diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzapls" "b/data_all_eng_slimpj/shuffled/split2/finalzzapls" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzapls" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nLeavitt path algebras are associative algebras constructed from directed graphs. They were introduced by G. Abrams and G. Aranda Pino in 2005 \\cite{aap05} and independently by P. Ara, M. Moreno and E. Pardo in 2007 \\cite{AMP}.\nThe field of Leavitt path algebras has connections to many other branches of mathematics, such as functional analysis, symbolic dynamics, K-theory and noncommutative geometry, cf. \\cite{AASbook}.\n\nThere have been a substantial number of papers devoted to the representation theory of Leavitt path algebras. P. Ara and M. Brustenga proved that the category of modules for a Leavitt path algebra $L(E)$ of a finite graph $E$ is equivalent to a quotient category of the category of modules for the path algebra $P(E)$ \\cite{AB}.\nD. Gon\\c{c}alves and D. Royer obtained modules for Leavitt path algebras by introducing the notion of an algebraic branching system \\cite{GR}. X. Chen used infinite paths in $E$ to obtain simple modules for the Leavitt path algebra $L(E)$ \\cite{C}. Numerous work followed, noteworthy the work of P. Ara and K. Rangaswamy producing new simple modules associated to infinite emitters and characterising those algebras which have countably (finitely) many distinct isomorphism classes of simple modules \\cite{ARa,R-2,R-3}. G. Abrams, F. Mantese and A. Tonolo studied the projective resolutions for these simple modules \\cite{AMT}, and P. \\'Anh and T. Nam provided another way to describe the so-called Chen and Rangaswamy simple modules \\cite{nam}. In the recent paper \\cite{HPS}, simple modules for \\textit{weighted} Leavitt path algebras were found. Moreover, the algebraic branching systems for unweighted graphs and the modules defined by them were classified.\n\nIn this paper, we obtain modules for a Leavitt path algebra $L(E)$ over a field $K$ by introducing the notion of an \\textit{extended} $E$-algebraic branching system. As a motivating example consider the graph\n\\[E:\\xymatrix{ v\\ar@[yellow]@(dr,ur)_{e}\\ar@[green]@(dl,ul)^{d}}.\\]\nSet\n\\begin{align*}\nX:=\\{v\\}\\cup \\{p,p^*\\mid p\\in E^{\\geq 1}\\}\\cup\\{pq^*\\mid ~&p=p_1\\dots p_m, q=q_1\\dots q_n\\in E^{\\geq 1},(p_m,q_n)\\neq (d,d)\\}.\n\\end{align*}\nThen $X$ is a linear basis for $L(E)$ by \\cite[Theorem 1]{zelmanov1}. For any $i\\in \\{d,e,d^*,e^*\\}$ let $Y_i$ be the subset of $X$ consisting of all elements whose last letter is $i$. Define the maps \n\\begin{align*}\n\\rho_d:&~X\\setminus (Y_{d^*}\\cup Y_{e^*})\\to Y_d,~x\\mapsto xd;\\\\\n\\rho_e:&~X\\setminus (Y_{d^*}\\cup Y_{e^*})\\to Y_e,~x\\mapsto xe;\\\\\n\\rho_{d^*}:&~X\\setminus Y_{d}\\to Y_{d^*},~x\\mapsto xd^*;\\\\\n\\rho_{e^*}:&~X\\to Y_{e^*},~x\\mapsto xe^{*}.\n\\end{align*}\nHere we use the convention that $vi=i$ for any $i\\in \\{d,e,d^*,e^*\\}$. Clearly the maps $\\rho_d$, $\\rho_e$, $\\rho_{d^*}$ and $\\rho_{e^*}$ are bijections. Let $\\pi:L(E)\\to \\operatorname{End}(L(E))^{\\operatorname{op}}$ be the right regular representation of $L(E)$. Then for any $x\\in X$ one has\n\\begin{align}\n\\pi(v)(x)&=x,\n\\\\\n\\pi(d)(x)&=\\begin{cases}\n\\rho_{d}(x),&\\text{ if }x\\in X\\setminus(Y_{d^*}\\cup Y_{e^*}),\\\\\n\\rho_{d^*}^{-1}(x),&\\text{ if }x\\in Y_{d^*},\\\\\n0,&\\text{ if }x\\in Y_{e^*},\n\\end{cases}\n\\\\\n\\pi(e)(x)&=\\begin{cases}\n\\rho_{e}(x),&\\text{ if }x\\in X\\setminus(Y_{d^*}\\cup Y_{e^*}),\\\\\n0,&\\text{ if }x\\in Y_{d^*},\\\\\n\\rho_{e^*}^{-1}(x),&\\text{ if }x\\in Y_{e^*},\n\\end{cases}\n\\\\\n\\pi(d^*)(x)&=\\begin{cases}\n\\rho_{d^*}(x),&\\text{ if }x\\in X\\setminus Y_{d},\\\\\n\\rho^{-1}_{d}(x)-\\rho_{e^*}(\\rho_{e}(\\rho_{d}^{-1}(x))),&\\text{ if }x\\in Y_{d},\n\\end{cases}\n\\\\\n\\pi(e^*)(x)&=\\rho_{e^*}(x).\n\\end{align}\nMore generally, suppose that\n\\begin{itemize}\n\\item $Y_d$, $Y_e$, $Y_{d^*}$, $Y_{e^*}$ are \\textit{any} pairwise disjoint subsets of a set $X$ and \n\\item $\\rho_d:X\\setminus (Y_{d^*}\\cup Y_{e^*})\\to Y_d$, $\\rho_e:X\\setminus (Y_{d^*}\\cup Y_{e^*})\\to Y_e$, $\\rho_{d^*}:X\\setminus Y_{d}\\to Y_{d^*}$, $\\rho_{e^*}:X\\to Y_{e^*}$ are \\textit{any} bijections.\n\\end{itemize}\nLet $V$ denote the $K$-vector space with basis $X$. Then it follows from the universal property of $L(E)$ that there is a unique representation $\\pi:L(E)\\to \\operatorname{End}(V)^{\\operatorname{op}}$ such that (1)-(5) hold. We call $\\mathcal{S}=(X,\\{Y_i\\}_{i\\in \\{d,e,d^*,e^*\\}},\\{\\rho_j\\}_{j\\in \\{d,e,d^*,e^*\\}})$ an \\textit{extended $E$-algebraic branching system} and the corresponding representation $\\pi:L(E)\\to \\operatorname{End}(V)^{\\operatorname{op}}$ the {\\it representation of $L(E)$ associated to $\\mathcal{S}$}. The usual $E$-algebraic branching systems are in 1-1 correspondence to the extended $E$-algebraic branching systems $\\mathcal{S}=(X,\\{Y_i\\},\\{\\rho_j\\})$ which have the property that $Y_d=Y_e=\\emptyset$ (for details see Example \\ref{exusual}). The notion of an extended $E$-algebraic branching system extends naturally to arbitary row-finite graphs $E$, see Section 4. \n\nThe rest of the paper is organised as follows. In Section 2, we recall some graph-theoretic notions and the definition of a Leavitt path algebra. In Section 3, we recall the notion of an algebraic branching system and the notion of a representation graph (the latter was recently introduced in \\cite{HPS}). Moreover, we recall the result from \\cite[A.5]{HPS} that for a graph $E$, the categories $\\ABS(E)$ of $E$-algebraic branching systems and $\\RG(E)$ of representation graphs for $E$ are equivalent. In Section 4, we introduce the main notion of this paper, namely the notion of an extended algebraic branching system. Moreover, we introduce the notion of an extended representation graph and show that for a given graph $E$, the categories $\\EABS(E)$ of extended $E$-algebraic branching systems and $\\ERG(E)$ of extended representation graphs for $E$ are equivalent. In Section 5, we classify the extended algebraic branching systems. For the usual algebraic branching systems we recover the classification obtained in \\cite[Section 4]{HPS}. Suppose that $M$ is a module of an algebra over a field. Recall that Schur's lemma says that if $M$ is simple, then the endomorphism ring of $M$ is a skew field. In Section 6, we obtain for any Leavitt path algebra $L(E)$ over a field $K$ a class of nonsimple $L(E)$-modules whose endomorphism rings are isomorphic to $K$ (hence these modules constitute counter-examples to the converse of Schur's lemma).\n\n\n\\section{Preliminaries}\nThroughout the paper $K$ denotes a fixed field. Rings and algebras are associative but not necessarily commutative or unital.\n$\\mathbb{N}$ denotes the set of positive integers.\n\n\\subsection{Graphs}\n\nA {\\it (directed) graph} is a quadruple $E=(E^0,E^1,s,r)$ where $E^0$ and $E^1$ are sets and $s,r:E^1\\rightarrow E^0$ maps. The elements of $E^0$ are called {\\it vertices} and the elements of $E^1$ {\\it edges}. If $e$ is an edge, then $s(e)$ is called its {\\it source} and $r(e)$ its {\\it range}. If $v$ is a vertex and $e$ an edge, we say that $v$ {\\it emits} $e$ if $s(e)=v$, and $v$ {\\it receives} $e$ if $r(e)=v$. A vertex is called a {\\it source} if it receives no edges, a {\\it sink} if it emits no edges, and an \\textit{infinite emitter} if it emits infinitely many edges. A vertex is called {\\it regular} if it is neither a sink nor an infinite emitter. The subset of $E^0$ consisting of all sinks is denoted by $E^0_{\\operatorname{sink}}$, and the subset consisting of all regular vertices by $E^0_{\\operatorname{reg}}$. A graph is called \\textit{nonempty} if it contains at least one vertex, and {\\it row-finite} if it does not contain any infinite emitters.\n\nLet $E$ and $F$ be graphs. A {\\it graph homomorphism} $\\phi: E\\to F$ consists of two maps $\\phi^0 : E^0\\to F^0$ and $\\phi^1 : E^1\\to F^1$ such that $s(\\phi^1(e)) = \\phi^0(s(e))$ and $r(\\phi^1(e)) = \\phi^0(r(e))$ for any $e\\in E^1$. If $v\\in E^0$ and $e\\in E^1$, we will usually write $\\phi(v)$ instead of $\\phi^0(v)$ and $\\phi(e)$ instead of $\\phi^1(e)$. A graph $G$ is called a {\\it subgraph} of a graph $E$ if $G^0\\subseteq E^0$, $G^1\\subseteq E^1$, $s_G=s_E|_{G^0}$ and $r_G=r_E|_{G^0}$.\n\nLet $E=(E^0,E^1,s_E,r_E)$ and $F=(F^0,F^1,s_F,r_F)$ denote graphs. The graph $E\\sqcup F=((E\\sqcup F)^0,(E\\sqcup F)^1,s,r)$ where $(E\\sqcup F)^0=E^0\\sqcup F^0$, $(E\\sqcup F)^1=E^1\\sqcup F^1$, $s|_{E^0}=s_E$, $s|_{F^0}=s_F$, $r|_{E^0}=r_E$ and $r|_{F^0}=r_F$ is called the \\textit{disjoint union} of $E$ and $F$.