathcal{K}$, if $|K'|=2$ or $|K'|=4$, then insert a new element $d$ such that $d$ covers the greatest element of $K'$ and set $\\ell'(d):=A_1$. Thus, $C_{\\upharpoonright K'}$ is replaced with a  labelled chain which is isomorphic to $A_3$ or $A_5$. Let $Y$ be the collection of all these new elements $d$. Set $J':=J\\cup Y$.   \nWe select an infinite descending sequence $(a_n)_{n<\\omega}$ of elements of $K\\cap J$ coinitial in $J$. To each map $f:\\mathbb{N}\\to\\{0,1\\}$ we associate a  labelled chain $C_f$ as follows. Let $f:\\mathbb{N}\\to\\{0,1\\}$. For $i\\in I\\setminus Y$ set $\\ell_f(i):=\\ell(i)$ and for $i\\in Y$ set $\\ell_f(i):=\\ell'(i)$.  For each $n<\\omega$, set $k(n):=2f(n)+4$ and insert elements $b_{1,n} < \\cdots < b_{k(n),n}$ such that $b_{1,n}$ covers $a_n$ and $b_{i+1,n}$ covers $b_{i,n}$ for each $1\\leq i\\leq k(n)-1$. Set $\\ell_f(b_{1,n}):=b_{1,n}$, $\\ell_f(b_{k(n),n}):=b_{k(n),n}$ and $\\ell_f(b_{i,n}):=A_1$ where $1<i<k(n)$. Indeed, the chain $b_{1,n}<\\cdots<b_{k(n),n}$ is the context chain of $B_{2f(n)+2}$. Let $b_n$ be the chain $b_{1,n}<\\cdots<b_{k(n),n}$. Then, $(b_n)_{n<\\omega}$ is a sequence of chains such that $b_{n+1}<b_n$.  Set $X:=\\bigcup_{n<\\omega} b_n$ which is coinitial in $J$. Set $\\Bar{J}:=J'\\cup X$, $F:=I\\setminus \\Bar{J}$ and $\\Bar{I}:=F\\cup \\Bar{J}$. Then, $C_f:=(\\Bar{I},\\ell_f)$ is a labelled chain. Thus, we obtain a sum $\\sum C_f$ which is a linear sum whose summands have property CCGC. That is, $\\sum C_f=+_{i\\in\\Bar{I}}P_i$ such that $K$ has no least element and $K$ is coinitial in $\\Bar{I}$.\n\nSuppose that $f, g$ are two maps from $\\mathbb{N}$ to $\\{0,1\\}$ such that $\\sum C_f\\cong \\sum C_g$ by some isomorphism $\\phi$. Since the summands of these linear sums have property CCGC,  by Lemma \\ref{Uniquesummand}, $\\phi$ induces an isomorphism $h$ on $\\Bar{I}$ preserving the labels, that is $\\ell_f(i)=\\ell_g(h(i))$ for each $i\\in \\Bar{I}$. If for each $n<\\omega$, $h(b_n)\\subset F$, then $h(\\Bar{I})\\subseteq F$ because $X$ is coinitial in $\\Bar{I}$. But this means that $h$ is not surjective, a contradiction. Therefore, $h(b_n)\\cap \\Bar{J}\\neq\\emptyset$ for some $n$. Since each $b_n$ is the context chain of $B_2$ or $B_4$ and since there is no  labelled chain $C_{\\upharpoonright K'}$, $K'\\in\\mathcal{K}$, in $C_{g\\upharpoonright\\Bar{J}}$ which is isomorphic to $A_2$ or $A_4$, it follows that $h(b_n)=b_m$ for some $m$ and consequently, for each $k\\geq 1$, $h(b_{n+k})=b_{m+k}$. Thus, $h$ may be regarded as an isomorphism from a final segment $D$ of $\\mathbb{N}$ onto another $D'$ such that $g(h(n))=f(n)$ for each $n\\in D$. By Lemma \\ref{Continuummaps}, there are $2^{\\aleph_0}$ maps in $\\{0,1\\}^\\mathbb{N}$ such that for two distinct maps $f$ and $g$, we have $\\sum C_f\\ncong \\sum C_g$. For each $f\\in\\{0,1\\}^\\mathbb{N}$, set $P_f:=\\sum C_f$. Then, the set $\\{P_f\\}_{f\\in\\{0,1\\}^\\mathbb{N}}$ has the desired properties. \n\\end{proof}\n\n\nIn lemma \\ref{ConstructionLS}, we observe that the order on $P_f:=\\sum C_f$ extends both $\\sum C_{f\\upharpoonright\\Bar{J}}$ and $\\sum C_{\\upharpoonright F}$ as follows: if $x\\in\\sum C_{f\\upharpoonright\\Bar{J}}$ and $y\\in \\sum C_{\\upharpoonright F}$, then $x\\in P_i$ and $y\\in P_j$ for some $i\\in \\Bar{J}$ and $j\\in F$. We have $i<j$ and since $r(i)=-1$, it follows that $x <_{P_f} y$. Therefore,  $\\sum C_f =\\sum C_{f\\upharpoonright\\Bar{J}}+\\sum C_{\\upharpoonright F}$.\n\n\n\nIn order to show that the $\\sum C_f$ in Lemma \\ref{ConstructionLS} are embeddable into $\\sum C$, first we prove the following claim.\n\n\\begin{claim} \\label{J'embedsJ}\nWe have $C_{\\upharpoonright J'}\\leq C_{\\upharpoonright J}$. Moreover, $C_{\\upharpoonright J'}$ is left indecomposable.\n\\end{claim}\n\n\\begin{proof}\nWe show that the following inequalities hold.\n$$ C_{\\upharpoonright J'}\\leq  \\underline{2}.C_{\\upharpoonright J}\\leq  C_{\\upharpoonright J}.$$\n\nFor the first inequality, recall $Y$ in Lemma \\ref{ConstructionLS}  which is countable. Let $d_0, \\ldots, d_n, \\ldots$ be an enumeration of $Y$. For each $n$, let $c_n\\in J$ be such that $d_n$ covers $c_n$. Therefore, $c_n\\in K$. Define an embedding $\\varphi:C_{\\upharpoonright J'}\\to\\underline{2}.C_{\\upharpoonright J}$ as follows. Define $f:J'\\to\\underline{2}.J$ by $f(j)=(0,j)$ if $j\\in J$ and $f(d_n)=(1,c_n)$ for each $n$. Since $\\ell'(d_n)$ is a 2-element antichain and $\\ell((1,c_n))=\\ell(c_n)$ is a direct sum of at least two non-empty posets, it follows that $\\ell'(d_n)$ embeds in $\\ell(f(d_n))$. Therefore, $C_{\\upharpoonright J'}\\leq \\underline{2}.C_{\\upharpoonright J}$. \n\nThe second inequality is because $\\underline{2}.C_{\\upharpoonright J}\\leq C_{\\upharpoonright J}$, a fact which follows from Lemma \\ref{ncopies} since $C_{\\upharpoonright J}$ is left indecomposable.  \n\nWe know that $C_{\\upharpoonright J}$ is left indecomposable. Let $L'+R'$ be a partition of $J'$ into an initial and a final segment and set $L:=L'\\setminus Y$ and $R:=R'\\setminus Y$. Then $L+R$ is a partition of $J$ into an initial and a final segment. By the inequality above and the fact that $C_{\\upharpoonright J}$ is left indecomposable, we have $C_{\\upharpoonright J'}\\leq C_{\\upharpoonright J}\\leq C_{\\upharpoonright L}\\leq C_{\\upharpoonright L'}$. Thus, $C_{\\upharpoonright J'}$ is left indecomposable. \n\\end{proof}\n\n\n\n\\begin{lemma} \\label{Noleastlinearcontinuum}\nLet $P=+_{i\\in I}P_i$ be a countable linear sum such that K is coinitial in I and has no least element. Then $Sib(P)=2^{\\aleph_0}$. \n\\end{lemma}\n\n\\begin{proof}\nLet $f:\\mathbb{N}\\to\\{0,1\\}$ and $C_f$ be the labelled chain constructed in Lemma \\ref{ConstructionLS}. We prove that $\\sum C_f\\leq \\sum C$. We have $C_f= C_{f\\upharpoonright \\Bar{J}}+C_{\\upharpoonright F}$ because $f$ does not change $F$. By the observation given after Lemma \\ref{ConstructionLS} we have $\\sum C_f =\\sum C_{f\\upharpoonright\\Bar{J}}+\\sum C_{\\upharpoonright F}$. It suffices to prove that $\\sum C_{f\\upharpoonright \\Bar{J}}\\leq \\sum C_{\\upharpoonright J}$.  We prove that \n$\\sum C_{f\\upharpoonright \\Bar{J}}\\leq \\sum C_{\\upharpoonright J'}$ because then the result follows from Claim \\ref{J'embedsJ}. \n\nBy Claim \\ref{J'embedsJ}, $C_{\\upharpoonright J'}$ is left indecomposable, so, we can write it as $\\sum^*_{n<\\omega} C_n$ where each $C_n$ is indecomposable and the set of $m$ such that $C_n\\leq C_m$ is infinite. \nFor each $n$, let $J_n$ be the domain of $C_n$. Set $L_n:=J_n\\cup\\bigcup_{a_m\\in J_n} b_m$ and $D_n:=(L_n,\\ell_{f\\upharpoonright L_n})$. Then we have $C_{f\\upharpoonright \\Bar{J}}=\\sum^*_{n<\\omega} D_n$. We define an embedding from $\\sum C_{f\\upharpoonright \\Bar{J}}$ into $\\sum C_{\\upharpoonright J'}$ by induction. Let $n$ be given. Suppose that $\\sum(D_{n-1}+\\cdots+D_0)$ has been embedded into $\\sum(C_{\\varphi(n-1)}+\\cdots+C_0)$. The labelled chain $D_n$ consists of $C_n$ plus finitely many elements, each labelled by a singleton or a 2-element antichain. Thus, $D_n$ can be written\n$D_{n,0}+\\alpha_{n,0}+D_{n,1}+\\cdots+\\alpha_{n,k_n-1}+D_{n,k_n}$\nwith $D_{n,0}+D_{n,1}+\\cdots+D_{n,k_n}=C_n$ and each $\\alpha_{n,i}$ is a finite chain whose elements are labelled by either a singleton or an antichain of size 2. Suppose that $\\sum(D_{n,i}+\\alpha_{n,i}+\\cdots+\\alpha_{n,k_n-1}+D_{n,k_n})$ has been embedded into $\\sum(C_{m+k}+\\cdots+C_m)$ for some $m > \\varphi(n-1)$ and some $k$. We know that the sequence $(a_n)_{n<\\omega}$ is coinitial in $J'$ and the $P_{a_n}$ are non-trivial. Pick a chain $a_{n_i}<\\cdots<a_{n_0}< J_{m+k}$ of elements of $(a_n)_{n<\\omega}$ such that its length equals to the size of the context chain of $\\alpha_{n,i-1}$.  Then, $\\alpha_{n,i-1}$ embeds in $P_{a_{n_i}}+\\cdots+P_{a_{n_0}}$.  We have $D_{n,i-1}\\leq C_n$. Further, $C_n$ embeds in infinitely many $C_m$. Therefore, we can find sufficiently large $l$ such that $J_l < a_{n_i}$ and $D_{n,i-1}$ embeds in $C_l$. Hence, $\\sum D_n$ embeds in $\\sum(C_{m+p}+\\cdots+C_m)$ for some $p$. Then by setting $\\varphi(n)=m+p$ we have $\\sum(D_n+\\cdots+D_0)\\leq \\sum(C_{\\varphi(n)}+\\cdots+C_0)$. This completes the induction step as well as the proof.\n\\end{proof}\n\nLet $P$ be a poset. It is easy to verify that $P$ and $P^*$ have the same sibling number.  With this observation, if in a countable linear sum $P=+_{i\\in I} P_i$, the chain $K$ has no greatest element and it is cofinal in $I$, then we may consider $P^*$ in which $K^*$ has no least element and it is coinitial in $I^*$. Then, by applying Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}, we get continuum many siblings for $P^*$ and thus for $P$. \n\nIn both Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}, $K$ has no least (greatest) element and it is coinitial (cofinal) in $I$. This is not necessarily the case in a general linear sum. However, the following lemma implies that we can always find some interval $J$ of $I$ in which $K$ is coinitial (cofinal) in $J$ with no least (greatest) element.  \n\n\\begin{lemma} \\label{Interval}\nLet I be an infinite chain. Then there exists an interval J of I such that J has no least or greatest element. \n\\end{lemma}\n\n\\begin{proof}\nFor the sake of a contradiction, assume that for each interval $J$ of $I$, $J$ has least and greatest element. Set $I_0:=I$ and let $x_0$, resp $y_0$, be the least, greatest, element of $I_0$. For $n\\geq 1$, set $I_{n+1}:=I_n\\setminus\\{x_n,y_n\\}$ and let $x_{n+1}$, resp $y_{n+1}$, be the least, resp greatest, element of $I_{n+1}$. Since $I$ is infinite, $I_n\\neq\\emptyset$ for every $n<\\omega$. Then, the initial interval $J:=\\{x\\in I : \\exists n, x < x_n\\}$ has no greatest element, a contradiction. \n\\end{proof}\n\n\n\n\\begin{proposition} \\label{Infinitesummand}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum with infinitely many non-trivial summands. Then, P has continuum many siblings. \n\\end{proposition}\n\n\\begin{proof}\nLet $K$ be the chain of indices $i\\in I$ such that $P_i$ is non-trivial. By assumption, $K$ is countably infinite. Let $I_1$, resp $I_2$, be the maximal interval of $I$ such that $I_1<_I K$, resp $K<_I I_2$. Then $I=I_1+\\hat{I}+I_2$ where $\\hat{I}$ is the minimal interval of $I$ containing $K$. By Lemma \\ref{Interval}, without loss of generality, there exists a final interval $J$ of $\\hat{I}$ and a final interval $\\hat{K}$ of $K$ such that $\\hat{K}$ has no least element and it is coinitial in $J$. Thus, if we set \n$$J:=\\{i\\in\\hat{I} : \\exists (k_n)_{n<\\omega},\\ \\forall n,\\  k_n\\in K\\ \\text{and}\\ k_{n+1} < k_n < i\\},$$\nthen $J\\neq\\emptyset$ and it is maximal among all final intervals of $\\hat{I}$ in which there is a descending sequence of elements of $K$ with no least element. Set $L:=\\hat{I}\\setminus J$. Then, $L\\cap K$ is well-ordered because otherwise there is a descending sequence of elements of $K$ in $L$ which contradicts the maximality of $J$. We have $\\hat{I}=L+J$ and also $I=I_1+L+J+I_2$. By Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}, the linear sum $Q:=+_{j\\in J} P_j$ has $2^{\\aleph_0}$ pairwise non-isomorphic siblings $\\{Q_\\alpha\\}_{\\alpha < \\mathfrak{c}}$ where $\\mathfrak{c}$ is the continuum. For each $\\alpha<\\mathfrak{c}$ set $\\hat{Q}_\\alpha:=Q_\\alpha+(+_{i\\in I_2} P_i)$. Suppose that $\\hat{Q}_\\alpha\\cong \\hat{Q}_\\beta$ for two $\\alpha\\neq \\beta$ by some isomorphism $\\phi$. Since the $\\hat{Q}_\\alpha$ are linear sums whose summands have property CCGC, by Lemma \\ref{Uniquesummand}, $\\phi$ induces an isomorphism $h$ on $J+I_2$. If for some $x\\in J$, $h(x)\\in I_2$, then $x\\notin K$. Since by assumption there is some $y\\in J\\cap K$ with $x<_I y$, we have $h(x)<_I h(y)$. But then $h(y)$ must be the index of a non-trivial summand in $+_{i\\in I_2} P_i$ which is not possible. Therefore, $h(J)=J$ meaning that $Q_\\alpha\\cong Q_\\beta$, a contradiction. Hence, the $\\hat{Q}_\\alpha$ are pairwise non-isomorphic. Next, for each $\\alpha<\\mathfrak{c}$ set $\\hat{P}_\\alpha:=(+_{i\\in L}P_i)+\\hat{Q}_\\alpha$. Suppose that $\\hat{P}_\\alpha\\cong \\hat{P}_\\beta$ for two $\\alpha\\neq\\beta$ by an isomorphism $\\phi$. By Lemma \\ref{Uniquesummand}, $\\phi$ induces an isomorphism $h$ on $L+J+I_2$. Suppose that for some $x\\in J$, $h(x)\\in L$. By definition of $J$, select a descending sequence $(k_n)_{n<\\omega}$ of elements of $K$ which has no least element in $J$ and $k_n < x$ for each $n$. Then $(h(k_n))_{n<\\omega}$ is a coinitial sequence of elements of $K$ in $L$ with no least element. But this is impossible because $L\\cap K$ is well-ordered. Thus, $h(J+I_2)=J+I_2$ meaning that $\\hat{Q}_\\alpha\\cong \\hat{Q}_\\beta$, a contradiction. Finally, for each $\\alpha<\\mathfrak{c}$ set $P_\\alpha:=(+_{i\\in I_1} P_i)+\\hat{P}_\\alpha$. By an argument similar to what we gave for the $\\hat{Q}_\\alpha$, it is proven that the $P_\\alpha$ are pairwise non-isomorphic siblings of $P$. \n\\end{proof}\n\n\n\n\nNow we are ready to conclude the main result of this section. \n\n\\begin{proposition} \\label{Linearsumsiblings}\nLet P be a countable linear sum of at least two non-empty NE-free posets with property CCGC. If each summand has one or infinitely many siblings, then, so does P. \n\\end{proposition}\n\n\\begin{proof}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum. \n\n\\noindent\\textbf{Case 1} Some summand $P_j$ has infinitely many siblings. Then either $P_j$ is the sum of a  labelled chain with no least element or $P_j$ is disconnected and has infinitely many siblings with property CCGC. In this case $Sib(P)=\\infty$ by Proposition \\ref{InfiniteDproperty}.  \n\n\\noindent\\textbf{Case 2} All summands of $P$ have only one sibling. \n\n\\noindent\\textbf{Subcase 2.1} $P$ has only finitely many non-trivial summands. In this case $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$ by Lemma \\ref{Finitesummand}. \n\n\\noindent\\textbf{Subcase 2.2} $P$ has infinitely many non-trivial summands. In this case $Sib(P)=2^{\\aleph_0}$ by Proposition \\ref{Infinitesummand}. \n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{AltThomasseNEP}}\n\nIn this short section, we prove that the alternate Thomass\\'{e} conjecture holds for a countable $NE$-free poset $P$, that is $P$ has one or infinitely many siblings.\n\n\\begin{proof} (of Theorem \\ref{AltThomasseNEP})\nLet $P$ be a countable $NE$-free poset. Note that the class of countable $NE$-free posets is w.q.o under embeddability and thus well-founded. Therefore, we can use induction. For each countable $NE$-free poset $Q$, let $\\mathcal{S}(Q)$ be the statement ``$Sib(Q)=1$ or $\\infty$\". The statement is true for a singleton. Let $P$ have more than one element and assume that $\\mathcal{S}(Q)$ holds for each $Q$ strictly embedding in $P$. By Theorem \\ref{Pstructure}, $P$ is either (1) a direct sum of at least two non-empty connected $NE$-free posets, or (2) a linear sum of at least two non-empty $NE$-free posets with property CCGC, or (3) the sum $\\sum C$ of a  labelled chain $C=(I,\\ell)$ with no least element such that each $\\ell(i)$ is the pair $(P_i,r(i))$ made of an $NE$-free poset $P_i$ and  $r(i)\\in\\{-1, 0, +1\\}$ where $r$ takes 0 and $\\pm 1$ densely. For case (3), by Theorem \\ref{Reduced} we have $Sib(P)=2^{\\aleph_0}$. Assume that $P$ satisfies (1), resp (2). Then, $P$ is disconnected, resp connected. If $P$ embeds in some of its components, resp disconnected summands, then $P$ is equimorphic to a connected, resp disconnected, $NE$-free poset and $Sib(P)=\\infty$ by Lemma \\ref{Connecteddisconnected}. Otherwise, each component, resp summand, of $P$ strictly embeds in $P$. By induction hypothesis and Proposition \\ref{Directsumsiblings}, resp Proposition \\ref{Linearsumsiblings}, $\\mathcal{S}(P)$ holds. \n\\end{proof} \n\n\n\n\\section{Finite Poset Substitution of Chains or Antichains}  \\label{Finiteposetsub}\n\n\nIn this section, we prove that a countable $NE$-free poset whose comparability graph has one sibling, is a finite poset substitution of chains or antichains and that it has one, countably many or else continuum many siblings. Moreover, the comparability graph of $NE$-free posets of this section is a finite graph substitution of cliques or independent sets. Thus, the result of this section is a generalisation of Thomass\\'e's conjecture for countable chains whose comparability graph is a clique. \n\n\nThe following theorem proven in \\cite{HPW} describes the structure of countable cographs $G$ with only one sibling which will be used in determining the structure of a poset $P$ with $CG(P)=G$.  \n\n\\begin{theorem} [\\cite{HPW}] \\label{Lexico}\nLet $G$ be a countable cograph. $G$ has only one sibling if and only if $G$ is a finite graph substitution of cliques or independent sets.\n\\end{theorem}  \n\nLet $P$ be a countable $NE$-free poset such that its comparability graph $G$ has one sibling. Then, by Theorem \\ref{Lexico}, $G=K[H_v\/v : v\\in K]$ where $K$ is a finite graph and each  $H_v$ is a clique or an independent set. We know that $G$ is a cograph, thus,\nby Proposition \\ref{Subcograph}, so is $K$. Throughout this section let $T:=T(K)$ be the decomposition tree  of $K$. For every $M\\in T$, denote by $P(M)$ the poset obtained by restricting $P$ to the graph blocks $H_v$ of $G$ where $v\\in M$. For every leaf $\\{u\\}$ of $T$ we write $P_u$ instead of $P(\\{u\\})$. We verify the following statement by induction on the vertices of $T$. For every $M\\in T$, let \n\\begin{center} \n     $\\mathcal{S}_{fps}(M)$ : $P(M)$ is a finite poset substitution of chains or antichains. \n\\end{center}\nThe statement $\\mathcal{S}_{fps}(M)$  also provides a structure of the poset $P(M)$ which will be used in the proof of the following statement: \n\\begin{center}\n    $\\mathcal{S}_{sib}(M)$ : $Sib(P(M))=1$ or $\\aleph_0$ or $2^{\\aleph_0}$.