\n\nLet $E$ be a graph. The graph $E_d=(E_d^0, E_d^1, s_d, r_d)$ where $E_d^0=E^0$, $E_d^1=\\{e,e^*\\mid e\\in E^1\\}$, and $s_d(e)=s(e),~r_d(e)=r(e),~s_d(e^*)=r(e),~r_d(e^*)=s(e)$ for any $e\\in E^1$ is called the {\\it double graph} of $E$. We sometimes refer to the edges in the graph $E$ as {\\it real edges} and to the additional edges in $E_d$ as {\\it ghost edges}. The graph $\\overline{E}$ obtained from $E_d$ by removing the real edges is called the \\textit{inverse graph} of $E$.\n\nA {\\it (finite) path} in a graph $E$ is a finite, nonempty word $p=x_1\\dots x_n$ over the alphabet $E^0\\cup E^1$ such that either $x_i\\in E^1~(i=1,\\dots,n)$ and $r(x_i)=s(x_{i+1})~(i=1,\\dots,n-1)$ or $n=1$ and $x_1\\in E^0$. By definition, the {\\it length} $|p|$ of $p$ is $n$ in the first case and $0$ in the latter case. The set of all paths in $E$ of length $n$ (resp. $\\geq n$) is denoted by $E^n$ (resp. $E^{\\geq n}$).\nFor a path $p=x_1\\dots x_n\\in E^n$ we set $s(p):=s(x_1)$ and $r(p):=r(x_n)$ using the convention $s(v)=r(v)=v$ for any $v\\in E^0$.\n\nA {\\it closed path} is a path $p\\in E^{\\geq 1}$ such that $s(p)=r(p)$.\nA {\\it cycle} is a closed path $x_1\\dots x_n$ such that $s(x_i)\\neq s(x_j)$ for any $i\\neq j$. An edge $e\\in E^1$ is called an {\\it exit} of a cycle $x_1\\dots x_n$ if there is an $i\\in \\{1,\\dots,n\\}$ such that $s(e)=s(x_i)$ and $e\\neq x_i$. A graph is called \\textit{cyclic} if it contains a cycle, and \\textit{acyclic} otherwise. \n\nA {\\it (left-)infinite path} in a graph $E$ is a left-infinite word $p=\\dots x_3x_2x_1$ over the alphabet $E^0\\cup E^1$ such that for any $n\\in\\mathbb{N}$ the suffix $x_n\\dots x_1$ is a path in $E$. We set $|p|:=\\infty$ and $r(p):=r(x_1)$. The set of infinite paths in $E$ is denoted by $E^\\infty$.\n\nWe say that a graph $E$ is \\textit{connected} if for any $u,v\\in E^0$ there is a path $p$ in the double graph $E_d$ such that $s_d(p)=u$ and $r_d(p)=v$. A maximal connected subgraph of $E$ is called a \\textit{connected component} of $E$. Clearly any graph is the disjoint union of its connected components.\n \n\n\n\n\n\\subsection{Leavitt path algebras}\n\\begin{definition}\\label{deflpa}\nLet $E$ be a graph. The $K$-algebra $L(E)$ presented by the generating set $\\{v,e,e^*\\mid v\\in E^0,e\\in E^1\\}$ and the relations\n\\begin{enumerate}[(i)]\n\\item $uv=\\delta_{uv}u\\quad(u,v\\in E^0)$,\n\\item $s(e)e=e=er(e),~r(e)e^*=e^*=e^*s(e)\\quad(e\\in E^1)$,\n\\item $e^*f= \\delta_{ef}r(e)\\quad(e,f\\in E^1)$,\n\\item $\\sum_{e\\in s^{-1}(v)}ee^*= v\\quad(v\\in E_{\\operatorname{reg}}^0)$\n\\end{enumerate}\nis called the {\\it Leavitt path algebra} of $E$. \n\\end{definition}\n\n\n\n\nLet $E$ be a graph and $A$ a $K$-algebra. An {\\it $E$-family} in $A$ is a subset $\\{\\sigma_v,\\sigma_e,\\sigma_{e^*}\\mid v\\in E^0, e\\in E^1\\}\\subseteq A$ such that\n\\begin{enumerate}[(i)]\n\\item $\\sigma_u\\sigma_v=\\delta_{uv}\\sigma_u\\quad(u,v\\in E^0)$, \n\\item\n$\\sigma_{s(e)}\\sigma_{e}=\\sigma_{e}=\\sigma_{e}\\sigma_{r(e)},~\\sigma_{r(e)}\\sigma_{e^*}=\\sigma_{e^*}=\\sigma_{e^*}\\sigma_{s(e)}\\quad(e\\in E^1)$,\n\\item $\\sigma_{e^*}\\sigma_{f}= \\delta_{ef}\\sigma_{r(e)}\\quad(e,f\\in E^1)$ and\n\\item $\\sum_{e\\in s^{-1}(v)}\\sigma_{e}\\sigma_{e^*}= \\sigma_{v}\\quad(v\\in E^0_{\\operatorname{reg}})$.\n\\end{enumerate}\nBy the defining relations of $L(E)$, there exists a unique $K$-algebra homomorphism $\\pi: L(E)\\rightarrow A$ such that $\\pi(v)=\\sigma_v$, $\\pi(e)=\\sigma_{e}$ and $\\pi(e^*)=\\sigma_{e^*}$ for all $v\\in E^0$ and $e\\in E^1$. We will refer to this as the {\\it universal property of $L(E)$}.\n\n\n\\section{Algebraic branching systems}\nUntil the end of this article $E$ denotes a fixed row-finite graph. For any $v\\in E^0_{\\operatorname{reg}}$ we choose an edge $e^v\\in s^{-1}(v)$. The edges $e^v~(v\\in E^0_{\\operatorname{reg}})$ are called {\\it special} and the other edges in $E$ are called {\\it nonspecial}.\n\n\\subsection{Algebraic branching systems}\n\n\\begin{definition}\\label{defabs}\nLet\n\\begin{itemize}\n\\item $\\{X_v\\}_{v\\in E^0}$ be a family of pairwise disjoint subsets of a set $X$,\n\\item $\\{Y_{e^*}\\}_{e\\in E^1}$ a family of pairwise disjoint sets such that $X_v=\\bigcup_{e\\in s^{-1}(v)}Y_{e^*}$ for any $v\\in E_{\\operatorname{reg}}^0$ and \n\\item $\\{\\rho_{e^*}\\}_{e\\in E^1}$ a family of maps such that $\\rho_{e^*}:X_{r(e)} \\to Y_{e^*}$ is a bijection for any $e\\in E^1$.\n\\end{itemize}\nThen $(X,\\{X_v\\}_{v\\in E^0},\\{Y_{e^*}\\}_{e\\in E^1},\\{\\rho_{e^*}\\}_{e\\in E^1})$\nis called an {\\it $E$-algebraic branching system}.\nIn this paper all $E$-algebraic branching systems are assumed to be \\textit{saturated}, i.e. $X=\\bigcup_{v\\in E^0}X_v$.\n\\end{definition}\n\n\\begin{remark}\nWe use the notation $Y_{e^*}$ (respectively $\\rho_{e^*}$) instead of $Y_{e}$ (respectively $\\rho_{e}$) because in this way it is easier to see that algebraic branching systems correspond to a certain type of extended algebraic branching systems, cf. Section 4.\n\\end{remark}\n\nWe denote by $\\ABS(E)$ the category of $E$-algebraic branching systems. A morphism $\\alpha:\\mathcal{S}\\to\\mathcal{S}'$ between two objects $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{e^*}\\},\\{\\rho_{e^*}\\})$ and $\\mathcal{S}'=(X',\\{X'_v\\},\\{Y'_{e^*}\\},$ $\\{\\rho'_{e^*}\\})$ in $\\ABS(E)$ is a map $\\alpha: X\\to X'$ such that $\\alpha(X_v)\\subseteq X'_v$ and $\\alpha (Y_{e^*})\\subseteq Y'_{e^*}$ for any $v\\in E^0$ and $e\\in E^1$, and $\\alpha$ is compatible with the bijections inside $\\mathcal{S}$ and $\\mathcal{S}'$. \n\nSuppose that $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{e^*}\\},\\{\\rho_{e^*}\\})$ is an $E$-algebraic branching system. Let $V=V(\\mathcal{S})$ denote the $K$-vector space with basis $X$. For any $v\\in E^0$ and $e\\in E^1$ define endomorphisms $\\sigma_v,\\sigma_e,\\sigma_{e^*}\\in \\operatorname{End}(V)$ by \n\n\\begin{align*}\n\\sigma_v(x)&=\\begin{cases}\nx,\\quad\\quad\\hspace{0.1cm}&\\text{ if }x\\in X_v,\\\\\n0,&\\text{ otherwise,}\n\\end{cases}\\\\\n\\sigma_{e}(x)&\n=\\begin{cases}\n\\rho^{-1}_{e^*}(x),&\\text{ if }x\\in Y_{e^*},\\\\\n0,&\\text{ otherwise,}\n\\end{cases}\\\\\n\\sigma_{e^*}(x)&=\\begin{cases}\n\\rho_{e^*}(x),\\hspace{0.1cm}&\\text{ if }x\\in X_{r(e)},\\\\\n0,&\\text{ otherwise,}\n\\end{cases}\n\\end{align*}\nfor any $x\\in X$. It follows from the universal property of $L(E)$ that there is a $K$-algebra homomorphism $\\pi:L(E)\\to \\operatorname{End}(V)^{\\operatorname{op}}$ such that $\\pi(v)=\\sigma_v$, $\\pi(e)=\\sigma_e$ and $\\pi(e^*)=\\sigma_{e^*}$ for any $v\\in E^0$ and $e\\in E^1$. We call $\\pi:L(E)\\to \\operatorname{End}(V)^{\\operatorname{op}}$ the {\\it representation of $L(E)$ associated to $\\mathcal{S}$}. \n\n\\begin{remark}\nLet $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{e^*}\\},\\{\\rho_{e^*}\\})$ be an $E$-algebraic branching system. Let $V$ denote the $K$-vector space with basis $X$ and $V'$ the $K$-vector space consisting of all maps from $X$ to $K$. In \\cite{GR}, the representation of $L(E)$ associated to $\\mathcal{S}$ was defined using $V'$ instead of $V$, but the possibility of using $V$ was mentioned in \\cite[Remark 2.3]{GR}. In \\cite{C} the representation of $L(E)$ associated to $\\mathcal{S}$ was defined using $V$. \n\\end{remark}\n\nIf $\\mathcal{S}$ is an object in $\\ABS(E)$, then the vector space $V(\\mathcal{S})$ becomes a right $L(E)$-module by defining $a.b:=\\pi(b)(a)$ for any $a\\in V(\\mathcal{S})$ and $b\\in L(E)$. We call this module the \\textit{$L(E)$-module associated to $\\mathcal{S}$}. If $\\alpha:\\mathcal{S}\\to\\mathcal{S}'$ is a morphism in $\\ABS(E)$, let $V(\\alpha):V(\\mathcal{S})\\to V(\\mathcal{S}')$ be the module homomorphism such that $V(\\alpha)(x)=\\alpha(x)$ for any $x\\in X$. We obtain a functor\n\\[V:\\ABS(E)\\to \\MOD(L(E))\\]\nwhere $\\MOD(L(E))$ denotes the category of right $L(E)$-modules.\n\n\\subsection{Representation graphs}\nOne can visualise branching systems using representation graphs, which are defined below. Recall from \\S 2.1 that $\\overline{E}$ denotes the inverse graph of $E$.\n\n\\begin{definition}\\label{defrg}\nA {\\it representation graph} for $E$ is a pair $(F,\\phi)$, where $F$ is a graph and $\\phi:F\\rightarrow \\overline{E}$ is a graph homomorphism such that the following hold for any $w\\in F^0$.