\n\\end{center}\nSince $T$ is finite, this proves $\\mathcal{S}_{fps}(M)$ and $\\mathcal{S}_{sib}(M)$ for all $M\\in T$. In particular, $P$ has one, countably many or continuum many siblings. \n\n\n \nWe say that a node $M$ of $T$ is \\textit{edge minimal} if $M$ does not have a module $\\{u, v\\}$ of $K$ satisfying the following:   \n\\begin{enumerate}\n    \\item $u$ and $v$ are adjacent and both $H_u$ and $H_v$ are cliques;\n    \\item $u$ and $v$ are non-adjacent and both $H_u$ and $H_v$ are independent sets.  \n\\end{enumerate} \nWe call $T$ \\textit{leaf minimal} if every node of $T$ is edge minimal. It follows that if $M\\in T$ is edge minimal, then for every $u, v\\in M$ with property (1) or (2), $\\{u,v\\}$ is not a module of $K$, thus, there exists some $w\\in K$ adjacent to one and only one of $u$ or $v$. Since $M$ is a module of $K$, we conclude that $w\\in M$ meaning that $\\{u,v\\}$ is not a module of $M$. Therefore, $M$ does not have two children $\\{u\\}, \\{v\\}$ as leaves of $T$ such that $u, v$ satisfy (1) or (2). \nWe say that the context graph $K$ of $G=K[H_v\/v : v\\in K]$ where the graph blocks are cliques or independent sets, is the \\textit{smallest} when $G$ cannot be represented as $K'[H'_v\/v : v\\in K']$ where $K'$ is a graph,  $|K'|<|K|$ and each $H'_v$ is a clique or an independent set. By the following lemma, it turns out that the smallest possible context graph $K$ for $G$ implies that $T$ is leaf minimal. \n\n\n\\begin{lemma} \\label{Kminimal} \nLet $G=K[H_v\/v : v\\in K]$ where $K$ is a finite cograph and each $H_v$ is either a clique or an independent set and $T$ be the decomposition tree of $K$. If $K$ is the smallest, then $T$ is leaf minimal. \n\\end{lemma}\n\n\\begin{proof} \nSuppose $T$ is not leaf minimal. Then some vertex $M\\in T$ is not edge minimal. Let $u, v\\in M$ form a module such that either they are adjacent and both $H_u, H_v$ are cliques or they do not form an edge and both $H_u, H_v$ are independent sets. In either case let $H_a$ be the graph obtained by restricting $G$ to the graph blocks $H_u$ and $H_v$. We know that in the first, resp second, case, $H_a$ is a clique, resp an independent set. Also, since $N:=\\{u, v\\}$ is a module, for $w\\in K\\setminus N$,  we have $uw\\in E(K)$ if and only if $vw\\in E(K)$. Let $K'$ be a graph whose vertex set is $V':=(K\\setminus N)\\cup \\{a\\}$ and its edge set is obtained by replacing all $wu, wv\\in E(K)$ with $wa$ where $w\\in K\\setminus N$. Let $x\\in G$. Then $x\\in H_w$ for some $w\\in K$. If $w\\in K\\setminus N$, then $w\\in V'$ and $x\\in H_w$; if $w=u$ or $w=v$, then $x\\in H_a$ and $a\\in V'$. Now, let $xy\\in E(G)$. Then either $x, y\\in H_w$, $w\\in K$, and $xy\\in E(H_w)$; or $x\\in H_w$, $y\\in H_{w'}$, $w, w'\\in K$ and $ww'\\in E(K)$. In the first case, if $w\\in K\\setminus N$, then $w\\in V'$ and $xy\\in E(H_w)$; and if $w=u$ ($w=v$), then $H_u$ ($H_v$) is a clique and we have $xy\\in E(H_a)$. In the second case when $w, w'\\in K\\setminus N$, then $w, w'\\in V'$ and $ww'\\in E(K')$; if $w\\in K\\setminus N$ and $w'\\in N$, then $wa\\in E(K')$; and finally if $w,w'\\in N$, then $x, y \\in H_a$ and $uv\\in E(K)$ which means that $xy\\in E(H_a)$.  Hence, $G=K'[H'_v\/v : v\\in K']$. Since $K'$ is a graph, $|K'|<|K|$ and each $H_w$, $w\\in K'$, is a clique or independent set, we get a contradiction because $K$ is the smallest context graph of $G$.     \n\\end{proof}     \n\n\nWe say that a node $M\\in T$ is \\textit{gs-connected}, resp \\textit{gs-disconnected}, if  $G(M)=M[H_v\/v : v\\in M]$ is connected, resp disconnected. Note that if $v(M)=1$, then each pair $\\{u, v\\}$ of elements of $M$ forms an edge meaning that $M$ is gs-connected. Similarly, if $v(M)=0$, then $M$ contains two non-adjacent elements $u, v$ meaning that $M$ is gs-disconnected.  So, when $M$ is gs-disconnected, then $P(M)$ has at least two components. \nWe will use the edge minimality of the nodes of $T$ to argue by induction on those vertices. \n\n\n\n\\begin{lemma} \\label{Disconnectedmodule}\nLet $M\\in T$ be such that $v(M)=1$ and $N$ a gs-disconnected child of $M$.\n\\begin{enumerate}\n    \\item If for some $y\\in P(M\\setminus N)$ and some $x\\in P(N)$ we have $x <_P y$, resp $y <_P x$, then $P(N) <_P \\{y\\}$, resp $\\{y\\} <_P P(N)$. \n    \\item If $N'$ is another gs-disconnected child of $M$, then we have $P(N) <_P P(N')$ or $P(N') <_P P(N)$. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1) \nSuppose some $y\\in P(M\\setminus N)$ and some $x\\in P(N)$ satisfy $x <_P y$. Since $N$ is gs-disconnected, there is some $z\\in P(N)$ such that $x$ and $z$ belong to different components of $P(N)$ that is $x \\perp z$. Also, $z\\in H_u$ and $y\\in H_v$ for some $u, v\\in M$. Since $v(M)=1$,  $u$ and $v$ are adjacent. Thus, $zy\\in E(G(M))$. In other words, $z$ and $y$ are comparable w.r.t $\\leq_P$. We have $z <_P y$ because otherwise we get $x <_P y <_P z$ which yields that $x$ and $z$ are comparable, a contradiction. It follows that for every element $z\\in P(N)$ in a component other than the component to which $x$ belongs, we have $z <_P y$. By a similar argument, for every element  $x\\in P(N)$ in a component other than the component to which $z$ belongs, we have $x <_P y$. Hence, $P(N) <_P \\{y\\}$ in this case. Similarly, if there is some $x\\in P(N)$ and $y\\in P(M\\setminus N)$ satisfying  $y <_P x$, then $\\{y\\} <_P P(N)$. \n\n(2) Now let $N'$ be another gs-disconnected child of $M$. Since $v(M)=1$, every element of $P(N)$ is comparable to every element of $P(N')$. Take some $x\\in P(N)$ and suppose that for some $y\\in P(N')$ we have $x <_P y$. Since $N'$ is gs-disconnected, (1) implies that $\\{x\\} <_P P(N')$. Since $N$ is gs-disconnected, take some $z\\in P(N)$ such that $x$ and $z$ belong to different components of $P(N)$. We have $z\\perp x$. Therefore, $\\{z\\} <_P P(N')$ because otherwise we get $x <_P y <_P z$, a contradiction. So, for every $z\\in P(N)$ in a component of $P(N)$ other than the component to which $x$ belongs, we have $\\{z\\} <_P P(N')$. Similarly, for every $x$ in a component of $P(N)$ other than the component $z$ belongs to, we have $\\{x\\} <_P P(N')$. It follows that $P(N) <_P P(N')$. The argument for the other case is similar.    \n\\end{proof}\n\nWith the above lemma at our disposal, we can prove $\\mathcal{S}_{fps}(K)$. The proof is by induction on the vertices of $T$.  \n \n\n\\begin{proposition} \\label{Finiteps} \nLet $P$ be a countable $NE$-free poset whose comparability graph is a finite graph substitution of cliques or independent sets. Then,  $P$ is a finite poset substitution of chains or antichains. \n\\end{proposition} \n\n\n\n\\begin{proof}\nSuppose $G:=CG(P)$ is of the form $K[H_v\/v : v\\in K]$ where $|K|<\\infty$ and each $H_v$ is a clique or independent set. We know that $K$ is a cograph. Assume that $K$ is the smallest. Let $T$ be the decomposition tree of $K$. By Lemma \\ref{Kminimal}, $T$ is leaf minimal. For every $M\\in T$ let $\\mathcal{S}_{fps}(M)$ be the statement ``$P(M)$ is a finite poset substitution of chains or antichains\". \n\nIt is easy to verify that a poset whose comparability graph is a clique, resp an independent set, is a chain, resp an antichain. \nTherefore, the statement is true for the leaves of $T$. Let $M\\in T$ be non-trivial and $M_1, \\ldots, M_k$ the children of $M$. Also, assume that $\\mathcal{S}_{fps}(M_i)$ holds for every $1\\leq i\\leq k$. We consider the following cases. \n\n\\noindent\\textbf{Case 1} $v(M)=0$. In this case $M$ is gs-disconnected. Let $x\\in P(M)$. Then there is some $u\\in M$ such that $x\\in H_u$.  Since the $M_i$ are the children of $M$, $x\\in P(M_i)$ for some $1\\leq i\\leq k$. Moreover, if $x\\in P(M_i)$ and $y\\in P(M_j)$, $i\\neq j$, then $x$ and $y$ are disconnected because $v(M)=0$. Thus, $P(M)=\\bigoplus_i P(M_i)$. That is $P(M)$ is the substitution of the $P(M_i)$ for a finite antichain. Since $\\mathcal{S}_{fps}(M_i)$ holds for every $i$, $\\mathcal{S}_{fps}(M)$ is true in this case.\n\n\\noindent\\textbf{Case 2} $v(M)=1$.   \nSince $M$ is edge minimal,  at most one of its children is a leaf $\\{u\\}$ such that $H_u$ is a clique, equivalently $P_u$ is a chain. If there is a leaf $u$ such that $P_u$ is a chain, we may suppose without loss of generality that $M_k=\\{u\\}$. Let $i, j\\neq k$ be given with $i\\neq j$. Then both $M_i$ and $M_j$ are gs-disconnected. By Lemma \\ref{Disconnectedmodule}, we have $P(M_i) <_P P(M_j)$ or $P(M_j) <_P P(M_i)$.  Without loss of generality assume that $P(M_i) <_P P(M_j)$ if $i < j < k$. So, $P(\\bigcup_{i<k}M_i)=+_{i<k}P(M_i)$. Again, by Lemma \\ref{Disconnectedmodule}, for every $i < k$,  $P(M_i)$ is in a gap of $P_u$. Set $P_u^1:=\\{x\\in P_u : \\{x\\} <_P P(M_1)\\}$, for every $1 < i < k$ set\n$$P_u^i:=\\{x\\in P_u : P(M_{i-1}) <_P \\{x\\} <_P P(M_i)\\}$$\nand also set $P_u^k:=\\{x\\in P_u : P(M_{k-1}) <_P \\{x\\} \\}$.  \nThen, we have\n$$P(M)=P_u^1 + P(M_1) + P_u^2 + P(M_2) + \\cdots + P(M_{k-1}) + P_u^k,$$\nwhere each $P_u^i$ is a chain. That is $P(M)$ is an alternate substitution of the $P(M_i)$ and the $P_u^i$ for a finite chain. \nSince $\\mathcal{S}_{fps}(M_i)$ holds for every $i$ and since each $P_u^i$ is a chain, $\\mathcal{S}_{fps}(M)$ is true in this case.  \n\nSince the statement is true for every $M\\in T$, $\\mathcal{S}_{fps}(K)$ is true which means that $P$ is a finite poset substitution of chains or antichains. \n\\end{proof}\n\nWe observed in Proposition \\ref{Finiteps} that for every $M\\in T$, $P(M)$ is a linear or direct sum of posets. Determining the structure of the siblings of $P(M)$ will enable us to obtain its sibling number. We show that a sibling of $P(M)$ has (up to isomorphism) the same context poset of $P(M)$ (a chain or antichain) replaced with siblings of poset blocks of $P(M)$. The following lemma shows that every embedding of $P(M)$ induces an injective mapping on the poset blocks of $P(M)$. Note that for each $u\\in M$, $P_u$ is a chain or an antichain. When we say $P_u$ and $P_v$, $u, v\\in M$, have the same {\\em type}, it means that either both $P_u$ and $P_v$ are chains or both are antichains. \n\n\n\\begin{lemma} \\label{Incomparableelement} \nLet $M\\in T$ be edge minimal. The images of two distinct poset blocks of $P(M)$ under any embedding of $P(M)$ do not intersect the same poset block. \n\\end{lemma}\n\n\\begin{proof}\nLet $h$ be an embedding of $P(M)$. It suffices to show that for two distinct $u, v\\in M$ and $x\\in P_u, y\\in P_v$, there exists some $z\\in P(M)$ other than $x, y$ such that $z$ is comparable to one and only one of $x$ or $y$. Then $h(z)$ is comparable to one and only one of $h(x)$ or $h(y)$ meaning that $h(P_u)$ and $h(P_v)$ cannot intersect the same poset block because $P(M)$ is a poset substitution by Proposition \\ref{Finiteps}.  \n\nNote that a singleton as a poset block of $P(M)$ is both a chain and an antichain. Let two distinct $u, v\\in M$ and $x\\in P_u, y\\in P_v$ be given. We consider the following cases:\n\n\\noindent\\textbf{Case 1} $u, v$ are adjacent.  If both $P_u, P_v$ are chains, then the edge minimality of $M$ implies that $\\{u, v\\}$ is not a module of $M$. So, there is some $w\\in M$ such that $w$ is adjacent to one of $u$ or $v$ and not to the other. Then picking some $z\\in P_w$ proves the statement. If both $P_u, P_v$ are antichains, then one of them is non-trivial since otherwise they can be regarded as chains. Assume, without loss of generality, that $P_u$ is non-trivial. Then there is some $z\\in P_u$ such that $z\\perp x$ but $z$ and $y$ are comparable. Now suppose, without loss of generality, that $P_u$ is an antichain and $P_v$ is a chain. $P_u$ must be non-trivial since otherwise it can be regarded as a chain. Again, take some $z\\in P_u$ with $z\\perp x$. We know that $z$ and $y$ are comparable. \n\n\\noindent\\textbf{Case 2} $u, v$ are non-adjacent. If both $P_u, P_v$ are antichains, then the edge minimality of $M$ implies that $\\{u,v\\}$ is not a module of $M$. So, there is some $w\\in M$ such that $w$ is adjacent to one of $u$ or $v$ but not to the other. Then pick some $z\\in P_w$ which is a witness to the statement. Note that this also proves the case both $P_u, P_v$ are singletons. So, assume that both $P_u, P_v$ are chains and one of them, say without loss of generality $P_v$, is non-trivial. Take some $y\\neq z\\in P_v$ comparable to $y$. However, we have $z\\perp x$. Finally, if $P_u$ and $P_v$ are of different types, then both of them are non-trivial. Therefore, there exists an element $z$ comparable to one and only one of $x$ or $y$. \n\\end{proof}\n\nBy the lemma above one can define a graph isomorphism on $M$. \n\n\\begin{lemma} \\label{Distinctobjects}\nLet $M\\in T$ be edge minimal. Each embedding of $P(M)$ induces a graph isomorphism of $M$. Moreover, the embedding preserves the types of the poset blocks of $P(M)$. \n\\end{lemma}  \n\n\\begin{proof} \nLet $h$ be an embedding of $P(M)$. First note that $M$ is finite. Therefore, there is a linear order $\\leq_M$ on $M$. Define the mapping $\\hat{h}:M\\to M$ as follows: for each $u\\in M$, select the least $v\\in M$ w.r.t $\\leq_M$ such that $h(P_u)\\cap P_v\\neq \\emptyset$ and define $\\hat{h}(u)=v$. If $h(P_u)\\cap P_v\\neq\\emptyset$ and $h(P_u)\\cap P_w\\neq\\emptyset$, then $v \\leq_M w$ and $w \\leq_M v$ which implies $v=w$. That is $\\hat{h}$ is well-defined. For $u\\neq v\\in M$, by Lemma \\ref{Incomparableelement}, $h(P_u)$ and $h(P_v)$ do not intersect the same poset block. Therefore, the mapping $\\hat{h}$ is injective and since $M$ is finite $\\hat{h}$ is onto. So, $\\hat{h}$ is a bijection on $M$. Further, $u, v\\in M$ are adjacent if and only if $\\hat{h}(u)$ and $\\hat{h}(v)$ are adjacent because $h$ preserves the order $\\leq_P$. Thus, $\\hat{h}$ is a graph isomorphism of $M$. \n\nSince $\\hat{h}$ is well-defined, it is not the case that for some $u\\in M$, $h(P_u)\\cap P_v\\neq \\emptyset$ and $h(P_u)\\cap P_w\\neq\\emptyset$ where $v\\neq w$. It follows that $h(P_u)\\subseteq P_{\\hat{h}(u)}$ for every $u\\in M$. The inclusion also implies that each poset block of $P(M)$ of any type is mapped into a poset block of $P(M)$ of the same type because a non-trivial chain cannot embed in an antichain and a non-trivial antichain cannot embed in a chain. \n\\end{proof}\n\nOne more fact resulting from the lemma above is that any embedding of $P(M)$, $M\\in T$, sends a singleton poset block of $P(M)$ to a singleton poset block because otherwise we get infinitely many chains or antichains as poset blocks of $P(M)$. \n\n\n\\begin{lemma} \\label{Bijectiononconnected}\nLet $M\\in T$ be edge minimal such that $v(M)=0$. Every embedding of $P(M)$ permutes the gs-connected children of $M$ and fixes the possible gs-disconnected child of $M$.  Hence, a sibling of $P(M)$ is a direct sum of the siblings of the $P(M_i)$ where $M_i$ is a child of $M$.  \n\\end{lemma}\n\n\\begin{proof}\nLet $h$ be an embedding of $P(M)$. \nLet $X:=\\{M_1, \\ldots, M_n\\}$ be the set of gs-connected children of $M$ and $\\{u\\}$ the possible gs-disconnected child of $M$. Note that if the gs-disconnected child $\\{u\\}$ of $M$ exists, then $P_u$ is non-trivial and also all $P(M_i)$ are non-trivial because otherwise the edge minimality of $M$ fails. Let $M_i\\in X$ be given and suppose that for some $x\\in P(M_i)$, $h(x)\\in P(M_j)$. For every $y\\in P(M_i)$, $y$ is connected to $x$, so $h(y)$ is connected to $h(x)$ because $h$ preserves the path connecting $x$ and $y$. Therefore,  $h(P(M_i))\\subseteq P(M_j)$ and since $P(M_i)$ is non-trivial, $h(P(M_i))\\cap P_u=\\emptyset$. Since $X$ is finite, there is a linear order $\\leq_X$ on $X$. For every $M_i\\in X$, select the least $M_j\\in X$ w.r.t $\\leq_X$ such that $h(P(M_i))\\cap P(M_j)\\neq\\emptyset$ and define $\\hat{h}(M_i)=M_j$. \nSince $M$ is gs-disconnected, it is not the case that for some $i$ and some $j\\neq k$, $h(P(M_i))\\cap P(M_j)\\neq \\emptyset$ and $h(P(M_i))\\cap P(M_k)\\neq\\emptyset$. So, $\\hat{h}$ is well-defined. Since $T$ is leaf minimal, the children of $M$ are edge minimal. Hence, each $P(M_i)$ is a finite poset substitution of chains or antichains. Let $\\lambda_i$ be the cardinal of $M_i$ for each $i$. We have $\\lambda_i > 0$ for every $i$. Now, suppose for two $i\\neq j$, $h(P(M_i)), h(P(M_j))\\subseteq P(M_k)$. By Lemma \\ref{Distinctobjects}, we have $\\lambda_i + \\lambda_j\\leq \\lambda_k$.  This means that $\\lambda_k > \\lambda_i, \\lambda_j$. A contradiction is immediate when $k=i$ or $k=j$ because then $\\lambda_k > \\lambda_k$. Assume that $k\\neq i, j$. Since $\\lambda_k > \\lambda_i, \\lambda_j$, $P(M_k)$ cannot be embedded into  $P(M_i)$ or $P(M_j)$. If $P(M_k)$ embeds in $P(M_k)$, then by Lemma \\ref{Distinctobjects}, $\\lambda_i+\\lambda_j+\\lambda_k \\leq \\lambda_k$, a contradiction to $\\lambda_i, \\lambda_j\\neq 0$.  It follows that $P(M_k)$ embeds into some $P(M_l)$ where $l\\neq i, j, k$. Continuing this, we get infinitely many gs-connected children of $M$, a contradiction. Thus, $\\hat{h}$ is injective and since $X$ is finite, $\\hat{h}$ is a bijection on $X$. It also follows that $h(P_u)\\subseteq P_u$. Define $\\hat{h}(\\{u\\})=\\{u\\}$.  \n\n\nSince $X$ is finite, for every $i$, there is some integer $m_i > 0$ such that $\\hat{h}^{m_i}(M_i)=M_i$. This means that for every $i$, $P(M_i)\\approx P(\\hat{h}.i(M_i))$ where $\\hat{h}.i$ is the orbit of $M_i$ under $\\hat{h}$. Hence, a sibling of $P(M)$ is of the form $\\bigoplus_iQ(M_i)\\oplus P'_u$ where $Q(M_i)\\approx P(M_i)$ for every $i$ and $P'_u\\cong P_u$ because $P_u$ has only one sibling.  \n\\end{proof} \n\nFor case $v(M)=1$ where $M\\in T$ is edge minimal, we provide a result similar to Lemma \\ref{Bijectiononconnected}.  \n\n\\begin{lemma} \\label{Bijectionondisconnected}\nLet $M\\in T$ be edge minimal such that $v(M)=1$ and $h$ an embedding of $P(M)$. \n\\begin{enumerate}\n    \\item For every gs-disconnected child $M_i$ of $M$ we have $h(P(M_i))\\subseteq P(M_i)$.