\n\\begin{enumerate}[(i)]\n\\item If $\\phi(w)$ receives an edge in $\\overline{E}$, then $|r^{-1}(w)|=1$.\n\\item $\\phi$ maps $s^{-1}(w)$ 1-1 onto $s_{\\overline{E}}^{-1}(\\phi(w))$. \n\\end{enumerate}\n\\end{definition}\n\n\\begin{example}\\label{ex-1}\nSuppose that $E$ is the graph $\\xymatrix{ v\\ar@[yellow]@(dr,ur)_{e}\\ar@[green]@(dl,ul)^{d}}$ whose inverse graph $\\overline{E}$ is $\\xymatrix{ v\\ar@[blue]@(dr,ur)_{e^*}\\ar@[red]@(dl,ul)^{d^*}}$. Then a representation graph $(F,\\phi)$ for $E$ is given by\n\\begin{equation*}\n\\xymatrix@R=0.5cm@C=0.5cm{\n&&&&&&&&&&&\\ar@[blue]@{<.}[dll]_{e^*}\\\\\n&&&&&&&&&v\\ar@[blue]@{<-}[ddllll]_{e^*}&&\\ar@[red]@{<.}[ll]^{d^*}\\\\\n&&&&&&&&&&&\\ar@[blue]@{<.}[dll]_{e^*}\\\\\n&v\\ar@[blue]@(dl,ul)^{e^*}&&&&v\\ar@[red]@{<-}[llll]^{d^*}&&&&v\\ar@[red]@{<-}[llll]^{d^*}&&\\ar@[red]@{<.}[ll]^{d^*}}\n\\end{equation*}\nwhere a label of a vertex (resp. edge) indicates the image of that vertex (resp. edge) under $\\phi$.\n\\end{example}\n\nWe denote by $\\RG(E)$ the category of representation graphs for $E$. A morphism $\\alpha:(F,\\phi)\\to (G,\\psi)$ in $\\RG(E)$ is a graph homomorphism $\\alpha:F\\to G$ such that $\\psi\\circ \\alpha=\\phi$.\n\nSuppose that $(F,\\phi)$ is an object in $\\RG(E)$. Let $W=W(F,\\phi)$ be the $K$-vector space with basis $F^0$. For any $v\\in E^0$ and $e\\in E^1$ define endomorphisms $\\sigma_v,\\sigma_e,\\sigma_{e^*}\\in \\operatorname{End}(W)$ by\n\\begin{align*}\n\\sigma_v(w)&=\\begin{cases}w,\\quad\\quad\\hspace{0.05cm}&\\text{if }\\phi(w)=v,\\\\0,& \\text{otherwise}, \\end{cases}\\\\\n\\sigma_e(w)&=\\begin{cases}s(f),\\quad&\\text{if }\\exists f\\in r^{-1}(w):\\phi(f)=e^*,\\\\0,& \\text{otherwise}, \\end{cases}\\\\\n\\sigma_{e^*}(w)&=\\begin{cases}r(f),\\quad&\\text{if }\\exists f\\in s^{-1}(w):\\phi(f)=e^*,\\\\0,& \\text{otherwise},\n \\end{cases}\n\\end{align*}\nfor any $w\\in F^0$. It follows from the universal property of $L(E)$ that there is a $K$-algebra homomorphism $\\pi:L(E)\\to \\operatorname{End}(W)^{\\operatorname{op}}$ such that $\\pi(v)=\\sigma_v$, $\\pi(e)=\\sigma_e$ and $\\pi(e^*)=\\sigma_{e^*}$ for any $v\\in E^0$ and $e\\in E^1$. We call $\\pi:L(E)\\to \\operatorname{End}(W)^{\\operatorname{op}}$ the {\\it representation of $L(E)$ associated to $(F,\\Phi)$}. \n\nIf $(F,\\phi)$ is an object in $\\RG(E)$, then the vector space $W(F,\\phi)$ becomes a right $L(E)$-module by defining $a.b:=\\pi(b)(a)$ for any $a\\in W(F,\\phi)$ and $b\\in L(E)$. We call this module the \\textit{$L(E)$-module associated to $(F,\\phi)$}. If $\\alpha:(F,\\phi)\\to (G,\\psi)$ is a morphism in $\\RG(E)$, let $W(\\alpha):W(F,\\phi)\\to W(G,\\psi)$ be the module homomorphism such that $W(\\alpha)(w)=\\alpha(w)$ for any $w\\in F^0$. We obtain a functor\n\\[W:\\RG(E)\\to \\MOD(L(E)).\\]\n\n\\begin{example}\\label{ex0}\nSuppose again that $E$ is the graph $\\xymatrix{ v\\ar@[yellow]@(dr,ur)_{e}\\ar@[green]@(dl,ul)^{d}}$ whose inverse graph $\\overline{E}$ is $\\xymatrix{ v\\ar@[blue]@(dr,ur)_{e^*}\\ar@[red]@(dl,ul)^{d^*}}$, and that $(F,\\phi)$ is the representation graph for $E$ given by\n\\begin{equation*}\n\\xymatrix@R=0.5cm@C=0.5cm{\n&&&&&&&&&&&\\ar@[blue]@{<.}[dll]_{e^*}\\\\\n&&&&&&&&&v_4\\ar@[blue]@{<-}[ddllll]_{e^*}&&\\ar@[red]@{<.}[ll]^{d^*}\\\\\n&&&&&&&&&&&\\ar@[blue]@{<.}[dll]_{e^*}\\\\\n&v_1\\ar@[blue]@(dl,ul)^{e^*}&&&&v_2\\ar@[red]@{<-}[llll]^{d^*}&&&&v_3\\ar@[red]@{<-}[llll]^{d^*}&&\\ar@[red]@{<.}[ll]^{d^*}.}\n\\end{equation*}\nRecall that $W(F,\\Phi)$ is the $K$-vector space with basis $F^0$. The action of $L(E)$ on $W(F,\\Phi)$ ``slides'' the vertices of $F$ along the edges. For example, $v_1. d^*=v_2$, $v_2.d^*=v_3$, $v_2. d=v_1$ and $v_2. e=0$. The $L(E)$-module $W(F,\\Phi)$ is isomorphic to the Chen simple module defined by the right-infinite path $eee\\dots $, see \\cite[Section 4]{HPS}. \n\\end{example}\n\n\\subsection{Algebraic branching systems vs. representation graphs}\nTo any object $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{e^*}\\},\\{\\rho_{e^*}\\})$ in $\\ABS (E)$ we associate the object $\\eta(\\mathcal{S})=(F,\\phi)$ in $\\RG(E)$ defined by\n\\begin{itemize}\n\\item $F^0=X$,\n\\item $F^1=\\{f_x\\mid x\\in \\bigcup_{e\\in E^1}Y_{e^*}\\}$,\n\\item $s_F(f_x)=\\rho_{e^*}^{-1}(x)$ if $x\\in Y_{e^*}$,\n\\item $r_F(f_x)=x$,\n\\item $\\phi^0(x)=v$ if $x\\in X_v$,\n\\item $\\phi^1(f_x)=e^*$ if $x\\in Y_{e^*}$.\n\\end{itemize}\nTo any morphism $\\alpha:\\mathcal{S}=(X,\\{X_v\\},\\{Y_{e^*}\\},\\{\\rho_{e^*}\\})\\to \\mathcal{S}'=(X',\\{X'_v\\},\\{Y'_{e^*}\\},\\{\\rho'_{e^*}\\})$ in $\\ABS (E)$ we associate the morphism $ \\eta(\\alpha):\\eta(\\mathcal{S})\\to \\eta(\\mathcal{S}')$ in $\\RG(E)$ defined by $ \\eta(\\alpha)^0(x)=\\alpha(x)$ for any $x\\in X$ and $ \\eta(\\alpha)^1(f_x)=f'_{\\alpha(x)}$ for any $x\\in \\bigcup_{e\\in E^1}Y_{e^*}$. In this way we obtain a functor $\\eta:\\ABS (E)\\to \\RG(E)$. \n\nConversely, to any object $(F,\\phi)$ in $\\RG(E)$ we associate an object $ \\theta(F,\\phi)=(X,\\{X_v\\},\\{Y_{e^*}\\},\\{\\rho_{e^*}\\})$ in $\\ABS (E)$ defined by\n\\begin{itemize}\n\\item $X=F^0$,\n\\item $X_v=\\{w\\in F^0\\mid \\phi(w)=v\\}$ for any $v\\in E^0$,\n\\item $Y_{e^*}=\\{w\\in F^0\\mid \\exists f\\in r_F^{-1}(w):\\phi(f)=e^*\\}$ for any $e\\in E^1$,\n\\item $\\rho_{e^*}(w)=r(f)$ for any $e\\in E^1$ and $w\\in X_{r(e)}$, where $f$ is the unique edge in $s^{-1}(w)\\cap \\phi^{-1}(e^*)$. \n\\end{itemize}\nTo any morphism $\\alpha:(F,\\phi)\\to (G,\\psi)$ in $\\RG(E)$ we associate the morphism $ \\theta(\\alpha): \\theta(F,\\phi)\\to \\theta(G,\\psi)$ in $\\ABS(E)$ defined by $ \\theta(\\alpha)(x)=\\alpha(x)$ for any $x\\in X=F^0$. In this way we obtain a functor $ \\theta:\\RG(E)\\to \\ABS (E)$. \n\nWe leave it to the reader to check that $ \\theta\\circ\\eta=\\operatorname{id}_{\\ABS (E)}$ and $ \\eta\\circ \\theta\\cong\\operatorname{id}_{\\RG(E)}$. Hence the categories $\\RG(E)$ and $\\ABS (E)$ are equivalent. Moreover, the diagrams\n\\[\\xymatrix{\\ABS (E)\\ar[dr]^V&\\\\&\\MOD L(E)& \\text{and}\\\\\\RG(E)\\ar[uu]^{\\theta}\\ar[ur]_W&\\\\}\n\\quad\\quad\n\\xymatrix{\\ABS (E)\\ar[dd]_{\\eta}\\ar[dr]^V&\\\\&\\MOD L(E)\\\\\\RG(E)\\ar[ur]_W&\\\\}\n\\]\nare commutative.\n\n\\section{Extended algebraic branching systems}\n\\subsection{Extended algebraic branching systems}\n\\begin{definition}\\label{defeabs}\nLet \n\\begin{itemize}\n\\item $\\{X_v\\}_{v\\in E^0}$ be a family of pairwise disjoint subsets of a set $X$,\n\\item $\\{Y_i\\}_{i\\in E^1\\cup (E^1)^*}$ a family of pairwise disjoint sets such that\n$Y_e\\subseteq X_{r(e)}$ and $Y_{e^*}\\subseteq X_{s(e)}$ for any $e\\in E^1$,\n\\item $\\{\\rho_j\\}_{j\\in E^1\\cup (E^1)^*}$ a family of maps such that \n\\begin{enumerate}[(i)]\n\\item $\\rho_{e}: X_{s(e)}\\setminus\\bigcup_{f\\in s^{-1}(s(e))}Y_{f^*}\\to Y_{e}$ is a bijection for any $e\\in E^1$,\n\\item $\\rho_{e^*}: X_{r(e)}\\to Y_{e^*}$ is a bijection for any nonspecial $e\\in E^1$,\n\\item $\\rho_{e^*}: X_{r(e)}\\setminus Y_{e}\\to Y_{e^*}$ is a bijection for any special $e\\in E^1$.\n\\end{enumerate}\n\\end{itemize}\nThen \\[(X, \\{X_v\\}_{v\\in E^0}, \\{Y_i\\}_{i\\in E^1\\cup (E^1)^*},\\{\\rho_j\\}_{j\\in E^1\\cup (E^1)^*})\\]\nis called an {\\it extended $E$-algebraic branching system}. In this paper all extended $E$-algebraic branching systems are assumed to be \\textit{saturated}, i.e. $X=\\bigcup_{v\\in E^0}X_v$.\n\\end{definition}\n\n\\begin{example}\\label{exusual}\nSuppose that $(X, \\{X_v\\}_{v\\in E^0}, \\{Y_{e^*}\\}_{e\\in E^1},\\{\\rho_{e^*}\\}_{e\\in E^1})$ is a usual $E$-algebraic branching system. For any $e\\in E^1$ let $Y_e=\\emptyset$ and $\\rho_{e}: \\emptyset=X_{s(e)}\\setminus\\bigcup_{f\\in s^{-1}(s(e))}Y_{f^*}\\to Y_{e}=\\emptyset$ be the unique map. Then $(X, \\{X_v\\}_{v\\in E^0}, \\{Y_i\\}_{i\\in E^1\\cup (E^1)^*},\\{\\rho_j\\}_{j\\in E^1\\cup (E^1)^*})$ is an extended $E$-algebraic branching system. In this way one obtains a 1-1 correspondence between the usual $E$-algebraic branching systems and the extended $E$-algebraic branching systems which have the property that $Y_e=\\emptyset$ for any $e\\in E^1$.\n\\end{example}\n\n\\begin{example}\nSet\n\\begin{align*}\nX:=\\{v\\mid v\\in E^0\\}\\cup\\{p,p^*\\mid p\\in E^{\\geq 1}\\}\\cup\\{pq^*\\mid \\hspace{0.12cm}&p=e_1\\dots e_m, q=f_1\\dots f_n\\in E^{\\geq 1},r(p)=r(q)\\\\&\\text {and either }e_m\\neq f_n\\text{ or }e_m=f_n\\text{ is nonspecial}\\}.\n\\end{align*}\nThen $X$ is a linear basis of $L(E)$ by \\cite[Theorem 1]{zelmanov1}. For any $v\\in E^0$ let $X_v$ be the subset of $X$ consisting of all elements whose range is $v$. For any $i\\in E^1\\cup (E^1)^*$ let $Y_i$ be the subset of $X$ consisting of all elements whose last letter is $i$. For any $j\\in E^1\\cup (E^1)^*$ let $\\rho_j$ be the map that adds the letter $j$ to the end of a path respectively replaces a letter from $E^0$ by $j$ (the domain and codomain of $\\rho_j$ are as in Definition \\ref{defeabs}). One checks easily that $(X,\\{X_v\\},\\{Y_i\\},\\{\\rho_j\\})$ is an extended $E$-algebraic branching system.\n\\end{example}\n\nWe denote by $\\EABS(E)$ the category of extended $E$-algebraic branching systems. A morphism $\\alpha:\\mathcal{S}\\to\\mathcal{S}'$ between two objects $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{i}\\},\\{\\rho_{j}\\})$ and $\\mathcal{S}'=(X',\\{X'_v\\},\\{Y'_{i}\\},$ $\\{\\rho'_{j}\\})$ in $\\EABS(E)$ is a map $\\alpha: X\\to X'$ such that $\\alpha (X_{v})\\subseteq X'_{v}$ and $\\alpha (Y_{i})\\subseteq Y'_{i}$ for any $v\\in E^0$ and $i\\in E^1\\cup (E^1)^*$, and $\\alpha$ is compatible with the bijections inside $\\mathcal{S}$ and $\\mathcal{S}'$ (in particular we require that $\\alpha(X_{s(e)}\\setminus\\bigcup_{f\\in s^{-1}(s(e))}Y_{f^*})\\subseteq X'_{s(e)}\\setminus\\bigcup_{f\\in s^{-1}(s(e))}Y'_{f^*}$ for any $e\\in E^1$ and $\\alpha(X_{r(e)}\\setminus Y_{e})\\subseteq X'_{r(e)}\\setminus Y'_{e}$ for any special $e\\in E^1$). \n\nSuppose that $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{i}\\},\\{\\rho_{j}\\})$ is an extended $E$-algebraic branching system. Let $V=V(\\mathcal{S})$ denote the $K$-vector space with basis $X$. For any $v\\in E^0$ and $e\\in E^1$ define endomorphisms $\\sigma_v,\\sigma_e,\\sigma_{e^*}\\in \\operatorname{End}(V)$ by \n\n\\begin{align*}\n\\sigma_v(x)&=\\begin{cases}\nx,&\\hspace{6.9cm}\\text{ if }x\\in X_v,\\\\\n0,&\\hspace{6.9cm}\\text{ otherwise,}\n\\end{cases}\\\\\n\\sigma_{e}(x)&=\\begin{cases}\n\\rho_{e}(x),&\\hspace{5.95cm}\\text{ if }x\\in X_{s(e)}\\setminus\\bigcup_{f\\in s^{-1}(s(e))}Y_{f^*},\\\\\n\\rho_{e^*}^{-1}(x),&\\hspace{5.95cm}\\text{ if }x\\in Y_{e^*},\\\\\n0,&\\hspace{5.95cm}\\text{ otherwise,}\n\\end{cases}\\\\\n\\sigma_{e^*}(x)&=\\begin{cases}\n\\rho_{e^*}(x),&\\text{ if }x\\in X_{r(e)}\\setminus Y_{e},\\\\\n\\rho_{e^*}(x),&\\text{ if }x\\in Y_{e}\\text{ and }e\\text{ is nonspecial},\\\\\n\\rho^{-1}_{e}(x)-\\sum_{d\\in s^{-1}(s(e))\\setminus\\{e\\}}\\rho_{d^*}(\\rho_{d}(\\rho_{e}^{-1}(x))),&\\text{ if }x\\in Y_{e}\\text{ and }e\\text{ is special},\\\\\n0,&\\text{ otherwise,}\n\\end{cases}\n\\end{align*}\nfor any $x\\in X$. It follows from the universal property of $L(E)$ that there is a $K$-algebra homomorphism $\\pi:L(E)\\to \\operatorname{End}(V)^{\\operatorname{op}}$ such that $\\pi(v)=\\sigma_v$, $\\pi(e)=\\sigma_e$ and $\\pi(e^*)=\\sigma_{e^*}$ for any $v\\in E^0$ and $e\\in E^1$. We call $\\pi:L(E)\\to \\operatorname{End}(V)^{\\operatorname{op}}$ the {\\it representation of $L(E)$ associated to $\\mathcal{S}$}. \n\nIf $\\mathcal{S}$ is an object in $\\EABS(E)$, then the vector space $V(\\mathcal{S})$ becomes a right $L(E)$-module by defining $a.b:=\\pi(b)(a)$ for any $a\\in V(\\mathcal{S})$ and $b\\in L(E)$. We call this module the \\textit{$L(E)$-module associated to $\\mathcal{S}$}. If $\\alpha:\\mathcal{S}\\to\\mathcal{S}'$ is a morphism in $\\EABS(E)$, let $V(\\alpha):V(\\mathcal{S})\\to V(\\mathcal{S}')$ be the module homomorphism such that $V(\\alpha)(x)=\\alpha(x)$ for any $x\\in X$. We obtain a functor\n\\[V:\\EABS(E)\\to \\MOD(L(E)).\\]\n\n\n\\subsection{Extended representation graphs}\n\nOne can visualise extended branching systems using extended representation graphs, which are defined below. Recall from \\S 2.1 that $E_d$ denotes the double graph of $E$.\n\n\\begin{definition}\\label{deferg}\nAn {\\it extended representation graph} for $E$ is a pair $(F,\\phi)$, where $F$ is a graph and $\\phi:F\\rightarrow E_d$ a graph homomorphism such that the following hold for any $w\\in F^0$.\n\\begin{enumerate}[(i)]\n\\item $w$ is either a source or receives a unique edge $f_w$.\n\\item If either $w$ is a source or $\\phi(f_w)$ is a nonspecial real edge, then $\\phi$ maps $s^{-1}(w)$ 1-1 onto $s_d^{-1}(\\phi(w))$.\n\\item If $\\phi(f_w)$ is a special real edge, then $\\phi$ maps $s^{-1}(w)$ 1-1 onto $s_d^{-1}(\\phi(w))\\setminus\\{(\\phi(f_w))^*\\}$.\n\\item If $\\phi(f_w)$ is a ghost edge, then $\\phi$ maps $s^{-1}(w)$ 1-1 onto $s_d^{-1}(\\phi(w))\\cap (E^1)^*$.\n\\end{enumerate}\n\\end{definition}\n\\vspace{0.1cm}\n\n\\begin{example}\\label{extrivial}\nThe usual representation graphs for $E$ are precisely the extended representation graphs $(F,\\phi)$ for $E$ having the property that $\\phi(f)$ is a ghost edge for any $f\\in F^1$.\n\\end{example}\n\n\\begin{example}\\label{ex1}\nSuppose that $E$ is the graph $\\xymatrix{ v\\ar@[yellow]@(dr,ur)_{e}\\ar@[green]@(dl,ul)^{d}}$ whose double graph $E_d$ is\n\\begin{equation*}\n\\xymatrix{ v\\ar@[yellow]@(dr,ur)_{e.}\\ar@[green]@(dl,ul)^{d}\\ar@[blue]@(dr,dl)^{e^*}\\ar@[red]@(ul,ur)^{d^*}}\n\\end{equation*} \nAs special edge choose $e=e^v$. Then an extended representation graph $(F,\\phi)$ for $E$ is given by\n\\begin{equation*}\n\\xymatrix@R=0.6cm@C=0.7cm{\n&&&&&&&&&\\\\\n&&&&&&&v\\ar@[red]@{.>}[u]^{d^*}\\ar@[blue]@{.>}[ur]_{e^*}&&\\\\\n&&&&&&&&&\\ar@[yellow]@{<.}[dll]_e\\\\\n&&&&&&&v\\ar@[yellow]@{<-}[ddllll]_e\\ar@[red][uu]^{d^*}&&\\ar@[green]@{<.}[ll]^d\\\\\n&&&&&&&&&\\ar@[yellow]@{<.}[dll]_e\\\\\n&&&v\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}&&&&v\\ar@[green]@{<-}[llll]^d\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}&&\\ar@[green]@{<.}[ll]^d.\\\\\n&&&&&&&&&\\\\\n&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\\\\n&&&&&&&&&\\\\\n&&&&&&&&&}\n\\end{equation*}\n\\end{example}\n\nWe denote by $\\ERG(E)$ the category of extended representation graphs for $E$. A morphism $\\alpha:(F,\\phi)\\to (G,\\psi)$ in $\\ERG(E)$ is a graph homomorphism $\\alpha:F\\to G$ such that $\\psi\\circ \\alpha=\\phi$.\n\nSuppose that $(F,\\phi)$ is an object in $\\ERG(E)$. Let $W=W(F,\\phi)$ be the $K$-vector space with basis $F^0$. For any $v\\in E^0$ and $e\\in E^1$ define endomorphisms $\\sigma_v,\\sigma_e,\\sigma_{e^*}\\in \\operatorname{End}(W)$ by\n\\begin{align*}\n\\sigma_v(w)&=\\begin{cases}w,\\quad\\quad\\quad\\quad~&\\text{if }\\phi(w)=v,\\\\0,& \\text{otherwise}, \\end{cases}\\\\\n\\sigma_{e}(w)&=\\begin{cases}\nr(f),\\quad&\\quad~~~\\text{ if }\\exists f\\in s^{-1}(w): \\phi(f)=e,\\\\\ns(f),\\quad&\\quad~~~\\text{ if }\\exists f\\in r^{-1}(w): \\phi(f)=e^*,\\\\\n0,&\\quad~~~\\text{ otherwise,}\n\\end{cases}\\\\\n\\sigma_{e^*}(w)&=\\begin{cases}\nr(f),\\quad&\\text{ if }\\exists f\\in s^{-1}(w): \\phi(f)=e^*,\\\\\ns(f)-T,\\quad&\\text{ if }e\\text{ is special and }\\exists f\\in r^{-1}(w): \\phi(f)=e,\\\\\n0,&\\text{ otherwise,}\n\\end{cases}\n\\end{align*}\nfor any $w\\in F^0$. Here $T=\\sum_p r(p)$ where $p$ ranges over all paths in $F$ starting in $s(f)$ such that $\\phi(p)=dd^*$ for some $d\\in s^{-1}(s(e))\\setminus\\{e\\}$. It follows from the universal property of $L(E)$ that there is a $K$-algebra homomorphism $\\pi:L(E)\\to \\operatorname{End}(W)^{\\operatorname{op}}$ such that $\\pi(v)=\\sigma_v$, $\\pi(e)=\\sigma_e$ and $\\pi(e^*)=\\sigma_{e^*}$ for any $v\\in E^0$ and $e\\in E^1$. We call $\\pi:L(E)\\to \\operatorname{End}(W)^{\\operatorname{op}}$ the {\\it representation of $L(E)$ associated to $(F,\\phi)$}. \n\nIf $(F,\\phi)$ is an object in $\\ERG(E)$, then the vector space $W(F,\\phi)$ becomes a right $L(E)$-module by defining $a.b:=\\pi(b)(a)$ for any $a\\in W(\\mathcal{S})$ and $b\\in L(E)$. We call this module the \\textit{$L(E)$-module associated to $(F,\\phi)$}. If $\\alpha:(F,\\phi)\\to (G,\\psi)$ is a morphism in $\\ERG(E)$, let $W(\\alpha):W(F,\\phi)\\to W(G,\\psi)$ be the module homomorphism such that $W(\\alpha)(w)=\\alpha(w)$ for any $w\\in F^0$. We obtain a functor\n\\[W:\\ERG(E)\\to \\MOD(L(E)).\\]\n\n\\subsection{Extended algebraic branching systems vs. extended representation graphs}\nTo any object $\\mathcal{S}=(X,\\{X_v\\},\\{Y_{i}\\},\\{\\rho_{j}\\})$ in $\\EABS (E)$ we associate the object $\\eta(\\mathcal{S})=(F,\\phi)$ in $\\ERG(E)$ defined by\n\\begin{itemize}\n\\item $F^0=X$,\n\\item $F^1=\\{f_x\\mid x\\in \\bigcup_{e\\in E^1}(Y_e\\cup Y_{e^*})\\}$,\n\\item $s_F(f_x)=\\rho_{e}^{-1}(x)$ if $x\\in Y_{e}$, respectively $s_F(f_x)=\\rho_{e^*}^{-1}(x)$ if $x\\in Y_{e^*}$,\n\\item $r_F(f_x)=x$,\n\\item $\\phi^0(x)=v$ if $x\\in X_v$,\n\\item $\\phi^1(f_x)=e$ if $x\\in Y_{e}$, respectively $\\phi^1(f_x)=e^*$ if $x\\in Y_{e^*}$.