\n    \\item For any chain $P_u^i$ in the representation $P_u^1 + P(M_1) + P_u^2 + P(M_2) + \\cdots + P(M_{k-1}) + P_u^k$ of $P(M)$ where $\\{u\\}$ is the unique possible gs-connected child of $M$, $h(P_u^i)\\subseteq P_u^i$. \n\\end{enumerate}\nHence, a sibling of $P(M)$ is of the form \n$$Q_u^1 + Q(M_1) + Q_u^2 + Q(M_2) + \\cdots + Q(M_{k-1}) + Q_u^k $$\nwhere $Q(M_i)\\approx P(M_i)$ and $Q_u^i\\approx P_u^i$ for every $i$. \n\\end{lemma}\n\n\\begin{proof}\nBy Proposition \\ref{Finiteps}, $P(M)$ is of the form \n$$P_u^1 + P(M_1) + P_u^2 + P(M_2) + \\cdots + P(M_{k-1}) + P_u^k\\ $$\nwhere each $M_i$ is a gs-disconnected child of $M$ and the $P_u^i$ are the intervals of the unique chain $P_u$ where $\\{u\\}$ is the possible gs-connected child of $M$.   \n\n(1) Let $i$ and $x\\in P(M_i)$ be given. Since $M_i$ is gs-disconnected, take some $y$ in a component of $P(M_i)$ other than the component to which $x$ belongs. So, $x\\perp y$. We have $h(x)\\perp h(y)$. This implies that $h(x)\\notin P_u^j$ where $1\\leq j \\leq k$. Suppose that $h(x)\\in P(M_j)$ where $i\\neq j$. Without loss of generality assume that $P(M_i) <_P P(M_j)$. Since $h(x)\\perp h(y)$, we have $h(y)\\in P(M_j)$. So, for every $y\\in P(M_i)$ in a component other than the component to which $x$ belongs, we have $h(y)\\in P(M_j)$. Exchanging the role of $x$ and $y$, for every $x\\in P(M_i)$ in a component other than the component to which $y$ belongs, we have $h(x)\\in P(M_j)$. It means that $h(P(M_i))\\subseteq P(M_j)$. Let $\\lambda_i, \\lambda_j$ be the number of elements of $M_i, M_j$, respectively. By Lemma \\ref{Distinctobjects} we have $\\lambda_i \\leq \\lambda_j$. Thus, it is not the case that $P(M_j)$ embeds in $P(M_j)$ by $h$ because then we get $\\lambda_i+\\lambda_j \\leq \\lambda_j$, a contradiction to $\\lambda_i > 0$. So, $h(P(M_j))\\subseteq P(M_l)$ where $j < l$. Continuing this, we get infinitely many gs-disconnected children of $M$, a contradiction.  \n\n(2) Let $i$ and some $x\\in P_u^i$ be given. Suppose $h(x)\\in P_u^j$ or $h(x)\\in P(M_j)$ where $j\\neq i$. Assume that $i < j$. The case $j<i$ is similar. Pick some gs-disconnected $M_l$ where $i \\leq l < j$. Since $M_l$ is gs-disconnected, $P(M_l)\\neq\\emptyset$. Pick some $y\\in P(M_l)$. We have $x <_P y$. By (1), $h(y)\\in P(M_l)$. It follows that $h(y) <_P h(x)$, equivalently, $y <_P x$, a contradiction. \nNow assume that $h(x)\\in P(M_i)$ (the argument for $P(M_{i-1})$ is similar). Since $M_i$ is gs-disconnected, take some $y\\in P(M_i)$ such that $y$ belongs to a component of $P(M_i)$ other than the component to which $h(x)$ belongs. We have $x <_P y$ implying that $h(x) <_P  h(y)$. Also, $h(x) <_P h^2(x)$. Moreover, by (1), $h_{\\upharpoonright P(M_i)}$ is an embedding from $P(M_i)$ into $P(M_i)$. This means that $h(x)$ and $y$ are mapped into the same component of $P(M_i)$. This contradicts Lemma \\ref{Bijectiononconnected} because $v(M_i)=0$. \n\\end{proof} \n\n\n\nNow we are ready to deduce Thomass\\'{e}'s Conjecture for a countable $NE$-free poset whose comparability graph has one sibling. Bear in mind that by Theorem \\ref{Dichchain}, a countable chain has either 1, $\\aleph_0$ or $2^{\\aleph_0}$ siblings. \n\n\n\\begin{theorem} \\label{GtoP} \nLet $P$ be a countable $NE$-free poset whose comparability graph has one sibling. Then $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$.  \n\\end{theorem}  \n\n\\begin{proof}\nSet $G:=CG(P)$. By Theorem \\ref{Lexico}, $G$ is of the form $K[H_v\/v : v\\in K]$ where $K$ is finite and each $H_v$ is a clique or an independent set. Indeed, $K$ is a cograph. Assume that $K$ is the smallest. Let $T$ be the decomposition tree of $K$. By Lemma \\ref{Kminimal},  $T$ is leaf minimal. For each $M\\in T$ let $\\mathcal{S}_{sib}(M)$ be the statement: ``$Sib(P(M))=1$  or $\\aleph_0$ or $2^{\\aleph_0}$\".   We prove that $\\mathcal{S}_{sib}(K)$ holds.\n\nFor every leaf $\\{v\\}$ of $T$, $P_v$ is a countable chain or antichain. Therefore, $\\mathcal{S}_{sib}(\\{v\\})$ holds since an antichain has only one sibling and the statement holds for a countable chain by Theorem \\ref{Dichchain}. Now, take $M\\in T$ such that $M$ is not a leaf of $T$ and suppose that for each child $N$ of $M$, $\\mathcal{S}(N)$ holds. If $v(M)=0$, then by Lemma \\ref{Bijectiononconnected},  each sibling of $P(M)$ is a finite direct sum of siblings of the $P(N)$ where $N$ is a child of $M$; and if $v(M)=1$, then by Lemma \\ref{Bijectionondisconnected}, each sibling of $P(M)$ is an alternate substitution of chains and the siblings of the $P(N)$ for a finite chain where $N$ is a non-trivial child of $M$. By the induction hypothesis and the fact that the statement holds for a countable chain,  $Sib(P(M))$ is the maximum of the sibling numbers of the components or the summands of $P(M)$. Thus, $\\mathcal{S}_{sib}(M)$ holds. Since the statement is true for every $M\\in T$,  $\\mathcal{S}_{sib}(K)$ holds.   \n\\end{proof}\n\n\n\n\\section{Open Cases} \n\nWe just proved the alternate Thomass\\'{e} conjecture for countable $NE$-free posets. We saw how the siblings of the sum of a labelled chain with no least element were constructed by its decomposition tree. The same technique was used for countable linear sums with infinitely many non-trivial summands. Also, there were other cases in both direct sums and linear sums in which we determined the exact sibling number. Moreover, Section \\ref{Finiteposetsub} asserts that Thomass\\'e's conjecture holds for countable $NE$-free posets whose comparability graph has one sibling. Posing all these restrictions, we may ask the following.\n\n\\begin{problem}\nLet P be a countable direct sum of connected $NE$-free posets or a countable linear sum of $NE$-free posets with property CCGC such that the comparability graph of $P$ has infinitely many siblings. Is it true that $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$?\n\\end{problem}\n\n\\vspace{0.5cm}\n\\noindent{\\bf \\large Acknowledgements}\n\nI would like to thank my PhD supervisors Professor Robert Woodrow and Professor Claude Laflamme for suggesting this problem and for their help and advice. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\n\\label{section:introduction}\n\nThe decomposition theorem for one-parameter persistence modules states that any such module decomposes uniquely as a direct sum of interval modules, which are completely characterized by their supports \\cite{gabriel,webb,zomorodian-carlsson,crawley-boevey}.\nThis theorem is of fundamental importance in topological inference and topological data analysis, where it is used, in particular, to distinguish between prominent topological features, which correspond to features of the process that generated the data, and spurious topological features, due to noise \\cite{edelsbrunner-letscher-zomorodian,edelsbrunner-harer,chazal-guibas-oudot-skraba}.\nThe consistency of this method is guaranteed by the stability theorem \\cite{cohensteiner-edelsbrunner-harer,damico-frosini-landi,chazal-silva-glisse-oudot,chazal-cohen-steiner-glisse-guibas-oudot,bauer-lesnick}, which states that decomposing a one-parameter persistence module into its indecomposable summands is a stable operation.\n\nMulti-parameter persistence modules arise naturally when dealing with data where both metric and density structure is relevant, or with otherwise parametrized data \\cite{carlsson-zomorodian,botnan-lesnick-2,biasotti-cerry-frosini-giorgi-landi,lesnick-wright,mcinnes-healy,vipond}.\nWhen one moves from one parameter to multiple parameters, the approach to understanding a persistence module by completely characterizing its indecomposable summands is less promising: classifying all indecomposable multi-parameter persistence modules is thought to be an infeasible task as it involves classifying all indecomposable representations of certain posets of wild representation type \\cite{Nazarova}.\nNevertheless, one can still consider computable descriptors\nderived from the decomposition into indecomposables \\cite{dey-xin}.\nA natural example is the descriptor given by the supports of the indecomposable summands, and a natural question is whether this descriptor is stable.\n\nFrom the results in this paper we can conclude that, for general two-parameter persistence modules, this descriptor is highly unstable.\nAs such, our results provide further motivation for approaches to understanding the structure and stability of multi-parameter persistence modules that do not rely on decomposing by arbitrary indecomposables\n\\cite{carlsson-zomorodian,\nlesnick-wright-2,\nscolamiero-chacholski-lundman-ramanujan-oberg,\nkim-memoli,\nasashiba2019approximation,\nmiller,\nmccleary2021edit,\nbotnan-oppermann-oudot,\nblanchette-brustle-hanson,\noudot-scoccola,\nasashiba-escolar-nakashima-yoshiwaki,\nbotnan-oppermann-oudot-scoccola,\ncacholski-guidolin-ren-scolamiero-tombari,\ncerri-difabio-ferri-frosini-landi,\nbiasotti-cerri-frosini-giorgi-landi},\nand give a sense of how complicated arbitrary indecomposables can be, complementing results such as the ones in \\cite{carlsson-zomorodian,buchet-escolar,moore}.\n\n\n\\subparagraph{Contributions}\nFor definitions, see \\cref{section:background}.\nThe following is our main result.\n\n\\begin{theorem}\n    \\label{theorem:main-theorem}\n    Consider the set of isomorphism classes of finitely presentable two-parameter persistence modules, metrized with the interleaving distance.\n    For every $\\epsilon > 0$ there exists a dense and open subset $O^\\epsilon$ with the property that every $N \\in O^\\epsilon$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n\\end{theorem}\n\nRecall that, in a topological space, a \\define{generic} property is one that holds for all points of some dense and open set.\nOur main result then implies that, for every $\\epsilon > 0$, the property of decomposing as a direct sum of an indecomposable and an $\\epsilon$-trivial module is generic.\n\nTo put this result into context, we mention a similar instance of a generic property of persistence modules. \nThe set of all finitely presentable one-parameter persistence modules that are \\emph{staggered} (meaning that the finite endpoints of the intervals in their barcode are all distinct) is dense but not open.\nNevertheless, for every $\\epsilon > 0$ there exists a dense and open set of modules that decomposes as $A \\oplus B$ with $A$ staggered and $B$ $\\epsilon$-trivial, as a direct consequence of the isometry theorem \\cite{lesnick,chazal-silva-glisse-oudot,bubenik-scott,bauer-lesnick}.\n\n\n\\smallskip\nWe prove \\cref{theorem:main-theorem} by combining the following two results, which are of independent interest.\nThe first one states that, in the space of finitely presentable two-parameter persistence modules, indecomposable modules are dense.\n\n\\begin{restatable}{proposition}{indecomposabledense}\n    \\label{theorem:indecomposables-dense}\n    Let $N : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable.\n    For every $\\epsilon > 0$, there exists an indecomposable persistence module $M : \\Rbf^2 \\to \\mathbf{vec}$ such that $d_I(M,N) \\leq \\epsilon$.\n\\end{restatable}\n\nThe second result relates the property of being close to an indecomposable to the property of decomposing as a direct sum of an indecomposable and a nearly trivial persistence module.\n\n\\begin{restatable}{proposition}{stabilityindecomposability}\n    \\label{proposition:stability-indecomposability}\n    Let $n \\geq 1$ and let $M : \\Rbf^n \\to \\mathbf{vec}$ be finitely presentable and indecomposable.\n    For every $\\epsilon > 0$ there exists $\\delta > 0$ such that every persistence module $N : \\Rbf^n \\to \\mathbf{vec}$ with $d_I(M,N) < \\delta$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n\\end{restatable}\n\n\\subparagraph{Discussion}\n\\cref{proposition:stability-indecomposability} motivates the following definition.\n\n\\begin{definition}\n    Let $\\epsilon > 0$.\n    A persistence module $M : \\Rbf^n \\to \\mathbf{vec}$ is \\define{$\\epsilon$-indecomposable} if we have $M \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n\\end{definition}\nOur main theorem then implies that, for every $\\epsilon > 0$, two-parameter persistence modules are generically $\\epsilon$-indecomposable.\nSince there are no general strong relationships between meager and measure-zero sets \\cite{oxtoby}, our results do not imply that random two-parameter persistence modules, according to some suitable probability measure, are $\\epsilon$-indecomposable.\nOne could then ask the following general questions:\nWhat are interesting and useful probability measures on spaces of multi-parameter persistence modules?\nAccording to these probability measures, how do random multi-parameter persistence modules decompose?\n\nWe prove \\cref{proposition:stability-indecomposability} in the generality of multi-parameter persistence modules, while we have given \\cref{theorem:indecomposables-dense} and consequently \\cref{theorem:main-theorem} only in the case of two-parameter persistence.\nWe believe the results hold for any number of parameters greater than one.\nSimilarly, the results extend to classes of modules more general than finitely presentable ones.\nWe leave these extensions for future work.\n\n\\subparagraph{Structure of the paper}\nIn \\cref{section:background}, we recall necessary background and introduce notation.\nIn \\cref{section:almost-indecomposable-open}, we prove \\cref{proposition:stability-indecomposability}.\nIn \\cref{section:indecomposables-dense}, we prove \\cref{theorem:indecomposables-dense} and \\cref{theorem:main-theorem}.\nIn \\cref{section:proof-main-lemma} we prove a key lemma that allows us to ``tack'' indecomposables together.\n\n\n\n\\ifarxivversion\n\\subparagraph{Acknowledgements}\nWe thank H\\r{a}vard Bjerkevik for interesting conversations and useful observations.\nL.~S.~thanks\nNicolas Berkouk,\nMathieu Carri\u00e8re,\nRen\u00e9 Corbet,\nChristian Hirsch,\nClaudia Landi,\nVadim Lebovici,\nDavid Loiseaux,\nEzra Miller,\nSteve Oudot,\nFran\u00e7ois Petit,\nand Alexander Rolle\nfor discussions during the \\emph{Metrics in Multiparameter Persistence workshop} (Lorentz Center, 2021).\nThis research has been conducted while U.~B.~was participating in the program \\emph{Representation Theory: Combinatorial Aspects and Applications};\nU.~B.~thanks the Centre for Advanced Study (CAS) at the Norwegian Academy of Science and Letters for their hospitality and support.\nThis research has been supported by the DFG Collaborative Research Center SFB\/TRR 109 \\emph{Discretization in Geometry and Dynamics}.\n\\fi\n\n\\section{Background}\n\\label{section:background}\n\nAlthough we use some notions from category theory, we only assume familiarity with basic concepts; in particular: categories, isomorphisms, functors, functor categories, direct sums, and kernels and cokernels.\nSee, e.g., \\cite{maclane} for an introduction.\n\nThroughout the paper, we fix a field $\\kbb$ and let $\\mathbf{vec}$ denote the category of finite dimensional $\\kbb$-vector spaces.\n\nAn \\define{extended metric space} consists of a set $A$ together with a function $d : A \\times A \\to \\Rbb \\cup \\{\\infty\\}$ such that, for all $a,b \\in A$ we have $d(a,b) = d(b,a) \\geq 0$ and $d(a,b) = 0$ if and only if $a=b$; and for all $a,b,c \\in A$ we have $d(a,c) \\leq d(a,b) + d(b,c)$, with the convention that $r + \\infty = \\infty + r = \\infty$ for all $r \\in \\Rbb \\cup \\{\\infty\\}$.\n\n\\subparagraph{Posets}\nWe let $\\Rbf$ denote the poset of real numbers with its standard ordering and reserve the notation $\\Rbb$ for the metric space of real numbers.\nLet $n \\in \\Nbb$ and consider the product poset $\\Rbf^n$.\nBy an abuse of notation, if $r \\in \\Rbf$, we interpret it as an element $r \\in \\Rbf^n$ all of whose coordinates are equal to $r$.\nThus, if for instance $s \\in \\Rbf^n$ and $r \\in \\Rbf$, then $s + r \\in \\Rbf^n$ denotes the element $(s_1 + r, s_2 + r, \\dots, s_n + r)$. \n\nWe interpret any poset $\\Pscr$ as a category with objects the elements of the poset and exactly one morphism $i \\to j \\in \\Pscr$ whenever $i \\leq j$.\n\n\\subparagraph{Persistence modules}\nLet $\\Pscr$ be a poset.\nA pointwise finite dimensional \\define{$\\Pscr$-persistence module} is a functor $M : \\Pscr \\to \\mathbf{vec}$.\nNote that all persistence modules in this paper are assumed to be pointwise finite dimensional and that, for the sake of brevity, we may omit this qualifier.\nWhen the indexing poset $\\Pscr$ is clear from the context, we may refer to a $\\Pscr$-persistence module simply as a persistence module or as a module.\nThe collection of all $\\Pscr$-persistence modules forms a category, where the morphisms are given by natural transformations.\n\nIf $M : \\Pscr \\to \\mathbf{vec}$ is a $\\Pscr$-persistence module and $i \\leq j \\in \\Pscr$, we let $\\phi^M_{i,j} : M(i) \\to M(j)$ denote the structure morphism corresponding to the morphism $i \\to j$ in $\\Pscr$ seen as a category.\n\nLet $\\Pscr$ be a poset and let $i \\in \\Pscr$.\nDefine the persistence module $\\Psf_i : \\Pscr \\to \\mathbf{vec}$ by\n\\[\n    \\Psf_i(j) =\n    \\begin{cases}\n        \\kbb & i \\leq j\\\\\n        0 & \\text{else},\n    \\end{cases}\n\\]\nwith all structure morphisms that are not forced to be zero being the identity $\\kbb \\to \\kbb$.\n\nA $\\Pscr$-persistence module is \\define{finitely presentable} if it is isomorphic to the cokernel of a morphism $\\bigoplus_{j \\in J} \\Psf_j \\to \\bigoplus_{i \\in I} \\Psf_i$, where $I$ and $J$ are finite multisets of elements of $\\Pscr$.\n\n\n\\subparagraph{Restrictions and extensions}\nIf $\\Qscr \\subseteq \\Pscr$ is an inclusion of posets and $M : \\Pscr \\to \\mathbf{vec}$ is a $\\Pscr$-persistence module, the \\define{restriction} of $M$ to $\\Qscr$, denoted $M|_\\Qscr : \\Qscr \\to \\mathbf{vec}$, is the $\\Qscr$-persistence module obtained by precomposing $M : \\Pscr \\to \\mathbf{vec}$ with the inclusion $\\Qscr \\to \\Pscr$.\n\nWe consider two main types of subposets of $\\Rbf^n$.\nIn one case, we let $\\{r_1 < r_2 < \\dots < r_k\\}$ be a finite set of real numbers and consider the product poset $\\{r_1 < r_2 < \\dots < r_k\\}^n \\subseteq \\Rbf^n$.