\n\\end{itemize}\nTo any morphism $\\alpha:\\mathcal{S}=(X,\\{X_v\\},\\{Y_{i}\\},\\{\\rho_{j}\\})\\to \\mathcal{S}'=(X',\\{X'_v\\},\\{Y'_{i}\\},\\{\\rho'_{j}\\})$ in $\\EABS (E)$ we associate the morphism $ \\eta(\\alpha):\\eta(\\mathcal{S})\\to \\eta(\\mathcal{S}')$ in $\\ERG(E)$ defined by $ \\eta(\\alpha)^0(x)=\\alpha(x)$ for any $x\\in X$ and $ \\eta(\\alpha)^1(f_x)=f'_{\\alpha(x)}$ for any $x\\in \\bigcup_{e\\in E^1}(Y_e\\cup Y_{e^*})$. In this way we obtain a functor $\\eta:\\EABS (E)\\to \\ERG(E)$. \n\nConversely, to any object $(F,\\phi)$ in $\\ERG(E)$ we associate an object $ \\theta(F,\\phi)=(X,\\{X_v\\},\\{Y_{i}\\},\\{\\rho_{j}\\})$ in $\\EABS (E)$ defined by\n\\begin{itemize}\n\\item $X=F^0$,\n\\item $X_v=\\{w\\in F^0\\mid \\phi(w)=v\\}$ for any $v\\in E^0$,\n\\item $Y_{e}=\\{w\\in F^0\\mid \\exists f\\in r_F^{-1}(w):\\phi(f)=e\\}$ for any $e\\in E^1$,\n\\item $Y_{e^*}=\\{w\\in F^0\\mid \\exists f\\in r_F^{-1}(w):\\phi(f)=e^*\\}$ for any $e\\in E^1$,\n\\item $\\rho_{e}(w)=r(f)$ for any $e\\in E^1$, where $f$ is the unique edge in $s^{-1}(w)\\cap \\phi^{-1}(e)$,\n\\item $\\rho_{e^*}(w)=r(f)$ for any $e\\in E^1$, where $f$ is the unique edge in $s^{-1}(w)\\cap \\phi^{-1}(e^*)$. \n\\end{itemize}\nTo any morphism $\\alpha:(F,\\phi)\\to (G,\\psi)$ in $\\ERG(E)$ we associate the morphism $ \\theta(\\alpha): \\theta(F,\\phi)\\to \\theta(G,\\psi)$ in $\\EABS(E)$ defined by $ \\theta(\\alpha)(x)=\\alpha(x)$ for any $x\\in X=F^0$. In this way we obtain a functor $ \\theta:\\ERG(E)\\to \\EABS (E)$. \n\nWe leave it to the reader to check that $ \\theta\\circ \\eta=\\operatorname{id}_{\\EABS (E)}$ and $ \\eta\\circ \\theta\\cong\\operatorname{id}_{\\ERG(E)}$. Hence the categories $\\ERG(E)$ and $\\EABS (E)$ are equivalent. Moreover, the diagrams\n\\[\\xymatrix{\\EABS (E)\\ar[dr]^V&\\\\&\\MOD L(E)& \\text{and}\\\\\\ERG(E)\\ar[uu]^{\\theta}\\ar[ur]_W&\\\\}\n\\quad\\quad\n\\xymatrix{\\EABS (E)\\ar[dd]_{\\eta}\\ar[dr]^V&\\\\&\\MOD L(E)\\\\\\ERG(E)\\ar[ur]_W&\\\\}\n\\]\nare commutative.\n\n\\section{Classification of the extended algebraic branching systems}\nThe goal of this section is to classify the extended $E$-algebraic branching systems. In order to do so, it suffices to classify the extended representation graphs for $E$ (in view of \\S4.3). \n\nIf $(F,\\phi)$ and $(G,\\psi)$ are extended representation graphs for $E$, then their \\textit{disjoint union} is the extended representation graph $(F\\sqcup G, \\phi\\sqcup \\psi)$ for $E$, where $(\\phi\\sqcup\\psi)|_{F}=\\phi$ and $(\\phi\\sqcup\\psi)|_{G}=\\psi$. If $(F,\\phi)$ is an extended representation graph for $E$ and $\\{F_i\\}_{i\\in\\Lambda}$ are the connected components of $F$, then clearly $(F,\\phi)$ is the disjoint union of the extended representation graphs $(F_i,\\phi|_{F_i})$ for $E$. Hence it suffices to classify the \\textit{connected} extended representation graphs for $E$.\n\n\\subsection{Basis paths}\n\\subsubsection{Finite basis paths}\nSet\n\\begin{align*}\nX:=\\{v\\mid v\\in E^0\\}\\cup\\{p,p^*\\mid p\\in E^{\\geq 1}\\}\\cup\\{pq^*\\mid \\hspace{0.12cm}&p=e_1\\dots e_m, q=f_1\\dots f_n\\in E^{\\geq 1},r(p)=r(q)\\\\&\\text {and either }e_m\\neq f_n\\text{ or }e_m=f_n\\text{ is nonspecial}\\}.\n\\end{align*}\nWe consider the elements of $X$ as paths in $E_d$ and call them \\textit{(finite) basis paths}.\nFor any $v\\in E^0$ we set $X_v:=\\{x\\in X\\mid s_d(x)=v\\}$. For a basis path $x=x_1\\dots x_l\\in X$ and $0\\leq n \\leq l$ we define $\\tau_{\\leq n}(x):=x_{1}\\dots x_n\\in X$ and $\\tau_{>n}(x):=x_{n+1}\\dots x_l\\in X$. Here we use the convention $\\tau_{\\leq 0}(x)=s_d(x)$ and $\\tau_{>l}(x)=r_d(x)$. \n\n \nFor any graph $F$, $w\\in F^0$ and $n\\geq 0$ we set $F^{\\geq n}_w:=\\{p\\in F^{\\geq n} \\mid s(p)=w\\}$. Suppose $(F,\\phi)$ is an extended representation graph for $E$. If $w\\in F^0$ and $f\\in F^1$, we say that \\textit{$w$ lies over $\\phi(w)$} and \\textit{$f$ lies over $\\phi(f)$}. If $w\\in F^0$ lies over $v$, then $\\phi$ defines a map $F^{\\geq 0}_w\\to (E_d)^{\\geq 0}_v$, which we also denote by $\\phi$.\n\n\\begin{lemma}\\label{lembase}\nLet $(F,\\phi)$ be an extended representation graph for $E$ and $w$ a vertex in $F$ lying over a vertex $v\\in E^0$. Then $(i)-(iii)$ below hold.\n\\begin{enumerate}[(i)]\n\\item If $w$ is a source or receives an edge lying over a nonspecial real edge, then $\\phi$ defines a bijection $F^{\\geq 0}_w\\to X_{v}$.\n\\item If $w$ receives an edge lying over a special real edge $e$, then $\\phi$ defines a bijection $F^{\\geq 0}_w\\to \\{x_1\\dots x_n\\in X_{v}\\mid x_1\\neq e^*\\}$.\n\\item If $w$ receives an edge lying over a ghost edge, then $\\phi$ defines a bijection $F^{\\geq 0}_w\\to \\{x_1\\dots x_n\\in X_{v}\\mid x_1\\in (E^1)^*\\}$.\n\\end{enumerate}\nIn particular, $\\phi$ maps paths in $F$ to basis paths in $E_d$. \n\\end{lemma}\n\\begin{proof}\nWe only prove (i) and leave (ii) and (iii) to the reader. So suppose that $w$ is a source or receives an edge lying over a nonspecial real edge. First we show that $\\phi(F^{\\geq 0}_w)\\subseteq X_v$. Clearly $\\phi(w)=v\\in X_v$. Now let $p=f_1\\dots f_n\\in F^{\\geq 1}_w$. In order to show that $\\phi(p)\\in X_v$, it clearly suffices to show that $\\phi(f_i)\\phi(f_{i+1})\\not\\in \\{ c^*d, e^u(e^u)^*\\mid c,d\\in E^1, u\\in E^0\\}$ for any $i\\in\\{1,\\dots,n-1\\}$. But that follows from Definition \\ref{deferg}(iii),(iv). \n\nNow we show that the map $F^{\\geq 0}_w\\to X_{v}$ defined by $\\phi$ is injective. We will use the fact that $\\phi$ is injective on all sets $s^{-1}(w')~(w'\\in F^0)$ (this follows from Definition \\ref{deferg}(ii)-(iv)). Suppose that $\\phi(p)=\\phi(q)$ where $p,q\\in F^{\\geq 0}_w$. Then $p$ and $q$ have the same length $n$ since $\\phi$ preserves lengths. If $n=0$, then $p=w=q$. Now assume that $p=f_1\\dots f_n,q=g_1\\dots g_n\\in F^{\\geq 1}_w$. Since $\\phi(p)=\\phi(q)$, we have $\\phi(f_i)=\\phi(g_i)~(i=1,\\dots,n)$. Clearly $f_1,g_1\\in s^{-1}(w)$. Since $\\phi$ is injective on $s^{-1}(w)$, it follows that $f_1=g_1$ and hence $f_2,g_2\\in s^{-1}(r(f_1))$. Since $\\phi$ is injective on $s^{-1}(r(f_1))$, it follows that $f_2=g_2$ and hence $f_3,g_3\\in s^{-1}(r(f_2))$. Proceeding like that we obtain $f_i=g_i~(i=1,\\dots,n)$ and hence $p=q$ as desired.\n\nIt remains to show that the map $F^{\\geq 0}_w\\to X_{v}$ defined by $\\phi$ is surjective. Clearly $X_v=\\bigcup_{n\\geq 0}X_v^n$ where for any $n\\geq 0$, $X_v^n$ is the subset of $X_v$ consisting of all elements of length $n$. Hence it suffices to show that $X_v^n\\subseteq \\phi(F^{\\geq 0}_w)$ for any $n\\geq 0$. \\\\\n\\\\\n\\underline{$n=0$:} Clearly $X_v^0=\\{v\\}=\\{\\phi(w)\\}\\subseteq\\phi(F^{\\geq 0}_w)$.\\\\\n\\\\\n\\underline{$n=1$:} \nLet $x\\in X_v^{1}=s_d^{-1}(v)$. It follows from Definition \\ref{deferg}(ii) that $w$ emits an edge $f$ lying over $x$. Hence $X_v^1\\subseteq \\phi(F^{\\geq 0}_w)$.\\\\\n\\\\\n\\underline{$n\\to n+1$:} \nAssume that $X_v^n\\subseteq \\phi(F^{\\geq 0}_w)$ for some $n\\geq 1$. Let $x=x_1\\dots x_nx_{n+1}\\in X_v^{n+1}$. By the induction assumption we know that there is a $p=f_1\\dots f_n\\in F^{\\geq 0}_w$ such that $\\phi(p)=x_1\\dots x_n$. It follows from Definition \\ref{deferg}(ii)-(iv) that $r(f_n)$ emits an edge $f_{n+1}$ lying over $x_{n+1}$. Hence $\\phi(f_1\\dots f_nf_{n+1})= x_1\\dots x_nx_{n+1}=x$ and thus we have shown that $X_v^{n+1}\\subseteq \\phi(F^{\\geq 0}_w)$.\n\\end{proof}\n\n\\subsubsection{Infinite basis paths}\nWe denote by $X^{\\infty}$ the set of all infinite paths $x=\\dots x_3x_2x_1$ in $E_d$ having the property that any finite suffix $x_n\\dots x_1$ lies in $X$. We call the elements of $X^{\\infty}$ \\textit{infinite basis paths}. We call an infinite basis path $x=\\dots x_3x_2x_1$ \\textit{real} if all of the edges $x_1,x_2, x_3\\dots$ are real, and \\textit{ghostly} if all of the edges $x_1,x_2, x_3\\dots$ are ghost edges. If $x=\\dots x_2x_1\\in X^{\\infty}$ and $n\\geq 0$, we set $\\tau_{\\leq n}(x):=x_{n}\\dots x_{1}\\in X$ and $\\tau_{>n}(x):=\\dots x_{n+2}x_{n+1}\\in X^{\\infty}$. Here we use the convention $\\tau_{\\leq 0}(x)=s_d(x)$. Two infinite basis paths $x,y\\in X^{\\infty}$ are called {\\it tail-equivalent}, denoted by $x\\sim_\\infty y$, if there are $m,n\\geq 0$ such that $\\tau_{>m}(x)=\\tau_{>n}(y)$. This defines an equivalence relation on $X^{\\infty}$\n\n\n\\subsubsection{Basis paths $x$ having the property that $x^2$ is again a basis path}\nWe denote by $X^c$ the set of all finite basis paths $x$ having the property that $x^2$ is again a basis path. Clearly $x\\in X^c$ if and only if $x$ is a closed path in $E_d$ that consists either only of real edges or only of ghost edges. We call an $x\\in X^c$ \\textit{real} if it consists only of real edges, and \\textit{ghostly} if it consists only of ghost edges. Two basis paths $x=x_1\\dots x_m,y=y_1\\dots y_n\\in X^c$ are called \\textit{equivalent}, denoted by $x\\sim_c y$, if $x_1\\dots x_m=y_{k+1}\\dots y_{n}y_1\\dots y_{k}$ for some $1\\leq k\\leq n$ (note that this implies $m=n$). This defines an equivalence relation on $X^c$.