\nWe refer to a subposet of $\\Rbf^n$ obtained in this way as a \\define{finite grid}.\nIn the other case, we let $\\{r_i\\}_{i \\in \\Zbb}$ be a countable set of real numbers without accumulation points and such that $r_i < r_{i+1}$ for all $i \\in \\Zbb$, and again consider the product poset $\\{r_i\\}^n \\subseteq \\Rbf^n$.\nWe refer to a subposet of $\\Rbf^n$ obtained in this way as a \\define{countable grid}.\nA \\define{regular grid} is any countable grid of the form $(s \\Zbf)^n = \\{s \\cdot m : m \\in \\Zbf\\}^n \\subseteq \\Rbf^n$ for $s > 0 \\in \\Rbb$, where $\\Zbf$ denotes the poset of integers.\nNote that, in the definitions of finite and countable grid, one could take a product of different subposets of $\\Rbf$, instead of an $n$-fold product of the same poset.\nSince we do not need this generality, we choose the more restrictive definition for simplicity.\n\n\nLet $\\Pscr \\subseteq \\Rbf^n$ be a finite or countable grid.\nGiven $M : \\Pscr \\to \\mathbf{vec}$ define $\\widehat{M} : \\Rbf^n \\to \\mathbf{vec}$\nby\n\\[\n    \\widehat{M}(r) =\n    \\begin{cases}\n        M\\left(\\sup\\{ p \\in \\Pscr : p \\leq r\\}\\right) & \\text{if there exists $p \\in \\Pscr$ such that $p \\leq r$}\\\\\n        0 & \\text{else};\n    \\end{cases}\n\\]\nfor its structure morphisms use the structure morphisms of $M$ and the fact that $\\sup\\{p \\in \\Pscr : p \\leq r\\} \\leq \\sup\\{p \\in \\Pscr : p \\leq s\\}$ whenever $r \\leq s$ and there exists $p \\in \\Pscr$ such that $p \\leq r$.\nWe refer to $\\widehat{M}$ as the \\define{extension} of $M$ along the inclusion $\\Pscr \\subseteq \\Rbf^n$.\nAs a side remark, we note that this notion of extension is an instance of the notion of left Kan extension from category theory, but we do not make use of this fact.\n\nIf $\\Pscr \\subseteq \\Rbf^n$ is a finite or countable grid and $M : \\Rbf^n \\to \\mathbf{vec}$, define the \\define{restriction-extension} of $M$ along $\\Pscr$, denoted $M_\\Pscr : \\Rbf^n \\to \\mathbf{vec}$, as the extension along $\\Pscr \\subseteq \\Rbf^n$ of the restriction $M|_\\Pscr : \\Pscr \\to \\mathbf{vec}$.\nGiven $M,N : \\Rbf^n \\to \\mathbf{vec}$ and a morphism $f : M \\to N$, there is a morphism $f_\\Pscr : M_\\Pscr \\to N_\\Pscr$ given by extending the restriction $f|_\\Pscr : M|_\\Pscr \\to N|_\\Pscr$.\nIt follows immediately that this construction is functorial in the sense that given modules $A,B,C : \\Rbf^n \\to \\mathbf{vec}$ and morphism $f : A \\to B$ and $g : B \\to C$, we have $g_\\Pscr \\circ f_\\Pscr = (g \\circ f)_\\Pscr : A_\\Pscr \\to C_{\\Pscr}$.\n\n\n\\begin{lemma}\n    \\label{lemma:fp-is-extension-finite-poset}\n    Let $n \\geq 1$ and let $M : \\Rbf^n \\to \\mathbf{vec}$.\n    Then $M$ is finitely presentable if and only if there exists a finite grid $\\Pscr \\subseteq \\Rbf^n$ such that $M \\cong M_\\Pscr$.\n\\end{lemma}\n\\begin{proof}\n    Assume that $M$ is finitely presentable, so that $M$ is isomorphic to the cokernel of a morphism $\\bigoplus_{j \\in J} \\Psf_j \\to \\bigoplus_{i \\in I} \\Psf_i$ with $I$ and $J$ finite subsets of $\\Rbf$.\n    Consider the set $S = \\{x_k \\in \\Rbf : x \\in I \\cup J, \\, 1 \\leq k \\leq n\\}$ of all the coordinates of the points in $I \\cup J$, and the finite grid $\\Pscr = S^n \\subseteq \\Rbf^n$.\n    It is straightforward to see that $M \\cong M_\\Pscr$.\n\n    Now assume that $M \\cong \\widehat{N}$ with $\\Pscr \\subseteq \\Rbf^n$ a finite grid and $N : \\Pscr \\to \\mathbf{vec}$.\n    Since the poset $\\Pscr$ has finitely many elements and $N$ is pointwise finite dimensional, there exists an epimorphism $e : \\bigoplus_{i \\in I} \\Psf_i \\to N$, for some finite multiset $I$ of elements of $\\Pscr$.\n    Similarly, if $K$ is the kernel of the morphism $e$, there must exist an epimorphism $\\bigoplus_{j \\in J} \\Psf_j \\to K$, with $J$ finite.\n    Putting these two morphisms together we see that $N$ is isomorphic to the cokernel of a morphism $\\bigoplus_{j \\in J} \\Psf_j \\to \\bigoplus_{i \\in I} \\Psf_i$.\n    It is easy to see that $M$ is then isomorphic to the cokernel of the induced morphism $\\bigoplus_{j \\in J} \\widehat{\\Psf_j} \\to \\bigoplus_{i \\in I} \\widehat{\\Psf_i}$, and that $\\widehat{\\Psf_r} = \\Psf_r : \\Rbf^n \\to \\mathbf{vec}$ for all $r \\in \\Pscr$.\n\\end{proof}\n\nIt is straightforward to see that, in a grid extension persistence module, the structure maps that do not cross the grid are isomorphisms, as recorded in the following lemma.\n\n\\begin{lemma}\n    \\label{lemma:structure-map-iso}\n    Let $\\Pscr$ be a finite or countable grid and let $M : \\Rbf^n \\to \\mathbf{vec}$ be isomorphic to the extension of a $\\Pscr$-persistence module.\n    Let $r < s \\in \\Rbf^n$ such that every $p \\in \\Pscr$ with $p \\leq s$ also satisfies $p \\leq r$.\n    Then the structure morphism $\\phi^M_{r,s} : M(r) \\to M(s)$ is an isomorphism.\n    \\qed\n\\end{lemma}\n\n\\subparagraph{Decomposition of persistence modules}\nThe proofs of the results in this section are standard, we include them in \\cref{appendix}, for completeness.\nLet $\\Pscr$ be a poset and let $M : \\Pscr \\to \\mathbf{vec}$.\nWe say that $M$ is \\define{decomposable} if there exist $A,B : \\Pscr \\to \\mathbf{vec}$ non-zero such that $M \\cong A \\oplus B$.\nIf $M$ is non-zero and not decomposable, we say that $M$ is \\define{indecomposable}.\n\n\nThe next two results follow from \\cite{botnan-crawleybovey} and the Krull--Remak--Schmidt--Azumaya theorem~\\cite{azumaya}.\n\n\n\\begin{restatable}{theorem}{theoremdecomposition}\n    \\label{theorem:decomposition}\n    Let $\\Pscr$ be a poset and let $M : \\Pscr \\to \\mathbf{vec}$ be a pointwise finite dimensional $\\Pscr$-persistence module.\n    There exists a set $I$ and an indexed family of indecomposable $\\Pscr$-persistence modules $\\{M_i\\}_{i \\in I}$ such that $M \\cong \\bigoplus_{i \\in I} M_i$.\n    Moreover, if $M \\cong \\bigoplus_{j \\in J} M_j$ for another indexed family of indecomposable $\\Pscr$-persistence modules $\\{M_j\\}_{j \\in J}$, then there exists a bijection $f : I \\to J$ such that $M_i \\cong M_{f(j)}$ for all $i \\in I$.\n\\end{restatable}\n\n\\begin{restatable}{lemma}{indecomposablelocalring}\n    \\label{lemma:indecomposable-local-ring}\n    Let $\\Pscr$ be a poset.\n    A persistence module $M : \\Pscr \\to \\mathbf{vec}$ is indecomposable if and only if its endomorphism ring $\\End(M)$ is local.\n\\end{restatable}\n\n\\begin{proof}\n    This is a direct consequence of \\cite[Theorem~1.1]{botnan-crawleybovey}.\n\\end{proof}\n\nThe following result states that direct sum decompositions behave well with respect to restrictions and extensions; its proof is straightforward.\n\n\\begin{lemma}\n    \\label{lemma:decomposition-restriction-extension}\n    Let $\\Pscr \\subseteq \\Rbf^n$ be a subposet.\n    Let $M : \\Rbf^n \\to \\mathbf{vec}$ decompose as $M \\cong \\bigoplus_{i\\in I} M_i$ for some indexed family $\\{M_i\\}_{i \\in I}$ of persistence modules.\n    Then $M|_\\Pscr \\cong \\bigoplus_{i\\in I} (M_i)|_\\Pscr$.\n    Similarly, if $\\Pscr$ is a finite or countable grid and $M : \\Pscr \\to \\mathbf{vec}$ decomposes as $M \\cong \\bigoplus_{i\\in I} M_i$ for some indexed family $\\{M_i\\}_{i \\in I}$ of $\\Pscr$-persistence modules, then $\\widehat{M} \\cong \\bigoplus_{i\\in I} \\widehat{M_i}$.\n    \\qed\n\\end{lemma}\n\nThe next result asserts that finitely presentable persistence modules decompose as a finite direct sum of finitely presentable modules.\nThis holds in the generality of persistence modules indexed by an arbitrary poset, but we do not need this generality.\n\n\\begin{restatable}{lemma}{decompositionfp}\n    \\label{lemma:decomposition-fp}\n    Let $n \\geq 1$ and let $M : \\Rbf^n \\to \\mathbf{vec}$.\n    If $M$ is finitely presentable, then $M \\cong \\bigoplus_{i \\in I} M_i$ with $\\{M_i\\}_{i \\in I}$ a finite family of finitely presentable indecomposable $\\Rbf^n$-persistence modules.\n\\end{restatable}\n\n\\subparagraph{Interleavings, interleaving distance, and space of persistence modules}\nLet $n \\in \\Nbb$.\nIf $M : \\Rbf^n \\to \\mathbf{vec}$ is a $\\Rbf^n$-persistence module and $r \\in \\Rbf$, the \\define{$r$-shift} of $M$ is the persistence module $M[r] : \\Rbf^n \\to \\mathbf{vec}$ with $M[r](s) = M[s+r]$ for all $s \\in \\Rbf^n$ and $\\phi^{M[r]}_{s,t} = \\phi^M_{s+r,t+r}$ for all $s\\leq t \\in \\Rbf^n$.\nIf $r \\geq 0 \\in \\Rbf$, then there is a natural transformation $\\eta^M_r : M \\to M[r]$ with component $M(a) \\to M[r](a) = M(a+r)$ given by the structure morphism $\\phi^M_{a,a+r}$.\n\nLet $M,N : \\Rbf^n \\to \\mathbf{vec}$ and $\\epsilon \\geq 0 \\in \\Rbf$.\nAn \\define{$\\epsilon$-interleaving} between $M$ and $N$ consists of a pair of morphisms $f : M \\to N[\\epsilon]$ and $g : N \\to M[\\epsilon]$ such that $g[\\epsilon] \\circ f = \\eta^M_{2\\epsilon}$ and $f[\\epsilon] \\circ f = \\eta^N_{2\\epsilon}$.\nThe \\define{interleaving distance} between $M$ and $N$ is\n\\[\n    d_I(M,N) = \\inf\\left(\\{\\epsilon \\geq 0 : \\text{ exists an $\\epsilon$-interleaving between $M$ and $N$ }\\} \\cup \\{\\infty\\}\\right) \\in \\Rbb \\cup \\{\\infty\\}.\n\\]\n\nBy composing interleavings, one shows that $d_I$ satisfies the triangle inequality.\nUsing \\cref{lemma:structure-map-iso}, one sees that if $M, N : \\Rbf^n \\to \\mathbf{vec}$ are finitely presentable and $d_I(M,N) = 0$, then $M$ and $N$ are isomorphic \\cite[Corollary~6.2]{lesnick}.\nThis implies that $d_I$ defines an extended metric on the set of isomorphism classes of finitely presentable $\\Rbf^n$-persistence modules.\nWe mention here that the collection of all isomorphism classes of finitely presentable $\\Rbf^n$-persistence modules is indeed a set, as opposed to a proper class, but we shall not delve into the details as they are standard and not relevant to the results presented here.\n\nWe consider the set of isomorphism classes of finitely presentable $\\Rbf^n$-persistence modules which we topologize with the topology generated by the open balls with respect to the interleaving distance.\n\nLet $\\epsilon > 0$.\nA persistence module $M : \\Rbf^n \\to \\mathbf{vec}$ is \\define{$\\epsilon$-trivial} if, for every $r \\in \\Rbf^n$, the structure morphism $\\phi^M_{r,r+\\epsilon} : M(r) \\to M(r + \\epsilon)$ is the zero morphism.\nNote that this is equivalent to $M$ being $\\epsilon\/2$-interleaved with the zero module.\nThus, if $d_I(M,0) < \\epsilon\/2$, then $M$ is $\\epsilon$-trivial.\n\n\n\\subparagraph{Shifts and restriction-extensions}\nIf $\\Pscr = (\\{s_i\\}_{i \\in \\Zbb})^n \\subseteq \\Rbf^n$ is a countable grid and $r \\in \\Rbf$, define the \\define{shifted} countable grid $\\Pscr + r = (\\{s_i + r\\}_{i \\in \\Zbb})^n \\subseteq \\Rbf^n$.\nLet $M : \\Rbf^n \\to \\mathbf{vec}$ and $s \\in \\Rbf$.\nIf $a \\in \\Rbf^n$, then $\\sup\\{ p + r : p + r \\leq a + s\\} = \\sup\\{p + r - s : p + r - s \\leq a\\} + s$.\nThus,\n\\[\n    M_{\\Pscr + r}[s] = M[s]_{\\Pscr + (r-s)},\n\\]\na fact that we use repeatedly in \\cref{section:almost-indecomposable-open}.\nIn particular, given $N : \\Rbf^n \\to \\mathbf{vec}$ and a morphism $f : M \\to N[s]$, we get a morphism\n\\begin{equation}\n    \\label{equation:shift-and-restriction-extension}\n    f_\\Pscr : M_{\\Pscr + r} \\to N[s]_{\\Pscr + r} = N_{\\Pscr + r + s}[s].\n\\end{equation}\n\n\\section{Nearly indecomposables are open}\n\\label{section:almost-indecomposable-open}\n\nBefore we can prove \\cref{proposition:stability-indecomposability}, we need a definition and a lemma.\n\n\\begin{definition}\n    Let $\\alpha > 0$.\n    A countable grid $\\Pscr = \\left(\\{s_i\\}_{i \\in \\Zbb}\\right)^n \\subseteq \\Rbf^n$ has \\define{mesh width $\\alpha$} if for all $i \\in \\Zbb$ we have $s_{i+1} - s_i \\leq \\alpha$.\n\\end{definition}\n\nNote that, for any fixed $\\alpha > 0$, every countable grid $S^n = \\left(\\{s_i\\}_{i \\in \\Zbb}\\right)^n$ is included in some countable grid with mesh width $\\alpha$, for example, the countable grid $(S \\cup (\\alpha\\Zbf))^n$.\n\n\\begin{lemma}\n    \\label{lemma:factor-bounded-grid}\n    Let $\\alpha > 0$, let $\\Pscr$ be a countable grid with mesh width $\\alpha$, and let $L : \\Rbf^n \\to \\mathbf{vec}$.\n    For every $r < \\alpha$ there exists a morphism $m : L_{\\Pscr + r}[r] \\to L_\\Pscr[\\alpha]$ rendering the following square commutative:\n    \\[\n    \\begin{tikzpicture}\n        \\matrix (m) [matrix of math nodes,row sep=4em,column sep=7em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n        {  L_{\\Pscr}  &  L_{\\Pscr}[\\alpha] \\\\\n           L[r]_\\Pscr & L_{\\Pscr + r}[r],\\\\};\n        \\path[line width=0.75pt, -{>[width=8pt]}]\n        (m-1-1) edge node [above] {$\\eta^{L_\\Pscr}_\\alpha$} (m-1-2)\n        (m-1-1) edge node [left] {$(\\eta^L_r)_\\Pscr$} (m-2-1)\n        (m-2-2) edge node [right] {$m$} (m-1-2)\n        (m-2-1) edge [-,double equal sign distance] (m-2-2)\n        ;\n    \\end{tikzpicture}\n    \\]\n\\end{lemma}\n\\begin{proof}\n    We start by defining the morphism $m$, which is equivalently a morphism $m : L_{\\Pscr+r} \\to L_\\Pscr[\\alpha-r]$.\n    If $a \\in \\Rbf^n$, then\n    $L_{\\Pscr + r}(a) = L(\\sup\\{ s + r : s \\in \\Pscr, s + r \\leq a\\})$ and\n    \\[\n        L_{\\Pscr}[\\alpha-r](a) = L_{\\Pscr}(a + \\alpha - r) = L(\\sup\\{s : s \\in \\Pscr, s \\leq a + \\alpha - r\\}).\n    \\]\n    Let $s_0$ such that $s_0 + r = \\sup\\{ s + r : s \\in \\Pscr, s + r \\leq a\\}$ so that $L_{\\Pscr + r}(a) = L(s_0 + r)$.\n    If $\\Pscr = (\\{s_i\\}_{i \\in \\Zbb})^n$, then $s_0 = (s_{i_1}, \\dots, s_{i_n})$.\n    Let $s_1 = (s_{i_1+1}, \\dots, s_{i_n+1})$.\n    Note that, since $\\Pscr$ has mesh width $\\alpha$, we have $s_1 - \\alpha \\leq s_0$.\n    Thus, $s_1 - \\alpha + r \\leq a$, which implies $s_1 \\leq a + \\alpha - r$.\n    This means that $s_1 \\leq \\sup\\{s : s \\in \\Pscr, s \\leq a + \\alpha - r\\}$.\n    We can then consider the composite:\n    \\[\n        L_{\\Pscr + r}(a) = L(s_0) \\to L(s_1) \\to L(\\sup\\{s : s \\in \\Pscr, s \\leq a + \\alpha - r\\}) = L_{\\Pscr}[\\alpha-r](a),\n    \\]\n    given by the structure morphisms of $L$.\n    Let us denote the above composite by $m_a$.\n    Since the morphism was constructed only using the structure morphisms of $L$, it is straightforward to check that the morphisms $m_a : L_{\\Pscr + r}(a) \\to L_{\\Pscr}[\\alpha-r](a)$ assemble into a natural transformation $m : L_{\\Pscr+r} \\to L_\\Pscr[\\alpha-r]$.\n    By the same reason, it follows that $m$ renders commutative the square in the statement.\n\\end{proof}\n\n\n\\stabilityindecomposability*\n\\begin{proof}[Proof outline]\\renewcommand{\\qedsymbol}{}\n    Let $M$ be as in the statement.\n    Let us start by giving an outline of the proof strategy.\n    We start by arguing that the persistence module $M$ is isomorphic to the extension of a persistence module over a sufficiently fine countable grid.\n    We then choose a sufficiently small $\\delta > 0$ and consider an arbitrary module $N$ at interleaving distance at most $\\delta$ from $M$.\n    Next, we use \\cref{lemma:structure-map-iso}, which says that the extension of a module over a grid can only change when one of its coordinates crosses a point in the grid; this is used to show that, after restricting to appropriate grids, a restriction of $M$ is indecomposable and isomorphic to a direct summand of a restriction of $N$.\n    This allows us to find a decomposition of $N$ into two summands, one of which is indecomposable and is related to a restriction of $M$.\n    Finally, we show that the other summand is close to the zero module in the interleaving distance.\n\\end{proof}\n\n\n\\begin{proof}\n    By \\cref{lemma:fp-is-extension-finite-poset}, the module $M$ is isomorphic to the extension of its restriction to some finite grid $\\Qscr = \\{r_1 < r_2 < \\dots < r_k\\}^n \\subseteq \\Rbf^n$.\n    Let\n    \\[\n        \\tau = \\min_{1\\leq i \\leq k-1} r_{i+1} - r_i\\;\\;, \\;\\; \\alpha = \\min\\left(\\epsilon\/4\\; , \\tau\\right)\\;\\;, \\;\\; \\delta < \\alpha\/4,\n    \\]\n    and let $\\Pscr = (\\{s_i\\}_{i \\in \\Zbb})^n$ be a countable grid with mesh width $\\alpha$ and containing $\\Qscr$.\n    Note that $M$ is isomorphic to the extension of its restriction to $\\Pscr$, that is $M \\cong M_\\Pscr$.\n\n    Assume given $N : \\Rbf^n \\to \\mathbf{vec}$ and a $\\delta$-interleaving $f : M \\to N[\\delta]$ and $g : N \\to M[\\delta]$ between $M$ and $N$.\n    The interleaving equalities $g[\\delta] \\circ f = \\eta^M_{2\\delta}$ and $f[2\\delta] \\circ g[\\delta] = \\eta^{N[\\delta]}_{2\\delta}$ induce, by \\cref{equation:shift-and-restriction-extension}, the following commutative diagram of $\\Rbf^n$-persistence modules:\n    \\begin{equation}\n        \\label{equation:decomposition-N}\n        \\begin{tikzpicture}[baseline=(current  bounding  box.center)]\n            \\matrix (m) [matrix of math nodes,row sep=4em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {      & N_{\\Pscr + \\delta}[\\delta]  & & N_{\\Pscr + 3\\delta}[3\\delta] \\\\\n                 M_{\\Pscr} &  & M_{\\Pscr + 2\\delta}[2\\delta] &  \\\\};\n            \\path[line width=0.75pt, -{>[width=8pt]}]\n            (m-2-1) edge node [above] {$\\cong$} node [below] {$(\\eta^M_{2\\delta})_{\\Pscr}$} (m-2-3)\n            (m-2-1) edge [>->] node [above left] {$f_{\\Pscr}$} (m-1-2)\n            (m-1-2) edge [->>] node [above right] {$g[\\delta]_{\\Pscr}$} (m-2-3)\n            (m-1-2) edge node [above] {$(\\eta^{N[\\delta]}_{2\\delta})_{\\Pscr}$} (m-1-4)\n            (m-2-3) edge node [below right] {$f[2\\delta]_\\Pscr$} (m-1-4)\n            ;\n        \\end{tikzpicture}\n    \\end{equation}\n    The bottom horizontal morphism is an isomorphism by \\cref{lemma:structure-map-iso} and the fact that $2\\delta < \\alpha\/2 \\leq \\tau\/2$.\n    This implies that the left diagonal morphism is a split monomorphism and that the middle diagonal morphism is a split epimorphism.\n\n    In particular, it follows that $M_{\\Pscr}$ is a direct summand of $N_{\\Pscr+\\delta}[\\delta]$, so that $N_{\\Pscr+\\delta}[\\delta] \\cong M_{\\Pscr} \\oplus X$ for some $X$.\n    At the same time, by \\cref{theorem:decomposition}, there exists a decomposition $N \\cong \\bigoplus_{i \\in I} N_i$ with $N_i$ indecomposable for all $i \\in I$.\n    Using \\cref{lemma:decomposition-restriction-extension}, we see that, by restricting to $\\Pscr$ and extending, we get a decomposition $N_{\\Pscr+\\delta} \\cong \\bigoplus_{i \\in I} (N_i)_{\\Pscr+\\delta}$, where now the summands $\\{(N_i)_{\\Pscr+\\delta}\\}_{i \\in I}$ may not be indecomposable anymore.\n    Nevertheless, since $M_\\Pscr \\cong M$ is indecomposable by assumption, there exists $i \\in I$ such that $(N_i)_{\\Pscr+\\delta}\\cong M_\\Pscr[-\\delta] \\oplus X'$ for some $X'$, using the fact that $N_{\\Pscr+\\delta}[\\delta] \\cong M_{\\Pscr} \\oplus X$ and \\cref{theorem:decomposition}.\n    We consider the decomposition\n    \\[\n        N_{\\Pscr + \\delta}\\;\\; \\cong \\;\\;(N_i)_{\\Pscr + \\delta} \\oplus \\bigoplus_{j \\in I \\setminus i} (N_j)_{\\Pscr + \\delta} \\;\\;\\cong \\;\\;(M_\\Pscr[-\\delta] \\oplus X') \\oplus \\bigoplus_{j \\in I \\setminus i} (N_j)_{\\Pscr + \\delta},\n    \\]\n    and let $A = N_i$ and $B = \\bigoplus_{j \\in I \\setminus i} N_j$.\n    Since this will be of use later, we note here that, by construction, $B_\\Pscr$ is isomorphic to a summand of $X$.\n    \n    We have thus decomposed $N \\cong A \\oplus B$ with $A$ indecomposable.\n    To conclude, it remains to show that $B$ is $\\epsilon$-trivial, and for this it is enough to show that $d_I(B,0) < \\epsilon\/2$.\n    By definition of $\\Pscr$, we have $d_I(B,B_\\Pscr) < \\alpha \\leq \\epsilon\/4$, which reduces the problem to showing that $d_I(B_\\Pscr,0) < \\epsilon\/4$, by the triangle inequality.\n    Since $B_\\Pscr$ is a summand of $X$, it is enough to prove that $d_I(X,0) < \\epsilon\/4$.\n    We will show that $\\eta^X_{\\alpha} : X \\to X[\\alpha]$ is the zero morphism, which implies that $d_I(X,0) \\leq \\alpha\/2 \\leq \\epsilon\/8 < \\epsilon\/4$.\n\n    By the naturality of shifting, that is, by the commutativity of the following square\n    \\[\n    \\begin{tikzpicture}\n        \\matrix (m) [matrix of math nodes,row sep=4em,column sep=7em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n        {  M_\\Pscr \\oplus X  &  M_\\Pscr[\\alpha] \\oplus X[\\alpha] \\\\\n           N_{\\Pscr + \\delta} [\\delta] & N_{\\Pscr + \\delta}[\\delta + \\alpha],\\\\};\n        \\path[line width=0.75pt, -{>[width=8pt]}]\n        (m-1-1) edge node [above] {$\\left(\\eta^{M_\\Pscr}_\\alpha, \\eta^X_\\alpha\\right)$} (m-1-2)\n        (m-1-1) edge node [left] {$\\cong$} (m-2-1)\n        (m-1-2) edge node [left] {$\\cong$} (m-2-2)\n        (m-2-1) edge node [above] {$\\eta^{N_{\\Pscr+\\delta}[\\delta]}_\\alpha$} (m-2-2)\n        ;\n    \\end{tikzpicture}\n    \\]\n    it is sufficient to show that the following composite is the zero morphism:\n    \\begin{equation}\n        \\label{equation:sufficient-zero-morphism}\n        X \\rightarrowtail M_{\\Pscr} \\oplus X \\xrightarrow{\\cong} N_{\\Pscr+\\delta}[\\delta] \\xrightarrow{\\eta^{N_{\\Pscr+\\delta}[\\delta]}_\\alpha} N_{\\Pscr+\\delta}[\\delta + \\alpha].\n    \\end{equation}\n    Note that, by the construction of the splitting $N_{\\Pscr+\\delta}[\\delta] \\cong M_{\\Pscr} \\oplus X$ and the commutativity of the right triangle in \\cref{equation:decomposition-N}, the following composite is the zero morphism\n    \\begin{equation}\n        \\label{equation:zero-morphism}\n        X \\rightarrowtail M_{\\Pscr} \\oplus X \\xrightarrow{\\cong} N_{\\Pscr+\\delta}[\\delta]\n        \\xrightarrow{(\\eta^{N[\\delta]}_{2\\delta})_{\\Pscr}} N_{\\Pscr + 3\\delta}[3\\delta].\n    \\end{equation}\n    Thus, it suffices to see that the right-most morphism in \\cref{equation:sufficient-zero-morphism} factors through the right-most morphism in \\cref{equation:zero-morphism}.\n    Letting $L = N[\\delta]$ and using \\cref{equation:shift-and-restriction-extension},\n    this translates to the claim that the morphism \n    $\\eta^{L_{\\Pscr}}_\\alpha : L_{\\Pscr} \\to L_{\\Pscr+\\delta}[\\delta + \\alpha]$ factors through $(\\eta^{L}_{2\\delta})_{\\Pscr} : L_{\\Pscr} \\to L_{\\Pscr + 2\\delta}[2\\delta]$, which follows from \\cref{lemma:factor-bounded-grid} by letting $r = 2\\delta = \\alpha\/2 < \\alpha$.\n\\end{proof}\n\n\\begin{corollary}\n    \\label{corollary:open-with-indecomposables}\n    Let $n \\geq 1$.\n    For every $\\epsilon \\geq 0$, there exists an open set $O^\\epsilon$ of isomorphism classes of finitely presentable $n$-parameter persistence modules, containing the set of all indecomposable finitely presentable modules as a subset, and such that every $N \\in O^\\epsilon$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $d_I(B,0) < \\epsilon$.\n\\end{corollary}\n\\begin{proof}\n    Fix $\\epsilon > 0$.\n    By \\cref{proposition:stability-indecomposability}, for every indecomposable finitely presentable $M : \\Rbf^n \\to \\mathbf{vec}$ there exists $\\delta^M > 0$ such that, for all finitely presentable $N : \\Rbf^n \\to \\mathbf{vec}$ in the open $\\delta^M$-ball around $M$ in the interleaving distance, it holds that $N$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n    It is then clear that the set\n    \\[\n        O^\\epsilon := \\bigcup_{\\substack{M \\text{ finitely presentable} \\\\ \\text{and indecomposable}}} \\{N : \\text{ $N$ is finitely presentable and } d_I(M,N) < \\delta^M \\}\n    \\]\n    satisfies the conditions in the statement.\n\\end{proof}\n\n\n\n\\section{Indecomposables are dense}\n\\label{section:indecomposables-dense}\n\n\nThe proof of \\cref{theorem:indecomposables-dense} depends on the following key lemma, which lets us ``tack together'' two indecomposable modules to obtain a third indecomposable module that is at small interleaving distance from the direct sum of the initial modules.\nThe proof is in \\cref{section:proof-main-lemma}.\n\n\\begin{restatable}[Tacking Lemma]{lemma}{mainlemma}\n    \\label{lemma:main-lemma-attaching}\n    Let $A,B : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable, indecomposable, and isomorphic to extensions of $\\Pscr$-persistence modules for some regular grid $\\Pscr \\subseteq \\Rbf^2$.\n    For every $\\delta > 0$ there exists a regular grid $\\Qscr \\supseteq \\Pscr$ and $M : \\Rbf^2 \\to \\mathbf{vec}$ with $M$ indecomposable, finitely presentable, isomorphic to the extension of a $\\Qscr$-persistence module, and such that\n    \\[\n        \\mathsf{im}\\left(\\eta^M_\\delta\\right) \\cong \\mathsf{im}\\left(\\eta^A_\\delta\\right) \\oplus \\mathsf{im}\\left(\\eta^B_\\delta\\right).\n    \\]\n\\end{restatable}\n\nWe also need the following two simple lemmas.\n\n\\begin{lemma}\n    \\label{lemma:image-to-interleaving}\n    Let $A : \\Rbf^n \\to \\mathbf{vec}$ and let $\\delta \\geq 0$.\n    Consider $\\mathsf{im}(\\eta^A_\\delta ) : \\Rbf^n \\to \\mathbf{vec}$, the image of the morphism of persistence modules $\\eta^A_\\delta : A \\to A[\\delta]$ given by the structure morphisms.\n    Then $A$ and $\\mathsf{im}(\\eta^A_\\delta)$ are $\\delta$-interleaved.\n    As a consequence, if $B : \\Rbf^n \\to \\mathbf{vec}$ satisfies $\\mathsf{im}(\\eta^A_\\delta) \\cong \\mathsf{im}(\\eta^B_\\delta)$, then $A$ and $B$ are $2\\delta$-interleaved.\n\\end{lemma}\nIn fact, the $2\\delta$ can be improved to $\\delta$ using the notion of asymmetric interleaving \\cite[Section~2.6.1]{lesnick-thesis}, but this is not necessary for our purposes.\n\\begin{proof}\n    The second statement follows directly from the first one.\n    By construction, there is a canonical epimorphism $A \\to \\mathsf{im}(\\eta^A_\\delta)$, and thus we get a morphism $f : A \\to \\mathsf{im}(\\eta^A_\\delta)[\\delta]$, by composing with $\\eta^{\\mathsf{im}(\\eta^A_\\delta)}_\\delta: \\mathsf{im}(\\eta^A_\\delta) \\to \\mathsf{im}(\\eta^A_\\delta)[\\delta]$.\n    Similarly, there is a canonical monomorphism $g : \\mathsf{im}(\\eta^A_\\delta) \\to A[\\delta]$.\n    A straightforward check shows that $f$ and $g$ form a $\\delta$-interleaving.\n\\end{proof}\n\n\\begin{lemma}\n    \\label{lemma:eta-direct-sum}\n    Let $A,B : \\Rbf^n \\to \\mathbf{vec}$ and $\\epsilon \\geq 0$.\n    Then $\\mathsf{im}\\left(\\eta^A_\\epsilon\\right) \\oplus\\mathsf{im}\\left(\\eta^B_\\epsilon\\right) \\cong \\mathsf{im}\\left(\\eta^{A \\oplus B}_\\epsilon\\right)$.\n    \\qed\n\\end{lemma}\n\n\\indecomposabledense*\n\\begin{proof}\n    We start by reducing the problem to the case in which the given module is an extension of a module defined over a regular grid.\n    Consider the regular grid $\\Pscr = ((\\epsilon\/2)\\Zbf)^2$ and note that $d_I(N_\\Pscr,N) \\leq \\epsilon\/2$.\n    Let $L = N_\\Pscr$.\n    It is easy to see that $L$ is finitely presentable.\n\n    If $L = 0$, then take $M$ to be the persistence module such that $M(x,y) = \\kbb$ if $0 \\leq x < \\epsilon$ and $y \\geq 0$, and $0$ otherwise, with all the structure morphisms that are not forced to be the zero morphism being the identity $\\kbb \\to \\kbb$.\n    It is straightforward to see that $M$ is indecomposable and that $d_I(M,0) = \\epsilon\/2$.\n    This case now follows from $d_I(N,M) \\leq d_I(N,L) + d_I(L,M) \\leq \\epsilon\/2 + d_I(0,M) \\leq \\epsilon$.\n\n    We now assume that $L$ is non-zero.\n    By \\cref{lemma:decomposition-fp}, there exist finitely presentable indecomposables $X_1, \\dots, X_k : \\Rbf^2 \\to \\mathbf{vec}$ such that $L \\cong X_1 \\oplus \\dots \\oplus X_k$.\n    Note that, for all $1 \\leq i \\leq k$, the persistence module $X_i$ is isomorphic to the extension of a $\\Qscr$-persistence module for any grid $\\Qscr \\supseteq \\Pscr$.\n    Let us define $M_i : \\Rbb^2 \\to \\mathbf{vec}$ for $1 \\leq i \\leq k$ inductively.\n    Let $M_1 = X_1$ and, for $2 \\leq i \\leq k$, let $M_{i}$ be obtained by setting $A = X_i$, $B = M_{i-1}$, and $\\delta = \\epsilon\/4$ in \\cref{lemma:main-lemma-attaching}.\n    Thus, $M_i$ is finitely presentable, indecomposable, and satisfies\n    \\[\n        \\mathsf{im}\\left(\\eta^{M_i}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^{X_i}_{\\epsilon\/4}\\right) \\oplus \\mathsf{im}\\left(\\eta^{M_{i-1}}_{\\epsilon\/4}\\right).\n    \\]\n    Let $M = M_k$.\n    It follows by induction and \\cref{lemma:eta-direct-sum} that \n    \\[\n        \\mathsf{im}\\left(\\eta^{M}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^{X_1}_{\\epsilon\/4}\\right) \\oplus \\dots \\oplus \\mathsf{im}\\left(\\eta^{X_k}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^{X_1 \\oplus \\dots \\oplus X_k}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^L_{\\epsilon\/4}\\right).\n    \\]\n    Using \\cref{lemma:image-to-interleaving}, we get that $M$ and $L$ are $\\epsilon\/2$-interleaved.\n    By the triangle inequality, we get $d_I(M,N) \\leq d_I(M,L) + d_I(L,N) \\leq \\epsilon$, as required.\n\\end{proof}\n\nWe can now give the proof of our main result.\n\n\\begin{proof}[Proof of \\cref{theorem:main-theorem}]\n    Given $\\epsilon > 0$, the set $O^\\epsilon$ of \\cref{corollary:open-with-indecomposables} is open and contains all indecomposables finitely presentable modules.\n    Since, when $n=2$, the set of indecomposables is dense by \\cref{theorem:indecomposables-dense}, the result follows.\n\\end{proof}\n\n\n\\section{Tacking indecomposables together}\n\\label{section:proof-main-lemma}\n\nIn this section, we prove \\cref{lemma:main-lemma-attaching} (the Tacking Lemma).\nWe first summarize the strategy.\n\n\n\\begin{proof}[Proof outline]\\renewcommand{\\qedsymbol}{}\nWe start by defining a persistence module $\\Gsf$ on a finite grid which allows us to ``tack together'' indecomposable modules.\nNow, fix modules $A$ and $B$ as in the statement.\nThe tacking works in three steps.\nIn \\cref{lemma:add-1-dim-corner}, we replace $A$ and $B$ by modules which admit a ``thin corner'' (\\cref{definition:1-dim-corner}); the purpose of this thin corner is that it allows us to attach a copy of $\\Gsf$ to each of the modules, which we do in the next step.\nIn \\cref{lemma:add-antenna-attachment}, we replace the modules by modules that admit a ``horizontal antenna attachment'' (\\cref{definition:horizontal-antenna-attachment}); the purpose of this horizontal antenna attachment is that it allows us to tack together the two modules with a third copy of $\\Gsf$. \nFinally, in \\cref{lemma:actual-tacking}, we construct $M$ by tacking together the two modules.\n\\cref{figure:diagram-proof-main-lemma} shows a diagrammatic description of these steps.\n\\end{proof}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{pictures\/diagram-steps.eps}\n    \\caption{A schematic summary of the main steps in the proof of the tacking lemma.\n    The transition from (0.) to (1.) corresponds to \\cref{lemma:add-1-dim-corner}; the transition from (1.) to (2.) corresponds to \\cref{lemma:add-antenna-attachment}; the transition from (2.) to (3.) corresponds to \\cref{lemma:actual-tacking}.}\n    \\label{figure:diagram-proof-main-lemma}\n\\end{figure}\n\n\\begin{definition}\n\\label{def:G}\n    Let $\\Pscr = \\{0, 1, 2, 3, 4\\}^2$ and define $\\Gsf : \\Pscr \\to \\mathbf{vec}$ as follows:\n    \\[ \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {  \\kbb & \\kbb   & \\kbb   & \\kbb   & \\kbb \\\\\n               \\kbb & \\kbb^2 & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n               0    & \\kbb   & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n               0    & 0      & \\kbb   & \\kbb^2 & \\kbb \\\\\n               0    & 0      & 0      & \\kbb   & \\kbb \\\\};\n            \\path[line width=0.5pt, -{>[width=6pt]}]\n            (m-1-1) edge [-,double equal sign distance] (m-1-2)\n            (m-1-2) edge [-,double equal sign distance] (m-1-3)\n            (m-1-3) edge [-,double equal sign distance] (m-1-4)\n            (m-1-4) edge [-,double equal sign distance] (m-1-5)\n\n            (m-2-2) edge [-,double equal sign distance] (m-2-3)\n            (m-2-3) edge [-,double equal sign distance] (m-2-4)\n\n            (m-3-3) edge [-,double equal sign distance] (m-3-4)\n\n            (m-5-5) edge [-,double equal sign distance] (m-4-5)\n            (m-4-5) edge [-,double equal sign distance] (m-3-5)\n            (m-3-5) edge [-,double equal sign distance] (m-2-5)\n            (m-2-5) edge [-,double equal sign distance] (m-1-5)\n\n            (m-4-4) edge [-,double equal sign distance] (m-3-4)\n            (m-3-4) edge [-,double equal sign distance] (m-2-4)\n\n            (m-3-3) edge [-,double equal sign distance] (m-2-3)\n\n            (m-2-1) edge node [left] {\\footnotesize $0$} (m-1-1)\n            (m-2-1) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-2-2)\n            (m-2-2) edge node [right] {\\footnotesize $(1,-1)$} (m-1-2)\n            (m-2-3) edge node [right] {\\footnotesize $(1,-1)$} (m-1-3)\n            (m-2-4) edge node [right] {\\footnotesize $(1,-1)$} (m-1-4)\n\n            (m-5-4) edge node [above] {\\footnotesize $0$} (m-5-5)\n            (m-5-4) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-4-4)\n            (m-4-4) edge node [above] {\\footnotesize $(1,-1)$} (m-4-5)\n            (m-3-4) edge node [above] {\\footnotesize $(1,-1)$} (m-3-5)\n            (m-2-4) edge node [above] {\\footnotesize $(1,-1)$} (m-2-5)\n\n            (m-3-2) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 0\\end{pmatrix}$} (m-2-2)\n            (m-3-2) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 0\\end{pmatrix}$} (m-3-3)\n\n            (m-4-3) edge node [left] {\\footnotesize $\\begin{pmatrix} 0\\\\ 1\\end{pmatrix}$} (m-3-3)\n            (m-4-3) edge node [above] {\\footnotesize $\\begin{pmatrix} 0\\\\ 1\\end{pmatrix}$} (m-4-4)\n\n            (m-3-1) edge (m-2-1)\n            (m-4-1) edge (m-3-1)\n            (m-5-1) edge (m-4-1)\n\n            (m-4-2) edge (m-3-2)\n            (m-5-2) edge (m-4-2)\n\n            (m-5-3) edge (m-4-3)\n\n            (m-5-1) edge (m-5-2)\n            (m-5-2) edge (m-5-3)\n            (m-5-3) edge (m-5-4)\n\n            (m-4-1) edge (m-4-2)\n            (m-4-2) edge (m-4-3)\n        \n            (m-3-1) edge (m-3-2)\n            ;\n        \\end{tikzpicture}\n    \\]\n\\end{definition}\n\n\n\n\\begin{restatable}{lemma}{Gindecomposable}\n    \\label{lemma:G-is-indecomposable}\n    The persistence module $\\Gsf$ is indecomposable.\n    More specifically, the morphism of rings $\\End(\\Gsf) \\to \\End(\\Gsf(4,4)) \\cong \\kbb$ given by evaluation at $(4,4)$ is an isomorphism.\n\\end{restatable}\n\n\\begin{proof}\n\n\n    Let $f : \\Gsf \\to \\Gsf$ be an endomorphism.\n    If $f(4,4) : \\kbb \\to \\kbb$ is given by multiplication by $\\alpha$, then, for all $0 \\leq j \\leq 4$, we have that $f(4,j) : \\kbb \\to \\kbb$ and $f(j,4) : \\kbb \\to \\kbb$ must also be given by multiplication by $\\alpha$.\n\n    Assume that $f(1,2) : \\kbb \\to \\kbb$ is given by multiplication by $\\beta$ and that $f(2,1) : \\kbb \\to \\kbb$ is given by multiplication by $\\gamma$.\n    Since the structure morphisms $\\kbb = \\Gsf(1,2) \\to \\Gsf(4,4) = \\kbb$ is the identity, we must have $\\beta = \\alpha$.\n    By an analogous argument, we have $\\gamma = \\alpha$.\n    This also implies that for all $(j,k)$ with $\\Gsf(j,k) = \\kbb^2$ the endomorphism $f(j,k)$ is given by multiplication by $\\alpha$.\n   \n    \n    To conclude, note that, by the definition of the structure morphisms $\\Gsf(0,3) \\to \\Gsf(1,3)$ and $\\Gsf(3,0) \\to \\Gsf(3,1)$, we have that $f(0,3)$ and $f(3,0)$ are also given by multiplication by $\\alpha$.\n    Thus, $\\End(\\Gsf) \\cong \\End(\\Gsf(4,4)) \\cong \\kbb$, and thus $\\Gsf$ is indecomposable, by \\cref{lemma:indecomposable-local-ring}.\n\\end{proof}\n\nThe crucial property of $\\Gsf$ of importance to us, other than being indecomposable, is the fact that there are two pairs of adjacent copies of the ground field $\\kbb$ connected by a zero map, on the bottom right at $(0,3)$ and $(0,4)$, and on the top left at $(3,0)$ and $(4,0)$.