\n\n\\subsection{The connected extended representation graphs containing a source}\n\nLet $v\\in E^0$. We define an extended representation graph $(F_v,\\phi_v)$ for $E$ by\n\\begin{align*}\nF_v^0=&\\{w_x\\mid x\\in X_v\\},\\\\\nF_v^1=&\\{f_x\\mid x\\in X_v\\setminus\\{v\\}\\},\\\\\ns_{F_v}(f_x)=&w_{\\tau_{\\leq |x|-1}(x)},\\\\\nr_{F_v}(f_x)=&w_x,\\\\\n\\phi^0_{v}(w_x)=&r_d(x),\\\\\n\\phi^1_{v}(f_x)=&\\tau_{>|x|-1}(x).\n\\end{align*}\nNote that $F_v$ is connected (since for any $x=x_1\\dots x_n\\in X_v\\setminus\\{v\\}$ there is a path from $w_v$ to $w_x$, namely $f_{x_1}f_{x_1x_2}\\dots f_{x_1\\dots x_n}$) and contains a unique source, namely $w_v$.\n\n\\begin{example}\nSuppose that $E$ is the graph\n\\begin{equation*}\n\\xymatrix{\n u \\ar@(ul,ur)^{d} \\ar@(dl,dr)_{e} \\ar[r]^{f} &v.\n}\n\\end{equation*}\nThen $(F_v,\\phi_v)$ is given by \n\\begin{equation*}\n\\xymatrix@=10pt{\n& & & & & & & & & &\\\\\n& & & &&u \\ar@{.>}[rr]_{e^*} \\ar@{.>}[urr]^{d^*} \\ar@{<-}[dll]_{d^*}& & &&& \\\\\n& v\\ar[rr]^{f^*}& &u & & && & & & \\\\\n& & & &&u\\ar@{.>}[rr]^{d^*} \\ar@{.>}[drr]_{e^*} \\ar@{<-}[ull]^{e^*}&&&&& \\\\\n& & & & & & & .& & &\\\\\n}\n\\end{equation*}\nNote that in this example $(F_v,\\phi_v)$ is a usual representation graph for $E$ as defined in Definition \\ref{defrg}. In general, $(F_v,\\phi_v)$ is a usual representation graph for $E$ if and only if $v$ is a sink in $E$.\n\\end{example}\n\n\\begin{example}\nThe extended representation graph in Example \\ref{ex1} is isomorphic to $(F_v,\\phi_v)$.\n\\end{example}\n\n\\begin{proposition}\\label{properg1a}\nLet $u,v\\in E^0$. Then $(F_u,\\phi_u)\\cong (F_v,\\phi_v)$ if and only if $u=v$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\alpha:(F_u,\\phi_u)\\to (F_v,\\phi_v)$ is an isomorphism of extended representation graphs. Let $w$ be the unique source in $F_u$ and $w'$ the unique source in $F_v$. Then $\\alpha(w)=w'$ (since a graph homomorphism maps a vertex which is not a source to a vertex which is not a source). It follows that $u=\\phi_u(w)=\\phi_v(\\alpha(w))=\\phi_v(w')=v$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemerg1}\nLet $(F,\\phi)$ be an extended representation graph for $E$, $w$ a source in $F$ and $p,q\\in F^{\\geq 0}_w$. Then $r(p)=r(q)$ if and only $p=q$. \n\\end{lemma}\n\\begin{proof}\nAssume that $r(p)=r(q)$. We have to show that this implies $p=q$. First suppose that the length of $p$ or $q$, say $p$, is zero. Then $p=w$. Since $w=r(p)=r(q)$ and $w$ is a source, it follows that the length of $q$ is also zero. Hence $p=q=w$.\n\nNow suppose that $p=f_1\\dots f_m,g_1\\dots g_n\\in F^{\\geq 1}_w$. Without loss of generality assume that $m\\geq n$. Since $r(p)=r(q)$ and any vertex in $F$ receives at most one edge, we obtain $f_m=g_n,\\dots, f_{m-n+1}=g_1$. Assume that $m>n$. Then $f_{m-n}$ is an edge in $F$ ending in $s(f_{m-n+1})=s(g_1)=w$, which contradicts the assumption that $w$ is a source. Hence $m=n$ and $p=q$.\n\\end{proof}\n\n\\begin{proposition}\\label{properg1}\nAny connected extended representation graph for $E$ that contains a source is isomorphic to one of the extended representation graphs $(F_v,\\phi_v)~(v\\in E^0)$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $(F,\\phi)$ is a connected extended representation graph for $E$ containing a source $w$ lying over a vertex $v\\in E^0$. By Lemma \\ref{lembase}(i) there is for any $x\\in X_v$ a unique path $p_x$ in $F^{\\geq 0}_{w}$ such that $\\phi(p_x)=x$. For any $x\\in X_v$ set $w_x:=r(p_x)$. For any $x\\in X_v\\setminus\\{v\\}$ let $f_x$ be the last edge of $p_x$. By Lemma \\ref{lemerg1} the vertices $w_x~(x\\in X_v)$ are pairwise distinct. It follows that the edges $f_x~(x\\in X_v\\setminus \\{v\\})$ are pairwise distinct (since $r(f_x)=r(p_x)=w_x$).\n\nLet $G$ be the connected subgraph of $F$ defined by $G^0=\\{w_x\\mid x\\in X_v\\}$ and $G^1=\\{f_x\\mid x\\in X_v\\setminus \\{v\\}\\}$. Then no vertex in $G^0$ receives an edge from $F^1\\setminus G^1$ (since $w_v=w$ is a source in $F$ and each $w_x$ where $x\\in X_v\\setminus \\{v\\}$ already receives the edge $f_x$). Now let $f\\in s_F^{-1}(w_x)$ where $x\\in X_v$. Then $p_xf\\in F^{\\geq 0}_w$. It follows from Lemma \\ref{lembase}(i) that $y:=\\phi(p_xf)\\in \\phi(F^{\\geq 0}_w)=X_v$. Clearly $p_y=p_xf$ and hence $f=f_y$. Hence no vertex in $G^0$ emits an edge from $F^1\\setminus G^1$. It follows that $G$ is a connected component of $F$. This implies that $F=G$ since $F$ is connected. Hence we have\n\\begin{align*}\nF^0=&\\{w_x\\mid x\\in X_v\\},\\\\\nF^1=&\\{f_x\\mid x\\in X_v\\setminus\\{v\\}\\},\\\\\ns(f_x)=&w_{\\tau_{\\leq |x|-1}(x)},\\\\\nr(f_x)=&w_x,\\\\\n\\phi(w_x)=&r_d(x),\\\\\n\\phi(f_x)=&\\tau_{>|x|-1}(x)\n\\end{align*}\nand thus $(F,\\phi)\\cong (F_v,\\phi_v)$.\n\\end{proof}\n\n\\subsection{The acyclic and connected extended representation graphs not containing a source}\nLet $x=\\dots x_3x_2x_1\\in X^{\\infty}$ denote an infinite basis path. For any $i\\in\\mathbb{N}$ we denote by $X_i$ the set of all finite basis paths $y=y_1\\dots y_n$ of length $\\geq 1$ such that $x_iy_1$ is a basis path and $y_1\\neq x_{i-1}$ if $i\\geq 2$. We define an extended representation graph $(F_{x},\\phi_{x})$ for $E$ by \n\\begin{align*}\nF_{x}^0&=\\{w_i\\mid i\\in \\mathbb{N}\\}\\sqcup \\{w_{i,y}\\mid i\\in \\mathbb{N}, y\\in X_i\\},\\\\\nF_{x}^1&=\\{f_i\\mid i\\in \\mathbb{N}\\}\\sqcup \\{f_{i,y}\\mid i\\in \\mathbb{N}, y\\in X_i\\},\\\\\ns_{F_{x}}(f_i)&=w_{i+1},\\\\\n\\quad r_{F_{x}}(f_i)&=w_{i},\\\\\ns_{F_{x}}(f_{i,y})&=\\begin{cases}w_{i},\\quad&\\text{if }|y|=1,\\\\w_{i,\\tau_{\\leq |y|-1}(y)},\\quad&\\text{if }|y|\\geq 2,\\end{cases}\\\\\nr_{F_{x}}(f_{i,y})&=w_{i,y},\\\\\n\\phi_{x}^0(w_i)&=r_d(x_i),\\\\ \n\\phi_{x}^0(w_{i,y})&=r_d(y),\\\\\n\\phi_{x}^1(f_i)&=x_i,\\\\\n\\phi_{x}^1(f_{i,y})&=\\tau_{>|y|-1}(y).\n\\end{align*}\nNote that $F_x$ is connected and does not contain a source. In order to see that $F_x$ is acyclic define a strict partial order $<$ on $F_x^0$ by $w_ij)$, $w_i}[rrrr]_{d^*}&&&&v\\ar[rrrr]_{d^*}&&&&v&&&&v\\ar@[green]@{<-}[llll]^{d^*}&&\\ar@[green]@{<.}[ll]^{d^*}.}\n\\end{equation*}\nNote that in this example $(F_{x},\\phi_{x})$ is a usual representation graph for $E$. In general, $(F_{x},\\phi_{x})$ is a usual representation graph for $E$ if and only if $x$ is ghostly.\n\\end{example}\n\n\\begin{example}\\label{ex4}\nSuppose that $E$ is the graph $\\xymatrix{ v\\ar@[yellow]@(dr,ur)_{e}\\ar@[green]@(dl,ul)^{d}}$ and choose as special edge $e=e^v$. Let $x$ be the real infinite basis path $\\dots ddd$. Then $(F_{x},\\phi_{x})$ is given by\n\\begin{equation*}\n\\xymatrix@R=0.7cm@C=0.7cm{\n&&&&&&&&&&&&&&\\\\\n&&&&&&&&v\\ar@[red]@{.>}[u]^{d^*}\\ar@[blue]@{.>}[ur]_{e^*}&&&&v\\ar@[red]@{.>}[u]^{d^*}\\ar@[blue]@{.>}[ur]_{e^*}&&\\\\\n&&&&&&&&&&\\ar@[yellow]@{<.}[dll]_e&&&&\\ar@[yellow]@{<.}[dll]_e\\\\\n&&&&&&&&v\\ar@[yellow]@{<-}[ddllll]_e\\ar@[red][uu]^{d^*}&&\\ar@[green]@{<.}[ll]^d&&v\\ar@[yellow]@{<-}[ddllll]_e\\ar@[red][uu]^{d^*}&&\\ar@[green]@{<.}[ll]^d\\\\\n&&&&&&&&&&&&&&\\ar@[yellow]@{<.}[dll]_e\\\\\n\\ar@{.>}[rrrr]_d&&&&v\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}\\ar[rrrr]_d&&&&v\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}&&&&v\\ar@[green]@{<-}[llll]^d\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}&&\\ar@[green]@{<.}[ll]^d.\\\\\n&&&&&&&&&&&&&&\\\\\n&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&v\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\\\\n&&&&&&&&&&&&&&\\\\\n&&&&&&&&&&&&&&}\n\\end{equation*}\n\\end{example}\n\n\\begin{proposition}\\label{properg2a}\nLet $x,z\\in X^\\infty$. Then $(F_x,\\phi_x)\\cong (F_z,\\phi_z)$ if and only if $x\\sim_\\infty z$.