\n\n\\begin{definition}\n    \\label{definition:1-dim-corner}\n    Let $X : \\Rbf^2 \\to \\mathbf{vec}$.\n    We say that $X$ \\define{admits a thin corner} if there exists $\\epsilon > 0$, $X' : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$, and $(x,y) \\in (\\epsilon \\Zbf)^2$ such that $X$ is isomorphic to the extension of $X'$, and we have $X'(x,y) = \\kbb$ and $X'(x-\\epsilon,y) = X'(x,y-\\epsilon) = 0$.\n\\end{definition}\n\nSee \\cref{figure:diagram-proof-main-lemma}(1.) for a schematic depiction of two modules admitting a thin corner.\n\n\\begin{lemma}\n    \\label{lemma:add-1-dim-corner}\n    Let $A : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable, indecomposable, and isomorphic to the extension of a $\\Pscr$-persistence module for some regular grid $\\Pscr \\subseteq \\Rbf^2$.\n    For every $\\delta > 0$, there exists $A_1 : \\Rbf^2 \\to \\mathbf{vec}$ indecomposable, finitely presentable, admitting a thin corner, isomorphic to an extension of a $\\Qscr$-persistence module for some regular grid $\\Qscr \\supseteq \\Pscr$, and such that $\\mathsf{im}\\left(\\eta^A_\\delta\\right) \\cong \\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big)$.\n\\end{lemma}\n\\begin{proof}\n    Assume that $\\Pscr = (\\epsilon \\Zbf)^2$.\n    Let $m \\in \\Nbb$ be such that $\\epsilon\/m < \\delta$.\n    Let $\\tau = \\epsilon\/m$.\n    Since $A$ is finitely presentable, there exists $(x,y) \\in (\\tau\\Zbf)^2$ such that $A(x,y) \\neq 0$ and $A(x-\\tau,y) = 0 = A(x,y-\\tau)$.\n    Thus, on the regular grid $((\\tau\/2)\\Zbf)^2$, and around index $(x,y)$ (highlighted entry), the module $A$ restricts as follows:\n    \\[ \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=1.5em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {  A(x-\\tau,y+\\tau) & A(x-\\tau\/2,y+\\tau) & A(x,y+\\tau) \n                   & A(x+\\tau\/2,y+\\tau)   & A(x+\\tau,y+\\tau) \\\\\n               0 & 0 & A(x,y+\\tau\/2)\n                   & A(x+\\tau\/2,y+\\tau\/2) & A(x+\\tau,y+\\tau\/2) \\\\\n               0    & 0      & |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]| A(x,y) & A(x+\\tau\/2,y) & A(x+\\tau,y) \\\\\n               0    & 0      & 0      & 0 & A(x+\\tau,y-\\tau\/2) \\\\\n               0    & 0      & 0      & 0   & A(x+\\tau,y-\\tau) \\\\};\n            \\path[line width=0.5pt, -{>[width=4pt]}]\n            (m-1-1) edge node [above] {$\\cong$} (m-1-2)\n            (m-2-1) edge node [above] {$\\cong$} (m-2-2)\n            (m-3-1) edge node [above] {$\\cong$} (m-3-2)\n            (m-4-1) edge node [above] {$\\cong$} (m-4-2)\n            (m-5-1) edge node [above] {$\\cong$} (m-5-2)\n\n            (m-1-2) edge (m-1-3)\n            (m-2-2) edge (m-2-3)\n            (m-3-2) edge (m-3-3)\n            (m-4-2) edge (m-4-3)\n            (m-5-2) edge (m-5-3)\n\n            (m-1-3) edge node [above] {$\\cong$} (m-1-4)\n            (m-2-3) edge node [above] {$\\cong$} (m-2-4)\n            (m-3-3) edge node [above] {$\\cong$} (m-3-4)\n            (m-4-3) edge node [above] {$\\cong$} (m-4-4)\n            (m-5-3) edge node [above] {$\\cong$} (m-5-4)\n\n            (m-1-4) edge (m-1-5)\n            (m-2-4) edge (m-2-5)\n            (m-3-4) edge (m-3-5)\n            (m-4-4) edge (m-4-5)\n            (m-5-4) edge (m-5-5)\n\n            (m-5-1) edge node [left] {$\\cong$} (m-4-1)\n            (m-5-2) edge node [left] {$\\cong$} (m-4-2)\n            (m-5-3) edge node [left] {$\\cong$} (m-4-3)\n            (m-5-4) edge node [left] {$\\cong$} (m-4-4)\n            (m-5-5) edge node [left] {$\\cong$} (m-4-5)\n\n            (m-4-1) edge (m-3-1)\n            (m-4-2) edge (m-3-2)\n            (m-4-3) edge (m-3-3)\n            (m-4-4) edge (m-3-4)\n            (m-4-5) edge (m-3-5)\n\n            (m-3-1) edge node [left] {$\\cong$} (m-2-1)\n            (m-3-2) edge node [left] {$\\cong$} (m-2-2)\n            (m-3-3) edge node [left] {$\\cong$} (m-2-3)\n            (m-3-4) edge node [left] {$\\cong$} (m-2-4)\n            (m-3-5) edge node [left] {$\\cong$} (m-2-5)\n\n            (m-2-1) edge (m-1-1)\n            (m-2-2) edge (m-1-2)\n            (m-2-3) edge (m-1-3)\n            (m-2-4) edge (m-1-4)\n            (m-2-5) edge (m-1-5)\n            ;\n        \\end{tikzpicture}\n    \\]\n    where the isomorphisms are due to the fact that $A$ is an extension of a persistence module on the regular grid $(\\tau\\Zbf)^2$.\n\n    Let $\\kbb \\to A(x,y)$ be any non-zero morphism.\n    We now define a persistence module $A'$, and then define $A_1$ to be an extension of $A'$.\n    Let $A' : ((\\tau\/2)\\Zbf)^2 \\to \\mathbf{vec}$ coincide with the restriction of $A$ to $((\\tau\/2)\\Zbf)^2$ at all places, except at $(x,y)$ where it takes the value $\\kbb$.\n    The structure morphisms $A'(x,y) \\to A'(x,y+\\tau\/2) = A(x,y+\\tau\/2) $ and $A'(x,y) \\to A'(x+\\tau\/2, y) = A(x+\\tau\/2, y)$ are defined by composing the corresponding structure morphisms of $A$ with the map $\\kbb \\to A(x,y)$.\n    Thus, around index $(x,y)$ (again highlighted), the module $A'$ looks as follows:\n    \\[  \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=1.5em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {  A(x-\\tau,y+\\tau) & A(x-\\tau\/2,y+\\tau) & A(x,y+\\tau) \n                   & A(x+\\tau\/2,y+\\tau)   & A(x+\\tau,y+\\tau) \\\\\n               0 & 0 & A(x,y+\\tau\/2)\n                   & A(x+\\tau\/2,y+\\tau\/2) & A(x+\\tau,y+\\tau\/2) \\\\\n               0    & 0      & |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]| \\kbb & A(x+\\tau\/2,y) & A(x+\\tau,y) \\\\\n               0    & 0      & 0      & 0 & A(x+\\tau,y-\\tau\/2) \\\\\n               0    & 0      & 0      & 0   & A(x+\\tau,y-\\tau) \\\\};\n            \\path[line width=0.5pt, -{>[width=4pt]}]\n            (m-1-1) edge (m-1-2)\n            (m-2-1) edge (m-2-2)\n            (m-3-1) edge (m-3-2)\n            (m-4-1) edge (m-4-2)\n            (m-5-1) edge (m-5-2)\n\n            (m-1-2) edge (m-1-3)\n            (m-2-2) edge (m-2-3)\n            (m-3-2) edge (m-3-3)\n            (m-4-2) edge (m-4-3)\n            (m-5-2) edge (m-5-3)\n\n            (m-1-3) edge (m-1-4)\n            (m-2-3) edge (m-2-4)\n            (m-3-3) edge (m-3-4)\n            (m-4-3) edge (m-4-4)\n            (m-5-3) edge (m-5-4)\n\n            (m-1-4) edge (m-1-5)\n            (m-2-4) edge (m-2-5)\n            (m-3-4) edge (m-3-5)\n            (m-4-4) edge (m-4-5)\n            (m-5-4) edge (m-5-5)\n\n            (m-5-1) edge (m-4-1)\n            (m-5-2) edge (m-4-2)\n            (m-5-3) edge (m-4-3)\n            (m-5-4) edge (m-4-4)\n            (m-5-5) edge (m-4-5)\n\n            (m-4-1) edge (m-3-1)\n            (m-4-2) edge (m-3-2)\n            (m-4-3) edge (m-3-3)\n            (m-4-4) edge (m-3-4)\n            (m-4-5) edge (m-3-5)\n\n            (m-3-1) edge (m-2-1)\n            (m-3-2) edge (m-2-2)\n            (m-3-3) edge (m-2-3)\n            (m-3-4) edge (m-2-4)\n            (m-3-5) edge (m-2-5)\n\n            (m-2-1) edge (m-1-1)\n            (m-2-2) edge (m-1-2)\n            (m-2-3) edge (m-1-3)\n            (m-2-4) edge (m-1-4)\n            (m-2-5) edge (m-1-5)\n            ;\n        \\end{tikzpicture}\n    \\]\n    We have the following isomorphisms of endomorphism rings:\n    \\[\n        \\End(A) \\cong \\End\\left(A|_{((\\tau\/2)(2\\Zbf+1))^2}\\right) = \\End\\left(A'|_{((\\tau\/2)(2\\Zbf+1))^2}\\right) \\cong \\End\\left(A'|_{((\\tau\/2)\\Zbf)^2}\\right).\n    \\]\n    The first isomorphism is due to the fact that $A$ is isomorphic to the extension of a $(\\tau\\Zbf)^2$-persistence module.\n    The equality is because\n    $A|_{((\\tau\/2)(2\\Zbf+1))^2} = A'|_{((\\tau\/2)(2\\Zbf+1))^2}$.\n    For the last isomorphism, note that the value of $A'$ at any element of the grid $((\\tau\/2)\\Zbf)^2$ other than $(x,y)$ is isomorphic, through a structure morphism of $A'$, to the value of $A'$ at an element of the grid $((\\tau\/2)(2\\Zbf+1))^2$, by construction.\n    Moreover, by construction, the structure morphism $A'(x,y) = \\kbb \\to A'(x+\\tau\/2, y+\\tau\/2)$ is non-zero, and thus a monomorphism; and $(x+\\tau\/2, y+\\tau\/2) \\in ((\\tau\/2)(2\\Zbf+1))^2$.\n    It follows that the action of an endomorphism of $A'$ is completely determined by the action of the endomorphism on the restriction $A'|_{((\\tau\/2)(2\\Zbf+1))^2}$, proving the last isomorphism.\n\n    Thus, $A'$ is indecomposable by \\cref{lemma:indecomposable-local-ring} and the fact that $A$ is indecomposable.\n    Define $A_1 = \\widehat{A'}$, the extension of $A'$ to $\\Rbf^2$.\n    Note that $A_1$ is indecomposable, finitely presentable, and satisfies $\\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big) = \\mathsf{im}\\left(\\eta^{A}_\\delta\\right)$.\n    Moreover, $A_1$ admits a thin corner, by construction.\n\\end{proof}\n\n\\begin{definition}\n    \\label{definition:horizontal-antenna-attachment}\n    Let $X : \\Rbf^2 \\to \\mathbf{vec}$ and $(x,y) \\in \\Rbf^2$.\n    We say that $X$ \\define{admits a horizontal antenna attachment} at $(x,y)$ if there exists $\\epsilon > 0$ and $X' : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$, such that $(x,y) \\in (\\epsilon \\Zbf)^2$, $X$ is isomorphic to the extension of $X'$, and we have $X'(x,y) = \\kbb$, $X'(x-k\\epsilon,y) = 0$ for all $k \\geq 1 \\in \\Nbb$, and the structure morphism $X'(x,y) \\to X'(x,y+\\epsilon)$ is zero.\n\\end{definition}\n\nFor example, the extension $\\widehat\\Gsf: \\Rbf^2 \\to \\mathbf{vec}$ admits a horizontal antenna attachment.\nSee \\cref{figure:diagram-step-3} for a schematic depiction of two modules admitting a horizontal antenna attachment.\n\n\\begin{lemma}\n    \\label{lemma:add-antenna-attachment}\n    Let $A : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable, indecomposable, and admitting a thin corner.\n    For every $\\delta > 0$, there exists $A_1 : \\Rbf^2 \\to \\mathbf{vec}$ indecomposable, finitely presentable, admitting a horizontal antenna attachment, and such that $\\mathsf{im}\\left(\\eta^A_\\delta\\right) \\cong \\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big)$.\n\\end{lemma}\n\n\\begin{proof}\n    Since $A$ admits a thin corner, we may assume the following:\n    there exists a regular grid $(\\epsilon \\Zbf)^2$ for some $\\epsilon > 0$ and $X : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$ such that $A \\cong \\widehat{X}$, and such that there exists $(x,y) \\in (\\epsilon \\Zbf)^2$ with $X(x,y) = \\kbb$ and $X(x-\\epsilon,y) = X(x,y-\\epsilon) = 0$.\n\n    Let $m \\in \\Nbb$ be such that $\\epsilon\/m < \\delta$ and let $\\tau = \\epsilon\/m$.\n    We make use of the regular grid $((\\tau\/5)\\Zbf)^2$.\n    By extending $X$ to $\\Rbf^2$ and restricting to $((\\tau\/5)\\Zbf)^2$, we get a module $X' : ((\\tau\/5)\\Zbf)^2 \\to \\mathbf{vec}$ that, when restricted to the finite grid\n    \\begin{align*}\n        \\Tscr = \\{(&x-\\tau\/5,x,x+\\tau\/5,x+2\\tau\/5,x+3\\tau\/5,x+4\\tau\/5)\\}\\\\\n        &\\times \\{(y-\\tau\/5,y,y+\\tau\/5,y+2\\tau\/5,y+3\\tau\/5,y+4\\tau\/5)\\},\n    \\end{align*}\n    looks as follows (index $(x,y)$ highlighted):\n    \\[\\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            { 0 & \\kbb & \\kbb   & \\kbb   & \\kbb   & \\kbb \\\\\n              0 & \\kbb & \\kbb & \\kbb & \\kbb & \\kbb \\\\\n              0 &  \\kbb    & \\kbb   & \\kbb & \\kbb & \\kbb \\\\\n             0 &   \\kbb    & \\kbb      & \\kbb   & \\kbb & \\kbb \\\\\n             0 &  |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]| \\kbb    & \\kbb      & \\kbb      & \\kbb   & \\kbb \\\\\n             0 &      0    & 0         & 0         & 0      &  0 \\\\};\n            \\path[line width=0.5pt, -{>[width=6pt]}]\n            (m-1-2) edge [-,double equal sign distance] (m-1-3)\n            (m-1-3) edge [-,double equal sign distance] (m-1-4)\n            (m-1-4) edge [-,double equal sign distance] (m-1-5)\n            (m-1-5) edge [-,double equal sign distance] (m-1-6)\n            (m-2-3) edge [-,double equal sign distance] (m-2-4)\n            (m-2-4) edge [-,double equal sign distance] (m-2-5)\n            (m-3-4) edge [-,double equal sign distance] (m-3-5)\n            (m-5-6) edge [-,double equal sign distance] (m-4-6)\n            (m-4-6) edge [-,double equal sign distance] (m-3-6)\n            (m-3-6) edge [-,double equal sign distance] (m-2-6)\n            (m-2-6) edge [-,double equal sign distance] (m-1-6)\n            (m-4-5) edge [-,double equal sign distance] (m-3-5)\n            (m-3-5) edge [-,double equal sign distance] (m-2-5)\n            (m-3-4) edge [-,double equal sign distance] (m-2-4)\n            (m-2-2) edge [-,double equal sign distance] (m-1-2)\n            (m-2-2) edge [-,double equal sign distance] (m-2-3)\n            (m-2-3) edge [-,double equal sign distance] (m-1-3)\n            (m-2-4) edge [-,double equal sign distance] (m-1-4)\n            (m-2-5) edge [-,double equal sign distance] (m-1-5)\n            (m-5-5) edge [-,double equal sign distance] (m-5-6)\n            (m-5-5) edge [-,double equal sign distance] (m-4-5)\n            (m-4-5) edge [-,double equal sign distance] (m-4-6)\n            (m-3-5) edge [-,double equal sign distance] (m-3-6)\n            (m-2-5) edge [-,double equal sign distance] (m-2-6)\n            (m-3-3) edge [-,double equal sign distance] (m-2-3)\n            (m-3-3) edge [-,double equal sign distance] (m-3-4)\n            (m-4-4) edge [-,double equal sign distance] (m-3-4)\n            (m-4-4) edge [-,double equal sign distance] (m-4-5)\n            (m-3-2) edge [-,double equal sign distance] (m-2-2)\n            (m-4-2) edge [-,double equal sign distance] (m-3-2)\n            (m-5-2) edge [-,double equal sign distance] (m-4-2)\n            (m-4-3) edge [-,double equal sign distance] (m-3-3)\n            (m-5-3) edge [-,double equal sign distance] (m-4-3)\n            (m-5-4) edge [-,double equal sign distance] (m-4-4)\n            (m-5-2) edge [-,double equal sign distance] (m-5-3)\n            (m-5-3) edge [-,double equal sign distance] (m-5-4)\n            (m-5-4) edge [-,double equal sign distance] (m-5-5)\n            (m-4-2) edge [-,double equal sign distance] (m-4-3)\n            (m-4-3) edge [-,double equal sign distance] (m-4-4)\n            (m-3-2) edge [-,double equal sign distance] (m-3-3)\n\n            (m-1-1) edge (m-1-2)\n            (m-2-1) edge (m-2-2)\n            (m-3-1) edge (m-3-2)\n            (m-4-1) edge (m-4-2)\n            (m-5-1) edge (m-5-2)\n            (m-6-1) edge (m-6-2)\n\n            (m-6-2) edge (m-6-3)\n            (m-6-3) edge (m-6-4)\n            (m-6-4) edge (m-6-5)\n            (m-6-5) edge (m-6-6)\n\n            (m-2-1) edge (m-1-1)\n            (m-3-1) edge (m-2-1)\n            (m-4-1) edge (m-3-1)\n            (m-5-1) edge (m-4-1)\n            (m-6-1) edge (m-5-1)\n\n            (m-6-2) edge (m-5-2)\n            (m-6-3) edge (m-5-3)\n            (m-6-4) edge (m-5-4)\n            (m-6-5) edge (m-5-5)\n            (m-6-6) edge (m-5-6)\n            ;\n        \\end{tikzpicture}\n    \\]\n    Let $A' : ((\\tau\/5)\\Zbf)^2 \\to \\mathbf{vec}$ coincide with $X'$ at all places, except at the subgrid $\\Tscr$, where we use a copy of the module $\\Gsf$ as follows (index $(x,y)$ highlighted):\n    \\[ \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            { 0 & \\kbb & \\kbb   & \\kbb   & \\kbb   & \\kbb \\\\\n              0 & \\kbb & \\kbb^2 & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n              0 & 0    & \\kbb   & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n              0 & 0    & 0      & \\kbb   & \\kbb^2 & \\kbb \\\\\n              0 & |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]|0    & 0      & 0      & \\kbb   & \\kbb \\\\\n              0 & 0    & 0      & 0      & 0      & 0    \\\\};\n            \\path[line width=0.5pt, -{>[width=6pt]}]\n            (m-1-2) edge [-,double equal sign distance] (m-1-3)\n            (m-1-3) edge [-,double equal sign distance] (m-1-4)\n            (m-1-4) edge [-,double equal sign distance] (m-1-5)\n            (m-1-5) edge [-,double equal sign distance] (m-1-6)\n\n            (m-2-3) edge [-,double equal sign distance] (m-2-4)\n            (m-2-4) edge [-,double equal sign distance] (m-2-5)\n\n            (m-3-4) edge [-,double equal sign distance] (m-3-5)\n\n            (m-5-6) edge [-,double equal sign distance] (m-4-6)\n            (m-4-6) edge [-,double equal sign distance] (m-3-6)\n            (m-3-6) edge [-,double equal sign distance] (m-2-6)\n            (m-2-6) edge [-,double equal sign distance] (m-1-6)\n\n            (m-4-5) edge [-,double equal sign distance] (m-3-5)\n            (m-3-5) edge [-,double equal sign distance] (m-2-5)\n\n            (m-3-4) edge [-,double equal sign distance] (m-2-4)\n\n            (m-2-2) edge node [left] {\\footnotesize $0$} (m-1-2)\n            (m-2-2) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-2-3)\n            (m-2-3) edge node [right] {\\footnotesize $(1,-1)$} (m-1-3)\n            (m-2-4) edge node [right] {\\footnotesize $(1,-1)$} (m-1-4)\n            (m-2-5) edge node [right] {\\footnotesize $(1,-1)$} (m-1-5)\n\n            (m-5-5) edge node [above] {\\footnotesize $0$} (m-5-6)\n            (m-5-5) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-4-5)\n            (m-4-5) edge node [above] {\\footnotesize $(1,-1)$} (m-4-6)\n            (m-3-5) edge node [above] {\\footnotesize $(1,-1)$} (m-3-6)\n            (m-2-5) edge node [above] {\\footnotesize $(1,-1)$} 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        (m-5-1) edge (m-5-2)\n            (m-6-1) edge (m-6-2)\n\n            (m-6-2) edge (m-6-3)\n            (m-6-3) edge (m-6-4)\n            (m-6-4) edge (m-6-5)\n            (m-6-5) edge (m-6-6)\n\n            (m-2-1) edge (m-1-1)\n            (m-3-1) edge (m-2-1)\n            (m-4-1) edge (m-3-1)\n            (m-5-1) edge (m-4-1)\n            (m-6-1) edge (m-5-1)\n\n            (m-6-2) edge (m-5-2)\n            (m-6-3) edge (m-5-3)\n            (m-6-4) edge (m-5-4)\n            (m-6-5) edge (m-5-5)\n            (m-6-6) edge (m-5-6)\n            ;\n            ;\n        \\end{tikzpicture}\n    \\]\n    We have the following isomorphisms of endomorphism rings:\n    \\[\n        \\End\\left(X'\\right) \\cong \\End\\left(X'|_{((\\tau\/5)(5\\Zbf + 4))^2}\\right) = \\End\\left(A'|_{((\\tau\/5)(5\\Zbf + 4))^2}\\right) \\cong \\End\\left(A'|_{((\\tau\/5)\\Zbf)^2}\\right).