\n\\end{proposition}\n\\begin{proof}\nWe leave it to the reader to show that $x\\sim_\\infty z$ implies $(F_x,\\phi_x)\\cong (F_z,\\phi_z)$. In order to show the converse, suppose that $\\alpha:(F_x,\\phi_x)\\to (F_z,\\phi_z)$ is an isomorphism of extended representation graphs. We write the vertices and edges in $F_z$ as $w_i'$, $w'_{i,y}$, $f'_i$ and $f'_{i,y}$ to distinguish them from the vertices and edges in $F_x$. Clearly $\\alpha$ maps the infinite path $\\dots f_3f_2f_1$ in $F_x$ to the infinite path $\\dots \\alpha(f_3)\\alpha(f_2)\\alpha(f_1)$ in $F_z$. Because of the strict partial order on $F_z^0$ described right above Example \\ref{ex3}, there is an $i\\in\\mathbb{N}$ such that $r(\\alpha(f_i))=w'_j$ for some $j\\in\\mathbb{N}$. Since for any $k\\in\\mathbb{N}$, $f'_k$ is the only edge in $F_z$ ending in $w'_k$, it follows that $\\dots \\alpha(f_{i+2})\\alpha(f_{i+1})\\alpha(f_{i})=\\dots f'_{j+2}f'_{j+1}f'_{j}$. By applying $\\phi_z$ to the last equation we obtain $\\dots x_{i+2}x_{i+1}x_{i}=\\dots z_{j+2}z_{j+1}z_{j}$. Thus $x\\sim_\\infty z$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemerg2}\nLet $(F,\\phi)$ be an acyclic extended representation graph for $E$, $w\\in F^0$ a vertex and $p,q\\in F^{\\geq 0}_w$. Then $r(p)=r(q)$ if and only $p=q$. \n\\end{lemma}\n\\begin{proof}\nAssume that $r(p)=r(q)$. We have to show that this implies $p=q$. First suppose that the length of $p$ or $q$, say $p$, is zero. Then $p=w$. Since $s(q)=w=r(p)=r(q)$ and $F$ is acyclic, it follows that the length of $q$ is also zero. Hence $p=q=w$.\n\nNow suppose that $p=f_1\\dots f_m,g_1\\dots g_n\\in F^{\\geq 1}_w$. Without loss of generality assume that $m\\geq n$. Since $r(p)=r(q)$ and any vertex in $F$ receives at most one edge, we obtain $f_m=g_n,\\dots, f_{m-n+1}=g_1$. Assume that $m>n$. Then $f_1\\dots f_{m-n}$ is a path in $F$ starting in $w$ and ending in $s(f_{m-n+1})=s(g_1)=w$, which contradicts the assumption that $F$ is acyclic. Hence $m=n$ and $p=q$.\n\\end{proof}\n\n\\begin{proposition}\\label{properg2}\nAny nonempty, acyclic and connected extended representation graph for $E$ that does not contain a source is isomorphic to one of the extended representation graphs $(F_{x},\\phi_{x})~(x\\in X^{\\infty})$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $(F,\\phi)$ is a nonempty, acyclic and connected extended representation graph for $E$ that does not contain a source. Choose a vertex $w\\in F^0$. Since $F$ does not contain sources, there is a (unique by Condition (i) in Definition \\ref{deferg}) infinite path $p=\\dots f_3f_2f_1$ in $F$ ending in $w$. Since $F$ is acyclic, the vertices $w_i:=r(f_i)~(i\\in \\mathbb{N})$ are pairwise distinct. It follows that the edges $f_i~(i\\in\\mathbb{N})$ are pairwise distinct.\n\nClearly $x:=\\phi(p)\\in X^\\infty$ by Lemma \\ref{lembase}. Define the sets $X_i~(i\\in\\mathbb{N})$ as in the first paragraph of \\S 5.3. Let $i\\in\\mathbb{N}$. Then, by Lemma \\ref{lembase}, there is for any $y\\in X_i$ a unique path $p_{i,y}$ in $F^{\\geq 1}_{w_i}$ such that $\\phi(p_{i,y})=y$. For any $i\\in\\mathbb{N}$ and $y\\in X_i$ set $w_{i,y}:=r(p_{i,y})$ and let $f_{i,y}$ be the last edge of $p_{i,y}$. Suppose that $w_{i,y}=w_{j,z}$ for some $i,j\\in\\mathbb{N}$, $y\\in X_i$ and $z\\in X_j$. Assume that $i>j$. Then the paths $p_{i,y}$ and $f_{i-1}\\dots f_jp_{j,z}$ have the same source and range. It follows from Lemma \\ref{lemerg2} that $p_{i,y}=f_{i-1}\\dots f_jp_{j,z}$. By applying $\\phi$ to the last equation we obtain $y_1=x_{i-1}$, which contradicts the assumption that $y\\in X_i$. We have shown that $i>j$ is impossible. By symmetry $i|y|-1}(y)\n\\end{align*}\nand thus $(F,\\phi)\\cong (F_x,\\phi_x)$.\n\\end{proof}\n\n\\subsection{The cyclic and connected extended representation graphs}\nLet $x=x_1\\dots x_m\\in X^c$, i.e. $x$ is a closed path in $E_d$ that consists either only of real edges or only of ghost edges. For any $1\\leq i\\leq m$ we denote by $X_i$ the set of all finite basis paths $y=y_1\\dots y_n$ of length $\\geq 1$ such that $x_iy_1$ is a basis path and $y_1\\neq x_{i+1}$ (here we use the convention $x_{m+1}=x_1$). We define an extended representation graph $(F_x,\\phi_x)$ for $E$ by \n\\begin{align*}\nF_x^0&=\\{w_i\\mid 1\\leq i\\leq m\\}\\sqcup \\{w_{i,y}\\mid 1\\leq i\\leq m, y\\in X_i\\},\\\\\nF_{x}^1&=\\{f_i\\mid 1\\leq i\\leq m\\}\\sqcup \\{f_{i,y}\\mid 1\\leq i\\leq m, y\\in X_i\\},\\\\\ns_{F_{x}}(f_i)&=w_{i-1},\\\\\n\\quad r_{F_{x}}(f_i)&=w_{i},\\\\\ns_{F_{x}}(f_{i,y})&=\\begin{cases}w_{i},\\quad&\\text{if }|y|=1,\\\\w_{i,\\tau_{\\leq |y|-1}(y)},\\quad&\\text{if }|y|\\geq 2,\\end{cases}\\\\\nr_{F_{x}}(f_{i,y})&=w_{i,y},\\\\\n\\phi_{x}^0(w_i)&=r_d(x_i),\\\\ \n\\phi_{x}^0(w_{i,y})&=r_d(y),\\\\\n\\phi_{x}^1(f_i)&=x_i,\\\\\n\\phi_{x}^1(f_{i,y})&=\\tau_{>|y|-1}(y)\n\\end{align*}\nusing the convention $w_0=w_{m}$. Note that $F_x$ is cyclic (since $c=f_1\\dots f_m$ is a cycle), connected and does not contain a source. Define a strict partial order $<$ on $F_x^0$ by $w_i}[u]^{d^*}\\ar@[blue]@{.>}[ur]_{e^*}&&\\\\\n&&&&&&&&&&&\\ar@[yellow]@{<.}[dll]_e\\\\\n&&&&&&&&&\\bullet\\ar@[yellow]@{<-}[ddllll]_e\\ar@[red][uu]^{d^*}&&\\ar@[green]@{<.}[ll]^d\\\\\n&&&&&&&&&&&\\ar@[yellow]@{<.}[dll]_e\\\\\n&\\bullet\\ar@[yellow]@(dl,ul)^e\\ar@[red][dd]^{d^*}&&&&\\bullet\\ar@[green]@{<-}[llll]^d\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}&&&&\\bullet\\ar@[green]@{<-}[llll]^d\\ar@[red][dd]^{d^*}\\ar@[blue][ddll]_{e^*}&&\\ar@[green]@{<.}[ll]^d.\\\\\n&&&&&&&&&&&\\\\\n&\\bullet\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\bullet\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\bullet\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\bullet\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\bullet\\ar@[red]@{.>}[d]^{d^*}\\ar@[blue]@{.>}[dl]_{e^*}&&\\\\\n&&&&&&&&&&&\\\\\n&&&&&&&&&&&}\n\\end{equation*}\n\\end{example}\n\n\\begin{proposition}\\label{properg3a}\nLet $x,z\\in X^c$. Then $(F_x,\\phi_x)\\cong (F_z,\\phi_z)$ if and only if $x\\sim_c z$.\n\\end{proposition}\n\\begin{proof}\nWe leave it to the reader to show that $x\\sim_c z$ implies $(F_x,\\phi_x)\\cong (F_z,\\phi_z)$. In order to show the converse, suppose that $\\alpha:(F_x,\\phi_x)\\to (F_z,\\phi_z)$ is an isomorphism of extended representation graphs. Write $x=x_1\\dots x_m$ and $z=z_1\\dots z_n$. We write the edges in $F_z$ as $f'_i$ and $f'_{i,y}$ to distinguish them from the edges in $F_x$. Clearly $\\alpha$ maps the cycle $f_1\\dots f_m$ in $F_x$ to the cycle $\\alpha(f_1)\\dots \\alpha(f_m)$ in $F_z$. By the last sentence before Example \\ref{ex1.75} there is a $1\\leq k\\leq n$ such that $\\alpha(f_1)\\dots \\alpha(f_m)=f'_{k+1}\\dots f'_nf'_1\\dots f'_k$. By applying $\\phi_z$ to the last equation we obtain $x_1\\dots x_m=z_{k+1}\\dots z_nz_1\\dots z_k$. Thus $x\\sim_c z$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemerg3}\nLet $(F,\\phi)$ be an extended representation graph for $E$, $w\\in F^0$ a vertex and $p,q\\in F^{\\geq 1}_w$. If $r(p)=r(q)$, then either $p=q$, $cp=q$ or $p=cq$ where $c$ is some closed path in $F$. \n\\end{lemma}\n\\begin{proof}\nAssume that $r(p)=r(q)$. Write $p=f_1\\dots f_m$ and $q=g_1\\dots g_n\\in F^{\\geq 1}_w$. Assume that $m\\geq n$. Since $r(p)=r(q)$ and any vertex in $F$ receives at most one edge, we obtain $f_m=g_n,\\dots, f_{m-n+1}=g_1$. If $m=n$, then $p=q$. Assume now that $m>n$. Then $c:=f_1\\dots f_{m-n}$ is a path in $F$ starting in $w$ and ending in $s(f_{m-n+1})=s(g_1)=w$. Clearly $p=cq$. Similarly one can show that $cp=q$ for some closed path $c$ if $mj$ is also impossible. It follows that $i=j$. Hence the paths $p_{i,y}$ and $p_{j,z}$ have the same source and range. By Lemma \\ref{lemerg3} we have \n\\begin{align}\np_{i,y}&=p_{j,z} \\text{ or}\\\\\ncp_{i,y}&=p_{j,z} \\text{ or}\\\\\np_{i,y}&=cp_{j,z}\n\\end{align}\nwhere $c$ is a closed path at $w_i$. Clearly $c$ must be a power of the cycle $f_{i+1}\\dots f_i$. If (10) holds, then we obtain $x_{i+1}=z_1$ by applying $\\phi$. But that contradicts the assumption that $z\\in X_j=X_i$. If (11) holds, then we obtain $y_1=x_{i+1}$. But that contradicts the assumption that $y\\in X_i$. Hence (9) holds. By applying $\\phi$ to (9) we obtain $y=z$ as desired. We have shown that the vertices $w_{i,y}~(1\\leq i \\leq m,y\\in X_i)$ are pairwise distinct. Similarly one can show that the vertices $w_{i,y}~(1\\leq i \\leq m,y\\in X_i)$ are distinct from the vertices $w_i~(1\\leq i \\leq m)$. Since $r(f_i)=w_i$ and $r(f_{i,y})=w_{i,y}$, it follows that the edges $f_{i,y}~(1\\leq i \\leq m,y\\in X_i)$ are pairwise distinct and distinct from the edges $f_i~(1\\leq i \\leq m)$. \n\nLet $G$ be the connected subgraph of $F$ defined by $G^0=\\{w_i\\mid 1\\leq i \\leq m\\}\\sqcup \\{w_{i,y}\\mid 1\\leq i \\leq m, y\\in X_i\\}$ and $G^1=\\{f_i\\mid 1\\leq i \\leq m\\}\\sqcup \\{f_{i,y}\\mid 1\\leq i \\leq m, y\\in X_i\\}$. One checks easily that no vertex in $G^0$ emits or receives an edge from $F^1\\setminus G^1$ (cf. the proof of Proposition \\ref{properg1}). It follows that $G$ is a connected component of $F$. This implies that $F=G$ since $F$ is connected. Hence we have\n\\begin{align*}\nF^0&=\\{w_i\\mid 1\\leq i\\leq m\\}\\sqcup \\{w_{i,y}\\mid 1\\leq i\\leq m, y\\in X_i\\},\\\\\nF^1&=\\{f_i\\mid 1\\leq i\\leq m\\}\\sqcup \\{f_{i,y}\\mid 1\\leq i\\leq m, y\\in X_i\\},\\\\\ns(f_i)&=w_{i-1},\\\\\n\\quad r(f_i)&=w_{i},\\\\\ns(f_{i,y})&=\\begin{cases}w_{i},\\quad&\\text{if }|y|=1,\\\\w_{i,\\tau_{\\leq |y|-1}(y)},\\quad&\\text{if }|y|\\geq 2,\\end{cases}\\\\\nr(f_{i,y})&=w_{i,y},\\\\\n\\phi(w_i)&=r_d(x_i),\\\\ \n\\phi(w_{i,y})&=r_d(y),\\\\\n\\phi(f_i)&=x_i,\\\\\n\\phi(f_{i,y})&=\\tau_{>|y|-1}(y)\n\\end{align*}\nand thus $(F,\\phi)\\cong (F_x,\\phi_x)$.