\n    \\]\n    The first isomorphism is due to the fact that $X'$ is isomorphic to the restriction of an extension of an $(\\epsilon\\Zbf)^2$-persistence module, and $\\epsilon$ is an integer multiple of $\\tau$, by construction.\n    The equality is because \n    $X'|_{((\\tau\/5)(5\\Zbf + 4))^2} = A'|_{((\\tau\/5)(5\\Zbf + 4))^2}$, by construction.\n    For the last isomorphism, note that the value of $A'$ at any element of the grid $((\\tau\/5)\\Zbf)^2$ not belonging to $\\Tscr$ is isomorphic, through a structure morphism of $A'$, to the value of $A'$ at an element of the grid $((\\tau\/5)(5\\Zbf+4))^2$, by construction.\n    Moreover, the action of an endomorphism of $A'$ on $A'|_\\Tscr$ is determined by the action of the endomorphism on $A'(x+4\\tau\/5,y+4\\tau\/5)$, by \\cref{lemma:G-is-indecomposable}; and $(x+4\\tau\/5,y+4\\tau\/5) \\in ((\\tau\/5)(5\\Zbf+4))^2$.\n    It follows that the action of an endomorphism of $A'$ is completely determined by the action of the endomorphism on the restriction $A'|_{((\\tau\/5)(5\\Zbf+4))^2}$, proving the last isomorphism.\n\n    Thus, $A'$ is indecomposable by \\cref{lemma:indecomposable-local-ring} and the fact that $A$, and hence $X'$, is indecomposable.\n    Define $A_1$ to be the extension of $A'$ to the full poset $\\Rbf^2$.\n    Note that $A_1$ is indecomposable, finitely presentable, and such that $\\mathsf{im}\\left(\\eta^{A}_\\delta\\right) = \\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big)$.\n    Moreover, $A_1$ admits a horizontal antenna attachment at $(x,y+3\\tau\/5)$, by construction.\n\\end{proof}\n\n\\begin{lemma}\n    \\label{lemma:sheaf-condition-persistence-module}\n    Let $\\epsilon > 0$ and let $a \\in \\epsilon \\Zbf$.\n    Let $s(a) = \\{(x,y) \\in (\\epsilon \\Zbf)^2 : x \\leq a\\}$, $g(a) = \\{(x,y) \\in (\\epsilon \\Zbf)^2 : x \\geq a\\}$, and $e(a) = \\{(x,y) \\in (\\epsilon \\Zbf)^2 : x = a\\}$, which are all subposets of $(\\epsilon \\Zbf)^2$.\n    Assume given persistence modules $X : s(a) \\to \\mathbf{vec}$ and $Y : g(a) \\to \\mathbf{vec}$ such that $X|_{e(a)} = Y|_{e(a)}$.\n    Then, there exists a unique persistence module $C : (\\epsilon\\Zbf)^2 \\to \\mathbf{vec}$ such that $C|_{s(a)} = X$ and $C|_{g(a)} = Y$.\n    If, moreover, $X$ is indecomposable and, for every indecomposable summand $I$ of $Y$, we have that $I|_{e(a)} \\neq 0$, then $C$ is indecomposable.\n\\end{lemma}\n\\begin{proof}\n    The statement about the existence and uniqueness of $C$ is clear.\n    Now, assume that $C \\cong S \\oplus T$.\n    On one hand, we have $X \\cong S|_{s(a)} \\oplus T|_{s(a)}$.\n    Since $X$ is indecomposable, without loss of generality, it must be the case that $T_{g(a)} = 0$.\n    In particular, $T_{e(a)} = 0$.\n    On the other hand, $Y \\cong S|_{g(a)} \\oplus T|_{g(a)}$, and, since all the indecomposable summands $I$ of $Y$ satisfy $I|_{e(a)} \\neq 0$, we must have $T|_{g(a)} = 0$.\n    Thus, $T = 0$ and $C$ does not admit any non-trivial decompositions.\n\\end{proof}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{pictures\/step-3.eps}\n    \\caption{A schematic summary of the main constructions in the proof of \\cref{lemma:actual-tacking}.}\n    \\label{figure:diagram-step-3}\n\\end{figure}\n\n\n\\begin{lemma}\n    \\label{lemma:actual-tacking}\n    Let $A,B : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable indecomposable, and isomorphic to extensions of $\\Pscr$-persistence modules for some regular grid $\\Pscr \\subseteq \\Rbf^2$.\n    Assume that $A$ and $B$ both admit horizontal antenna attachments at points in the grid $\\Pscr$.\n    For every $\\delta > 0$, there exists a regular grid $\\Qscr \\supseteq \\Pscr$ and $M : \\Rbf^2 \\to \\mathbf{vec}$ with $M$ is indecomposable, finitely presentable, isomorphic to the extension of a $\\Qscr$-persistence module, and such that\n    \\[\n        \\mathsf{im}\\left(\\eta^M_\\delta\\right) \\cong \\mathsf{im}\\left(\\eta^A_\\delta\\right) \\oplus \\mathsf{im}\\left(\\eta^B_\\delta\\right).\n    \\]\n\\end{lemma}\n\\begin{proof}\n    Without loss of generality, we may assume that $A$ and $B$ are extensions of modules $A'$ and $B'$ defined over a regular grid $\\Qscr = (\\epsilon\\Zbf)^2$ with $\\epsilon < \\delta\/5$.\n    Let $(x,y) \\in \\Qscr$ and $(w,z) \\in \\Qscr$ be horizontal antenna attachments for $A$ and $B$, respectively.\n    Without loss of generality, we may assume that $y \\geq z$.\n    Choose $a,b \\in \\epsilon\\Zbf$ with $a < \\min(x,w)$ and $b > y$.\n    Let the subposets $s(a)$, $g(a)$, and $e(a)$ of $(\\epsilon\\Zbf)^2$ be as in \\cref{lemma:sheaf-condition-persistence-module}.\n\n    We now use \\cref{lemma:sheaf-condition-persistence-module} to construct a persistence module $M$ as in the statement.\n    In order to do this, we construct persistence modules $X$ and $Y$ over the grids $s(a)$ and $g(a)$, respectively.\n    \\cref{figure:diagram-step-3} is a schematic summary of these constructions.\n\n    Define a persistence module $Y_1 : g(a) \\to \\mathbf{vec}$ as follows:\n    \\[\n    Y_1(c,d) =\n    \\begin{cases}\n        \\kbb & \\text{if $d = y$ and $a \\leq c \\leq x$} \\\\\n        A'(c,d) & \\text{else}\n    \\end{cases}\n    \\]\n    The structure morphisms are taken to be those of $A'$ wherever that makes sense.\n    The structure morphisms $\\kbb = Y_1(c,y) \\to Y_1(c+\\epsilon,y) = \\kbb$ for $a \\leq c \\leq x-\\epsilon$ are taken to be the identity, and the structure morphisms $Y_1(c,y) \\to Y_1(c,y+\\epsilon)$ and $Y_1(c,y-\\epsilon) \\to Y_1(c,y)$ are taken to be zero.\n    It is straightforward to check that $Y_1$ is a well-defined persistence module and that $\\End(Y_1) \\cong \\End(A')$.\n    Thus, $Y_1$ is indecomposable.\n    Define $Y_2$ in an entirely analogous way using $B'$ and its horizontal antenna attachment.\n    Let $Y = Y_1 \\oplus Y_2$.\n\n    Consider the following subgrid of $\\epsilon \\Zbf$:\n    \\[\n        \\Tscr = \\{(a-5\\epsilon,a-4\\epsilon,a-3\\epsilon,a-2\\epsilon,a-\\epsilon)\\} \\times \\{(b, b+\\epsilon, b+2\\epsilon, b+3\\epsilon, b+4\\epsilon)\\}.\n    \\]\n    Let us assume that $y > z$; the case $y=z$ is similar.\n    Define a persistence module $X : s(a) \\to \\mathbf{vec}$ as follows:\n    \\[\n    X(c,d) =\n    \\begin{cases}\n        \\Gsf\\left(\\frac{(c,d)-(a-5\\epsilon,b)}{\\epsilon}\\right) & \\text{if $(c,d) \\in \\Tscr$} \\\\\n        \\kbb & \\text{if $c=a-\\epsilon$ and $y \\leq d \\leq b$} \\\\\n        \\kbb & \\text{if $c=a-2\\epsilon$ and $z \\leq d \\leq b$} \\\\\n        \\kbb & \\text{if $(c,d)=(a-\\epsilon,z)$} \\\\\n        0 & \\text{else}\n    \\end{cases}\n    \\]\n    For the structure morphisms corresponding to pairs of elements of $\\Tscr$ we use the structure morphisms of $\\Gsf$; for the remaining vertical structure morphisms we use the identity of $\\kbb$ when possible and the zero morphism otherwise; having done this, for the remaining horizontal morphisms we use the identity of $\\kbb$ when possible (taking commutativity into account) and the zero morphism otherwise (in particular, all horizontal maps with second coordinate different from $y$ and $z$).\n    \n    It is again straightforward to check that $X$ is a well-defined persistence module and that $\\End(X) = \\End(\\Gsf)$.\n    Thus, $X$ is indecomposable.\n\n    Finally, define $C : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$ using \\cref{lemma:sheaf-condition-persistence-module}; it follows that $C$ is indecomposable.\n    Finally, extend $C$ to the poset $\\Rbf^2$ to get a module $M$, which satisfies the conditions in the statement by construction.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of \\cref{lemma:main-lemma-attaching}]\n    Let $A$ and $B$ be as in the statement.\n    Using \\cref{lemma:add-1-dim-corner}, we may assume that $A$ and $B$ admit a thin corner, and using \\cref{lemma:add-antenna-attachment}, we may assume that $A$ and $B$ admit horizontal antenna attachments.\n    \\cref{lemma:actual-tacking} now finishes the proof.\n\\end{proof}\n\n\\ifarxivversion\n\\bibliographystyle{alpha}\n\\fi\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{introduction}\\label{sec_1}\nUltrafast pump-probe spectroscopy is a good tool to investigate the nonequilibrium properties of a given system since a pump pulse triggers ultrafast processes and a subsequent probe pulse monitors the pump-induced dynamical processes~\\cite{Mukamel1995, Diels1996, Krausz2009, Giannetti2016}.\nEspecially, by using femtosecond pulses, nonequilibrium dynamics of electrons can be detected since the timescale of the motion of electrons is of the order of a femtosecond.\nHowever, increasing the resolution of optical measurements in both the time and energy domains is difficult and limited by the uncertainty principle.\n\nRecently, ultrafast spectroscopic techniques have been advanced by using a transform-limited pulse, i.e., a pulse that has the minimum possible duration for a given spectral bandwidth, and have opened a new door to make both temporal and spectral resolutions as high as possible~\\cite{Diels1996}.\nThese techniques can disclose new ultrafast nonequilibrium phenomena.\nIn fact, by applying these techniques, interference in the energy domain has been observed in atomic systems and nanometric tips~\\cite{Wollenhaupt2002, Lindner2005, Kiffner2006, Milosevic2006, Kruger2018}.\nThis interference is applied to control the atomic storage medium for recording the information of optical pulses~\\cite{Leung1982, Carlson1983, Hemmer1994, Fleischauer2002, Ohmori2009}.\nHowever, as far as we know, there has been no such report on transient interference of pump-probe spectroscopy of band and Mott insulators both experimentally and theoretically.\n\nIn this paper, we investigate ultrafast pump-probe spectroscopy of band and Mott insulators and propose transient interference between temporary well-separated pulses in electron systems as in the case of atomic systems.\nWe formulate such transient interference in pump-probe spectroscopy of a two-band model.\nWe find that the existence of a continuum structure in the excitation spectrum is important for generating the transient interference since the continuum structure acts as a medium for storing the spectral information of the pump pulse and for creating interference between temporary well-separated pump and probe photons.\nThe information persists due to a memory effect, i.e., a relaxation process of electron systems.\nAs a result, the time-domain pump-probe spectrum depends on both probe energy $\\omega$ and the central frequency of the pump and probe pulses $\\Omega$ and thus oscillates with a frequency\n\\begin{align}\n\\label{probe_dep}\n\\omega_0 = \\omega - \\Omega.\n\\end{align}\nIn order to demonstrate the transient interference in the presence of electron correlation, we perform numerical calculations of the pump-probe spectrum in a one-dimensional (1D) half-filled Hubbard model.\nMoreover, we find that bosons coupled to electrons in the two-band model make an additional contribution to the interference.\nBased on the result, we speculate that the transient interference will be observed in Mott insulators strongly correlated to magnons.\nFor the observation of the proposed transient interference, high resolution of measurements of both time and energy is required in ultrafast pump-probe spectroscopy.\nRecently, oscillations of electronic states above the charge-transfer gap in a two-dimensional (2D) Mott insulator Nd$_2$CuO$_4$ were observed on the reflectivity changes detected by pump-probe measurement with ultrashort pulses~\\cite{Miyamoto2018}. \nThe time and energy resolutions of the measurement are as high as $10 \\text{fs}$ and $0.01 \\text{eV}$, respectively.\nBy extracting the oscillatory components from the pump-probe spectrum, the oscillation component with the frequency indicated by Eq.~(\\ref{probe_dep}) was found~\\cite{Miyamoto2018}.\nWe propose that the transient interference will be one of the possible origins of the observed oscillations.\n\nThis paper is organized as follows. We introduce a two-band model, which is a minimal model to describe the interference effect by two photon pulses through an electron system, and show the pump-probe absorption spectrum in Sec.~\\ref{sec_2}. In Sec.~\\ref{sec_3}, we calculate the time-dependent optical conductivity at half filling just after pumping. The effect of bosons coupled to electrons on the pump-probe spectrum is discussed in Sec.~\\ref{sec_4}. Finally, a summary is given in Sec.~\\ref{sec_5}.\n\n\\section{Two-band model}\\label{sec_2}\nWe first introduce a two-band model, which is the minimal model to describe the interference effect by two photon pulses through an electron system, and analytically calculate the pump-probe absorption spectrum.\nWith the assumption of dipole transitions, the Hamiltonian of the two-band model under the time-dependent electric field reads\n\\begin{align*}\n\\mathcal{H} =& \\sum_{\\bm{k}} \\varepsilon_{\\bm{k}} c_{\\text{c}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} + \\sum_{\\bm{k}} \\varrho_{\\bm{k}} c_{\\text{v}\\bm{k}}^\\dag c_{\\text{v}\\bm{k}} \\nonumber\\\\\n&-\\sum_{\\bm{k}} \\left( d_{\\text{cv}} \\mathcal{E}(t) c_{\\text{c}\\bm{k}}^\\dag c_{\\text{v}\\bm{k}} + d_{\\text{cv}}^* \\mathcal{E}(t) c_{\\text{v}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} \\right), \\nonumber\n\\end{align*}\nwhere $c_{\\text{c(v)}\\bm{k}}$ is an annihilation operator for fermions in the conduction (valence) band with momentum $\\bm{k}$. The energies of the conduction and valence bands are $\\varepsilon_{\\bm{k}} = \\varepsilon + \\frac{\\hbar ^2 \\bm{k}^2}{2m_{\\text{c}}}$ and $ \\varrho_{\\bm{k}} = \\varrho + \\frac{\\hbar ^2 \\bm{k}^2}{2m_{\\text{v}}}$,\nwhere $\\varepsilon$ and $\\varrho$ are the minimum and maximum of the conduction and valence bands, respectively, and $m_c$ and $m_v$ are the effective masses of electrons in the conduction and valence bands, respectively.\nWe introduce the interband dipole matrix element $d_{\\text{cv}}$ and external electric field $\\mathcal{E}(t)$. Hereafter, we set $\\hbar=1$.\n\nBy taking the long-wave-length limit of the electric field, the optical Bloch equation is written as~\\cite{HaugKoch}\n\\begin{align} \\label{Eq_bloch_eq_1}\n&\\left( \\frac{\\partial}{\\partial t} + i\\{ \\varepsilon _{\\bm{k}}-\\varrho_{\\bm{k}}-i \\gamma \\} \\right) p_{\\text{vc}}^0(\\bm{k},t) = d_{\\text{cv}} \\mathcal{E}(t) \\{ 1-2f_c(\\bm{k}) \\}\n\\end{align}\nand\n\\begin{align} \\label{Eq_bloch_eq_2}\n\\left(  \\frac{\\partial}{\\partial t} +  \\Gamma \\right) f_{\\text{c}}(\\bm{k},t) = -2\\text{Im}\\left[ d_{\\text{cv}} \\mathcal{E}(t) p_{\\text{vc}}^{0*}(\\bm{k},t) \\right],\n\\end{align}\nwhere $f_{\\text{c}}(\\bm{k}) = \\langle c_{\\text{c}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} \\rangle$ and $p_{\\text{vc}}^0(\\bm{k}) = \\langle c_{\\text{v}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} \\rangle$, with $\\langle \\cdots \\rangle$ representing the expectation value.\nWe introduce a phenomenological damping rate $\\Gamma$ for $f_{\\text{c}}$ and dephasing rate $\\gamma$ for $p_{\\text{vc}}^0$.\nWe consider an electric field $\\mathcal{E}(t) = \\frac{1}{2}\\left(\\mathcal{\\tilde{E}}(t)e^{-i\\Omega t} + \\mathcal{\\tilde{E}}^*(t)e^{i\\Omega t}\\right)$, where $\\mathcal{\\tilde{E}}(t)=2\\left\\{ \\mathcal{\\tilde{E}}_{\\text{p}}(t)e^{i\\bm{k}_p\\cdot \\bm{r}} + \\mathcal{\\tilde{E}}_{\\text{t}}(t)e^{i\\bm{k}_t\\cdot \\bm{r}} \\right\\}$, and the electric field and wave vector of the pump (probe) pulse are $\\mathcal{\\tilde{E}}_{\\text{p}}$ and $\\bm{k}_{\\text{p}}$ ($\\mathcal{\\tilde{E}}_{\\text{t}}$ and $\\bm{k}_{\\text{t}}$), respectively.\nIntroducing an expansion parameter $\\lambda$ through $\\mathcal{E}(t) \\rightarrow \\lambda \\mathcal{E}(t)$, we obtain $p_{\\text{vc}}^0 = \\lambda p_{\\text{vc}}^{0(1)} + \\lambda^2 p_{\\text{vc}}^{0(2)} + \\lambda^3 p_{\\text{vc}}^{0(3)} + \\cdots,\\; f_{\\text{c}} = \\lambda f_{\\text{c}}^{(1)} + \\lambda^2 f_{\\text{c}}^{(2)} + \\lambda^3 f_{\\text{c}}^{(3)} + \\cdots$.\nThe shape of the probe pulse is represented by the delta function $\\mathcal{\\tilde{E}}_{\\text{t}}(t) = \\mathcal{\\tilde{E}}_{\\text{t}} \\delta (t -\\tau)$ $(\\tau >0)$, where $\\tau$ is the delay time between the pump and probe pulses.\nThe pump-induced absorption change is given by $\\alpha = -\\textrm{Im}\\left[ d_{cv}^*\\chi(\\bm{k},\\omega) \\right].$\nTaking $\\mathcal{\\tilde{E}}_{\\text{p}}(t)=\\mathcal{\\tilde{E}}_{\\text{p}}e^{-\\sigma|t|}$ and with the rotating-wave approximation, the probe susceptibility is given by (see Appendix A)\n\\begin{align} \\label{chi0}\n&\\chi (\\bm{k},\\omega)\\simeq \\frac{p_{\\text{vc}}^{0(3)}(\\bm{k},\\omega)}{\\mathcal{E}_{\\text{t}}(\\omega)}\\nonumber \\\\\n&=\\frac{8d_{\\text{cv}}\\left|d_{\\text{cv}}\\right| {}^2 \\left| \\mathcal{\\tilde{E}}_p\\right| {}^2  e^{-\\left(\\sigma - \\gamma \\right)\\tau} e^{i \\tau  \\left( - \\Omega +\\varepsilon _k-\\varrho   _k\\right)}  \\Gamma \\sigma }{\\left(i \\gamma +\\omega -\\varepsilon _k+\\varrho _k\\right)(i\\Gamma +i\\sigma + \\omega -\\Omega ) v_{\\bm{k}}^+ u_{\\bm{k}}^+ u_{\\bm{k}}^-}+\\cdots,\n\\end{align}\nwhere $u_{\\bm{k}}^{\\pm}=i\\gamma \\pm i\\sigma + \\Omega - \\varepsilon _{\\bm{k}} + \\varrho _{\\bm{k}}$ and $v_{\\bm{k}}^+ = i\\gamma +i\\Gamma -i\\sigma + \\Omega - \\varepsilon _{\\bm{k}}+ \\varrho _{\\bm{k}}$. \nIn the limit $\\gamma\\rightarrow 0$, the pole of the energy denominator $\\omega = \\varepsilon_{\\bm{k}} - \\varrho _{\\bm{k}}$ in the third term of $\\chi (\\bm{k},\\omega)$ gives rise to an oscillatory behavior of $e^{i(\\omega -\\Omega)\\tau}$ with decay $e^{-(\\sigma-\\gamma)\\tau}$.\nThis is the oscillation component indicated by Eq.~(\\ref{probe_dep}).\nSince the timescale where the oscillation persists is on the order of $\\gamma^{-1}$, real-time ultrafast dynamics should be observed with high accuracy~\\cite{Rhodes2013}.\n\nIn order to maintain the oscillation in the two-band model, we have to select a proper set of parameters that leads to the coherence and memory effect in the energy domain.\nFirst of all, we examine the coherence in the energy domain.\nWhen $\\sigma \\gg 1\/\\tau$, i.e., the pulse duration is much shorter than the time delay $\\tau$, we obtain $\\Delta t \\sim 0$, where $\\Delta t$ is the uncertainty in the time domain.\nSimultaneously, the energy uncertainty $\\Delta E \\sim \\infty$, leading to low energy resolution.\nAs a result, the interference in the energy domain is invisible.\nThis corresponds to the fact that the interference pattern vanishes in Young's double-slit experiment if the path of light is measured~\\cite{Englert1996, Durr1998}.\nIn fact, if the electric field of the pump pulse is represented by the $\\delta$ function, $p_{vc}^{0(3)}(\\bm{k},\\omega)$ completely cancels out $\\mathcal{E}_t(\\omega)$, which means that $\\chi(\\bm{k},\\omega)$ does not have the interference term $e^{i(\\omega -\\Omega)\\tau}$ (see Appendix A).\nIn contrast, when $\\sigma \\lesssim 1\/\\tau$, the coherence in the energy domain is obtained, which leads to the interference in energy space.\n\nSecond, we examine the memory effect.\nWhen $\\sigma \\ll \\gamma$, i.e., the pulse duration is longer than the dephasing time, $\\Delta t \\sim \\infty$ and $\\Delta E\\sim 0$ are simultaneously obtained.\nThis leads to the relaxation that holds true as long as electrons have well-defined energies, and their energy changes are slow with the timescale of $1\/\\Delta \\epsilon$, where $\\Delta \\epsilon$ is the characteristic energy exchange in a scattering event~\\cite{Aihara1982, Kuznetsov1991, Kuznetsov1991_2, Rossi2002, SchaeferWegener}.\nWhen $\\sigma \\gtrsim \\gamma$, the relaxation involving electrons with ill-defined energies starts to contribute to the memory effect.\nTherefore, if $\\sigma$ and $1\/\\tau$ are carefully controlled to realize $1\/\\tau \\gtrsim \\sigma \\gtrsim \\gamma$, both coherence in the energy domain and the memory effect are relevant, and the interference in the energy domain is maintained for the time $\\gamma^{-1}$.\nUsually, $\\gamma$ of a given system cannot be changed. However, if we make use of the quantum Zeno effect~\\cite{Misra1977, Itano1990, Kaulakys1997, Streed2006}, we might be able to control $\\gamma$, which can help us to observe our finding.\n\n\\section{Hubbard model}\\label{sec_3}\nPump-probe spectroscopy has been performed in strongly correlated systems to investigate exotic phenomena~\\cite{Okamoto2010, Okamoto2011, Filippis2012, Matsueda2012, Zala2013, Golez2014, Eckstein2014, Novelli2014, Prelovsek2015, Giannetti2016, Bittner2017, Bittner2018, Miyamoto2018}.\nEven in correlated electron systems, there is a continuum structure in the excitation spectrum. This indicates that interference effects similar to those in the two-band model may be realized, which will be demonstrated by using a 1D half-filled Hubbard model, which is given by\n\\begin{align}\nH=-t_\\mathrm{h}\\sum_{i,\\sigma} \\left( c^\\dagger_{i,\\sigma} c_{i+1,\\sigma} + \\mathrm{H.c.}\\right) + U\\sum_i n_{i,\\uparrow}n_{i,\\downarrow},\n\\label{H}\n\\end{align}\nwhere $c^\\dagger_{i\\sigma}$ is the creation operator of an electron with spin $\\sigma$ at site $i$, $n_{i,\\sigma}=c^\\dagger_{i,\\sigma}c_{i,\\sigma}$, $n_i=\\sum_\\sigma n_{i,\\sigma}$, and $t_\\mathrm{h}$ and $U$ are the nearest-neighbor hopping and on-site Coulomb interaction, respectively. Taking $t_\\mathrm{h}$ to be the unit of energy ($t_\\mathrm{h}=1$), we use $U=10$.\n\nWe investigate the probe-energy dependence of the optical conductivity of a Hubbard open chain with $L=10$, where $L$ is the number of sites.\nWe assume that both the pulses have the same shape of the vector potential given by $A(t)=A_0 e^{-(t-t_0)^2\/(2t_\\mathrm{d}^2)} \\cos[\\Omega(t-t_0)]$.\nWe set $A_0=0.1$, $t_0=3.0$, $t_\\mathrm{d}=0.5$, and $\\Omega=E_g =7.1$ for the pump pulse and $A_0=0.001$, $t_0=\\tau+3.0$, $t_\\mathrm{d}=0.02$, and $\\Omega=E_g =7.1$ for the probe pulse unless otherwise specified, where $E_g$ is the energy of the Mott gap.\nAn external spatially homogeneous electric field applied along the chain in the Hamiltonian can be incorporated via the Peierls substitution in the hopping terms as $c_{i,\\sigma}^\\dag c_{i+1,\\sigma} \\rightarrow e^{iA(t)}c_{i,\\sigma}^\\dag c_{i+1,\\sigma}$.\nUsing the method discussed in Refs.~\\cite{Lu2015, Shao2016}, we obtain the optical conductivity in the nonequilibrium system, $\\sigma(\\omega,\\tau) = \\frac{j_\\text{probe}(\\omega,\\tau) }{i(\\omega +i\\eta)LA_\\text{probe}(\\omega)}$, where $j_\\text{probe}(\\omega,\\tau)$ is the Fourier transform of the current induced by the probe pulse and $A_{\\text{probe}}(\\omega)$ is the Fourier transform of the vector potential of the probe pulse (see Appendix B for details).\n\nTo trace the temporal evolution of the system, we employ the time-dependent Lanczos method to evaluate $|\\psi (t)\\rangle$. \nHere $|\\psi(t+\\delta{t})\\rangle\\simeq\\sum_{l=1}^{M}{e^{-i\\epsilon_l\\delta{t}}}|\\phi_l\\rangle\\langle\\phi_l|\\psi(t)\\rangle$, where $\\epsilon_l$ and $|\\phi_l\\rangle$ are eigenvalues and eigenvectors of the tridiagonal matrix generated in the Lanczos iteration, respectively, $M$ is the dimension of the Lanczos basis, and $\\delta t$ is the minimum time step. We set $M=50$ and $\\delta t=0.02$.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=20pc]{Fig_OC_L10p_wpump10_wprobe_t_high.eps}\n    \\caption{$\\text{Re} \\sigma (\\omega,\\tau)$ in the 1D half-filled Hubbard chain with $L=10$ and $U=10$, before pumping ($\\tau<0$) and after pumping ($\\tau=10$,\\;20,\\;30, and 40). Since the system is weakly excited, the dashed line for $\\tau<0$ is almost overlapped by the solid lines above $\\omega =7$.}\n    \\label{band}\n\\end{figure}\n\nFigure~\\ref{band} shows the real part of the time-dependent optical conductivity $\\text{Re} \\sigma(\\omega,\\tau)$ of the Hubbard model.\nPhotoinduced decreases in the spectral weights at absorption peaks above the Mott gap are small since the system is weakly excited. \nThe pump photon excites carriers into an optically allowed odd-parity state.\nThe probe pulse couples in part to the odd-parity state, resulting in an excitation from the optically allowed state to an optically forbidden even-parity state.\nIn 1D Mott insulators with open boundary conditions, the optically forbidden state is located slightly above the optically allowed state~\\cite{Mizuno2000}.\nLow-energy in-gap excitation comes from the excitation from the optically allowed to forbidden state~\\cite{Lu2015}.\nInside the Mott gap, we find photoinduced low-energy spectral weights at $\\omega \\simeq1.2$, 2.2, and 3.3.\nThese energies correspond to the energy differences between the optically allowed populated state at $\\omega=7.1$ and the optically forbidden states.\n\nFigures~\\ref{probe_energy_dep_w0}(a)-\\ref{probe_energy_dep_w0}(e) show the $\\tau$ dependence of $\\text{Re} \\sigma(\\omega,\\tau)$ above the Mott gap with probe energy $\\omega=7.10$, $7.92$, $8.98$, $10.08$, and $11.18$, respectively, whose energies agree with the peak energies of the absorption spectrum in Fig.~\\ref{band}.\nWe find that the frequencies of the oscillations depend on $\\omega$.\nThe larger $\\omega$ is, the larger the frequency is, which is consistent with our argument in the two-band model discussed above.\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=20pc]{OC_power.eps}\n    \\caption{$\\text{Re} \\sigma(\\omega,\\tau)$ in the 1D half-filled Hubbard chain with $L=10$ and $U=10$ for (a) $\\omega=7.1$, (b) $\\omega=7.92$, (c) $\\omega=8.98$, (d) $\\omega=10.08$, and (e) $\\omega=11.18$. The power spectra of $\\text{Re} \\sigma (\\omega,\\tau)$ for (f) $\\omega=7.1$, (g) $\\omega=7.92$, (h) $\\omega=8.98$, (i) $\\omega=10.08$, and (j) $\\omega=11.18$.}\n    \\label{probe_energy_dep_w0}\n\\end{figure}\n\nIn order to further examine the probe-energy dependence, we show the power spectra of $\\text{Re} \\sigma(\\omega,\\tau)$ with respect to $\\tau$ in Figs.~\\ref{probe_energy_dep_w0}(f)-\\ref{probe_energy_dep_w0}(j) for $\\omega=7.1$, 7.92, 8.98, 10.08, and 11.18, respectively.\nWe discuss two possible contributions to the power spectra.\nThe first one is the contribution from the Rabi oscillation, whose frequencies are related to the low-energy in-gap states at $\\omega=$1.2, 2.2, and 3.3.\nIn fact, we find the Rabi-oscillation contributions to the spectral weights at $\\omega_0=$1.2, 2.2, and 3.3 in Figs.~\\ref{probe_energy_dep_w0}(f)-\\ref{probe_energy_dep_w0}(j).\nSince our system is of finite size, energy levels are discretized.\nTherefore, there are oscillations with resonant frequencies between the levels.\nIn the thermodynamic limit, the number of the levels is infinite, and thus we expect that the contributions from a huge number of such resonances with various frequencies cancel out, giving rise to an inifinite number of infinitesimal weights in the power spectra.\nThus, we consider that the Rabi-oscillation contribution to the power spectra is visible only in finite-size systems and negligible in the thermodynamic limit.\n\nThe second one is the contribution from the interference effect, which gives rise to the $\\omega$ dependence of the pump-probe spectra as discussed in the two-band model. \nThe oscillations with the frequencies $\\omega-\\Omega$ appear at $\\omega_0=7.92-7.10=0.82$, $8.98-7.10=1.88$, $10.08-7.10=2.98$, and $11.18-7.10=4.08$.\nThese energies correspond to the energy difference between the levels at $\\omega=\\Omega=7.1$ and the excited states above the Mott gap, all of which belong to the same electronic states with odd parity.\nWe consider that this origin makes the dominant contribution to the power spectra in the thermodynamic limit.\nIn order to induce the transient interference, we should use the pump pulse whose spectrum covers some energy levels. Then we can store the information of the pump pulse in electronic states with a wide range of energies above the Mott gap.\n\nAccording to the two possible contributions to the power spectra, in Fig.~\\ref{probe_energy_dep_w0}(g), for example, we find peak structures at $\\omega_0=0.82$, 1.2, and 2.2.\nThe peak structures at $\\omega_0=1.2$ and 2.2 come from the Rabi oscillation of the two odd- and even-parity states.\nOn the other hand, the origin of the structure at $\\omega_0=0.82$ is the interference because $\\omega_0=0.82$ corresponds to one of the energy differences between the odd-odd states mentioned above.\nSimilarly, Figs.~\\ref{probe_energy_dep_w0}(h)-\\ref{probe_energy_dep_w0}(j) are understood in the same way (see Appendix B for details).\n\n\\section{electron-boson coupling in the two-band model} \\label{sec_4}\n\nFinally, we discuss the effect of bosons coupled to electrons on the probe-energy-dependent oscillation.\nNonequilibrium electron dynamics coupled to a boson driven by a laser has been extensively studied.\nFurthermore, since non-Markovian relaxation is important in electron systems coupled to a bosonic environment, open quantum systems with non-Markovian properties have been studied for a long time~\\cite{Jaynes1963, Caldeira1985, Leggett1987, BreuerPetruccione, Zurek2003, Reiter2014, Seetharam2015, Nazir2016, deVega2017}.\nThe additional Hamiltonian due to boson degrees of freedom is \n\\begin{align}\\label{H_ph}\n\\mathcal{H}_{\\text{ph}} = \\sum_{\\bm{q}} \\omega_{\\bm{q}} a_{\\bm{q}}^\\dag a_{\\bm{q}} + \\sum_{\\bm{k},\\bm{q}} g_{\\bm{q}} (a_{-\\bm{q}}^\\dag + a_{\\bm{q}}) (c_{\\text{c}\\bm{k+q}}^\\dag c_{\\text{c}\\bm{k}} + c_{\\text{v}\\bm{k+q}}^\\dag c_{\\text{v}\\bm{q}}),\n\\end{align}\nwhere $a_{\\bm{q}}$ is an annihilation operator for bosons with momentum $\\bm{q}$, $\\omega_{\\bm{q}}$ is the boson frequency, and $g_{\\bm{q}}$ is an electron-boson coupling constant.\n\nWe examine the two-band model with electron-boson coupling under the application of the exponential pump pulse.\nTotal polarization is given by $p_{\\text{vc}}^{}(\\bm{k},t)= p_{\\text{vc}}^{0}(\\bm{k},t) + p_{\\text{vc}}^{b}(\\bm{k},t)$, where $p_{\\text{vc}}^{0}(\\bm{k},t)$ is from the one-particle contribution, as discussed above, and $p_{\\text{vc}}^{b}(\\bm{k},t)$ is from the electron-boson coupling.\nSolving the kinetic equation with $\\mathcal{H}_{\\text{ph}}$ (see Appendix A), the probe susceptibility is given by\n\\begin{align} \\label{Eq_chi_b_main}\n&\\chi ^{b}(\\bm{k},\\omega)\\simeq \\frac{p_{\\text{vc}}^{b(3)}(\\bm{k},\\omega)}{\\mathcal{E}_{\\text{t}}(\\omega)}\\nonumber =\\sum _{\\bm{q}} g_{\\bm{q}}^2\\mathcal{N}_{\\bm{q}} \\cdot 4i\\sigma d_{\\text{cv}}\\left| d_{\\text{cv}}\\right|{}^2 \\left| \\mathcal{\\tilde{E}}_p\\right| {}^2 \\nonumber \\\\\n&\\times \\Biggl[\n\\frac{e^{-\\tau \\left(\\sigma-\\gamma\\right)} e^{i \\tau  \\left( -\\Omega +\\varepsilon_{\\bm{k}} -\\varrho _{\\bm{k}}\\right)} \\left(-i\\gamma -2i \\Gamma - \\omega + \\varepsilon _{\\bm{k}}- \\varrho _{\\bm{k}}\\right) }{\\left(i \\gamma +\\omega -\\varepsilon _{\\bm{k}}+\\varrho _{\\bm{k}}\\right){}^2 \\left(i \\gamma +\\omega -\\varepsilon _{\\bm{k+q}}+\\varrho _{\\bm{k}}+\\omega _{\\bm{q}}\\right) v_{\\bm{k}}^+}\\nonumber \\\\\n&\\times \\frac{\\left(2 i \\gamma +2 \\omega -\\varepsilon _{\\bm{k}}-\\varepsilon _{\\bm{k+q}}+\\varrho _{\\bm{k}}+\\varrho _{\\bm{k-q}}+2 \\omega _{\\bm{q}}\\right)}{(i\\Gamma +i\\sigma + \\omega -\\Omega ) \\left(i \\gamma +\\omega -\\varepsilon _{\\bm{k}}+\\varrho _{\\bm{k-q}}+\\omega _{\\bm{q}}\\right) u_{\\bm{k}}^+u_{\\bm{k}}^-}\\Biggr]\\nonumber \\\\\n&+\\cdots,\n\\end{align}\nwhere $\\mathcal{N}_{\\bm{q}}=\\frac{1}{e^{\\omega_{\\bm{q}}\/k_BT}-1}$.\nIn the limit $\\gamma \\rightarrow 0$, the pole of the energy denominator $\\omega = \\varepsilon_{\\bm{k}} - \\varrho _{\\bm{k}}$ gives rise to an oscillatory behavior of $e^{i(\\omega -\\Omega)\\tau}$ with decay $e^{-(\\sigma -\\gamma)\\tau}$, which is the same behavior as the third term in Eq.~(\\ref{chi0}).\nTherefore, the information of pump and probe pulses is transmitted with the help of boson-assisted electron scattering, which gives one of the possible origins of the transient interference.\n\nIn Mott insulators, magnons are strongly coupled to photo-excited electrons in 2D Mott insulators, in contrast to the 1D Mott insulator where spin and charge degrees of freedom are separated.\nTherefore, the interference proposed in this work will be easily realized in the 2D Mott insulators.\nWe thus speculate that the oscillations observed by the pump-probe spectroscopy of the 2D Mott insulator Nd$_2$CuO$_4$~\\cite{Miyamoto2018} come from the interference effect. \nIn order to confirm this speculation, we need to investigate theoretically the pump-probe spectrum of the 2D half-filled Hubbard model, but it remains for a future work.\n\n\\section{summary}\\label{sec_5}\nIn summary, we suggested the transient interference in the energy domain between temporary well-separated light pulses using electronic states of band and Mott insulators as a medium, which manifests as the oscillation of the pump-probe spectrum whose frequency is indicated by Eq.~(\\ref{probe_dep}).\nThis interference could be observed only after recent developments of ultrafast spectroscopic techniques.\nThe transient interference reflects the universal property of interference between two photon pulses mediated by electron systems, which does not depend on the details of the electron systems. Therefore, the interference is also realized in the presence of electron correlation since there is a continuum structure. We examined this by calculating the pump-probe spectrum in the 1D half-filled Hubbard model.\nTo verify our prediction, we suggested an experiment for Nd$_2$CuO$_4$ with varying pump-pulse duration and delay. \nSince our theory predicts the transient oscillation even in the 1D Mott insulators, we proposed a pump-probe experiment in Sr$_2$CuO$_3$.\nFurthermore, we found that bosons coupled to electrons in the two-band model make the additional contribution to the transient interference.\nBased on the result, both magnons coupled to electrons and the continuum structure in electronic excitation spectrum would be possible origins of the oscillation observed in Nd$_2$CuO$_4$.\n\n\\begin{acknowledgments}\nWe would like to thank H. Okamoto, T. Miyamoto, I. Eremin, and P. Prelov\\v{s}ek for fruitful discussions. This work was supported by CREST (Grant No. JPMJCR1661), the Japan Science and Technology Agency, the creation of new functional devices and high-performance materials to support next-generation industries (CDMSI), and the challenge of basic science exploring extremes through multi-physics and multi-scale simulations to be tackled by using a post-K computer.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Conclusion and Future Work}\n\\label{sec:conclusions}\n\nWe presented the first distributed optimal primal-dual resource allocation\nalgorithm for uplink OFDM systems. The key features of the proposed\nalgorithm include: (a) incorporating practical OFDM system\nconstraints such as self-noise and maximum SNR constraints, (b)\nreduced primal-dual algorithm which eliminates unnecessary variable\nupdates, (c) distributed implementation by the end users and base\nstation, (d) simple local updates with limited message passing, (e)\nglobal convergence despite of the existence of multiple global\noptimal solutions, and (f) fast convergence compared with the\nstate-of-art centralized optimal algorithm. Currently we are\nextending this work in several directions, including proving the\ntheoretical convergence speed of the algorithm.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}