\n\\end{proof}\n\n\\subsection{Summary}\n\nThe theorem below follows from Propositions \\ref{properg1a}, \\ref{properg1}, \\ref{properg2a}, \\ref{properg2}, \\ref{properg3a} and \\ref{properg3}.\n\n\\begin{theorem}\\label{thmm1}\nLet $R$ (resp. $S$) be a complete set of representatives for the $\\sim_\\infty$-equivalence classes (resp. $\\sim_c$-equivalence classes). Then $\\{(F_{v},\\phi_{v}),(F_{x},\\phi_{x}),(F_{y},\\phi_{y})\\mid v\\in E^0, x\\in R, y\\in S\\}$ is a complete set of representatives for the isomorphism classes of the nonempty and connected extended representation graphs for $E$.\n\\end{theorem}\n\nWe call a $\\sim_\\infty$-equivalence class or $\\sim_c$-equivalence class \\textit{ghostly} if it contains a ghostly element (note that in this case any element of this class is ghostly). In view of Example \\ref{extrivial} we obtain the following classification of the usual representation graphs (which, in turn, yields a classification of the usual algebraic branching systems in view of \\S 3.3).\n\n\n\\begin{theorem}\\label{corm1}\nLet $R_{\\operatorname{ghost}}$ (resp. $S_{\\operatorname{ghost}}$) be a complete set of representatives for the ghostly $\\sim_\\infty$-equivalence classes (resp. ghostly $\\sim_c$-equivalence classes). Then $\\{(F_{v},\\phi_{v}),(F_{x},\\phi_{x}),(F_{y},\\phi_{y})\\mid v\\in E^0_{\\operatorname{sink}}, x\\in R_{\\operatorname{ghost}}, y\\in S_{\\operatorname{ghost}}\\}$ is a complete set of representatives for the isomorphism classes of the nonempty and connected representation graphs for $E$.\n\\end{theorem}\n\n\\section{A class of counter-examples to the converse of Schur's lemma}\n\n\\begin{theorem}\\label{thmschur}\nLet $x$ be a cycle in $E$ which consists only of special edges and has an exit. Then the $L(E)$-module $W:=W(F_x,\\phi_x)$ is not simple and $\\operatorname{End}_{L(E)}(W)\\cong K$.\n\\end{theorem}\n\\begin{proof}\nWrite $x=x_1\\dots x_m$ and choose an exit $e$ of $x$. Then $s(e)=s(x_i)$ and $e\\neq x_i$ for some $1\\leq i\\leq m$. Since $x_i$ is special, $e$ is not special. Clearly $e\\in X_{i-1}$ (recall that $X_{i-1}$ is the set of all basis paths $y=y_1\\dots y_n$ of length $\\geq 1$ such that $x_{i-1}y_1$ is a basis path and $y_1\\neq x_{i}$). Let $V$ be the $K$-span of the vertices $w_{i-1,y}~(y\\in X_{i-1}, y_1=e)$. One checks easily that $V$ is a submodule of $W$. Since $V$ contains $w_{i-1,e}$ but not $w_{i-1}$, it is a proper submodule of $W$. Hence $W$ is not simple.\n\nIt remains to show that $\\operatorname{End}_{L(E)}(W)\\cong K$. Let $\\theta\\in \\operatorname{End}_{L(E)}(W)$. Then \n\\begin{equation}\n\\theta(w_m)=\\sum_{1\\leq i\\leq m}k_iw_i+\\sum_{1\\leq i\\leq m,y\\in X_i}k_{i,y}w_{i,y}\n\\end{equation}\nfor some $k_i,k_{i,y}\\in K~(1\\leq i\\leq m, y\\in X_{i})$ of which only a finite number are nonzero. Clearly $\\theta(w_m)=\\theta(w_m.x)=\\theta(w_m).x$. Hence we obtain\n\\begin{equation}\n\\sum_{1\\leq i\\leq m}k_iw_i+\\sum_{1\\leq i\\leq m,y\\in X_i}k_{i,y}w_{i,y}=\\sum_{1\\leq i\\leq m}k_iw_i.x+\\sum_{1\\leq i\\leq m,y\\in X_i}k_{i,y}w_{i,y}.x.\n\\end{equation}\nClearly \n\\begin{equation}\nw_i.x=\\delta_{i,m}w_i\n\\end{equation}\nfor any $1\\leq i\\leq m$. On the other hand, we have for any $1\\leq i\\leq m$ and $y\\in X_i$\\newpage \n\\begin{equation}\nw_{i,y}.x=\\begin{cases}\nw_{i,px},\\quad &\\text{if }y=p\\text{ and }r(p)=s(x),\\\\\nw_{i,q^*},\\quad &\\text{if }y=q^*x^*,\\\\\nw_{i,pq^*},\\quad &\\text{if }y=pq^*x^*,\\\\\nw_{i,px_{k+1}\\dots x_m},\\quad &\\text{if }y=px_k^*\\dots x_1^*\\text{ for some }1\\leq k \\leq m,\\\\\n0,\\quad &\\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere $p,q\\in E^{\\geq 1}$. Note that $y_1\\neq x_k^*$ for any $1\\leq k\\leq m$, since $x_iy_1$ is a basis path, $x$ is a cycle and $x_i$ is special. It follows from (13), (14) and (15) that \n\\begin{equation}\nk_i=0 ~(1\\leq i\\leq m-1)\n\\end{equation}\nand \n\\begin{equation}\n\\sum_{y\\in X_i}k_{i,y}w_{i,y}=\\sum_{y\\in X_i}k_{i,y}w_{i,y}.x~(1\\leq i\\leq m).\n\\end{equation}\nFix a $1\\leq i \\leq m$ and set $Y_i:=\\{y\\in X_i\\mid k_{i,y}\\neq 0\\}$. Note that $Y_i$ is a finite set. It follows from equation (17) that\n\\begin{equation}\n\\sum_{y\\in Y_i}k_{i,y}w_{i,y}=\\sum_{y\\in Y_i}k_{i,y}w_{i,y}.x.\n\\end{equation}\nClearly $w_{i,y}.x\\neq 0$ for any $y\\in Y_i$, otherwise the right hand side of (18) would have less summands than the left hand side. Similarly $w_{i,y}.x\\neq w_{i,z}.x$ for any $y\\neq z\\in Y_i$. For any $n\\geq 0$ we denote by $Y_i^n$ the subset of $Y_i$ consisting of all $y$ which have precisely $n$ letters that are ghost edges (of course these must be the last $n$ letters). We show by induction that $Y_i^n=\\emptyset$ for any $n\\geq 0$.\\\\\n\\\\\n\\underline{$n=0$}: Assume that $Y^0_i$ is not the empty set. Then we can choose an element $y\\in Y^0_i$ with maximal length. Clearly $y=p$ for some $p\\in E^{\\geq 1}$ and hence $w_{i,y}.x=w_{i,px}$ by (15). Therefore $w_{i,px}$ has to appear with nonzero coefficient also on the left hand side of (18), i.e. $k_{i,px}\\neq 0$. It follows that $px\\in Y^0_i$, which is impossible since $y=p$ has maximal length among the elements of $Y^0_i$. Thus $Y^0_i=\\emptyset$.\\\\\n\\\\\n\\underline{$n\\to n+1$}: Assume that $Y^{0}_i=\\dots=Y^n_i=\\emptyset$ for some $n\\geq 0$. Suppose that $Y^{n+1}_i\\neq\\emptyset$. Choose a $y\\in Y^{n+1}_i$. By (15), $w_{i,y}.x=w_{i,z}$ for some $z\\in X_i$. Clearly $z$ has less letters which are ghost edges than $y$. By the induction assumption it follows that $z\\not\\in Y_i$ and hence $k_{i,z}=0$. But since $w_{i,y}.x=w_{i,z}$, the vertex $w_{i,z}$ has to appear with nonzero coefficient on the left hand side of (18), a contradiction. Thus $Y^{n+1}_i=\\emptyset$.\n\nWe have shown that $Y_i^n=\\emptyset$ for any $n\\geq 0$. Since $Y_i=\\bigcup_{n\\geq 0}Y_i^n$, it follows that $Y_i=\\emptyset$. Hence \n\\begin{equation}\nk_{i,y}=0~(1\\leq i\\leq m,y\\in X_i).\n\\end{equation} \nIt follows from (12), (16) and (19) that $\\theta(w_m)=k_m w_m$. Hence for any $\\theta\\in\\operatorname{End}_{L(E)}(W)$ there is a $k(\\theta)\\in K$ such that $\\theta(w_m)=k(\\theta) w_m$. Let $\\xi:\\operatorname{End}_{L(E)}(W)\\to K$ be the map defined by $\\xi(\\theta)=k(\\theta)$. One checks routinely that $\\xi$ is a ring isomorphism.\n\\end{proof}\n\n\\begin{remark}\nThe following example shows that the assumption in Theorem \\ref{thmschur} that the cycle $x$ has an exit is necessary. Suppose that $E$ is the graph $\\xymatrix{v\\ar@[yellow]@(dr,ur)_{e}}$ and $x=e$. Then $(F_x,\\phi_x)$ is given by $\\xymatrix{v\\ar@[yellow]@(dr,ur)_{e}}$. Clearly the module $W(F_x,\\phi_x)$ is simple.\n\\end{remark}\n\nSuppose $x$ and $x'$ are cycles in $E$ which consist only of special edges and have an exit. If $x\\sim_c x'$, then $(F_x,\\phi_x)\\cong(F_{x'},\\phi_{x'})$ by Proposition \\ref{properg3a}. It follows that the modules $W(F_x,\\phi_x)$ and $W(F_{x'},\\phi_{x'})$ are isomorphic since $W$ is a functor. One can ask if $W(F_x,\\phi_x)$ and $W(F_{x'},\\phi_{x'})$ can be isomorphic if $x\\not\\sim_c x'$. The theorem below gives a partial answer to this question. \n\n\n\\begin{theorem}\\label{thmschur2}\nLet $x$ and $x'$ be cycles in $E$ which do not have a common vertex. Then the $L(E)$-modules $W(F_x,\\phi_x)$ and $W(F_{x'},\\phi_{x'})$ are not isomorphic.\n\\end{theorem}\n\\begin{proof}\nSuppose that there is an isomorphism $\\theta:W(F_{x'},\\phi_{x'})\\to W(F_x,\\phi_x)$. Write $x=x_1\\dots x_m$ and $x'=x'_1\\dots x'_{m'}$. Moreover, write the vertices in $F_{x'}$ as $w'_i$ and $w'_{i,y}$ to distinguish them from the vertices in $F_x$. Clearly \n\\begin{equation}\n\\theta(w'_{m'})=\\sum_{1\\leq i\\leq m}k_iw_i+\\sum_{1\\leq i\\leq m,y\\in X_i}k_{i,y}w_{i,y}\n\\end{equation}\nfor some $k_i,k_{i,y}\\in K~(1\\leq i\\leq m, y\\in X_{i})$ of which only a finite number are nonzero. Since $\\theta(w'_{m'})=\\theta(w'_{m'}.x')=\\theta(w'_{m'}).x'$, it follows that\n\\begin{equation}\n\\sum_{1\\leq i\\leq m}k_iw_i+\\sum_{1\\leq i\\leq m,y\\in X_i}k_{i,y}w_{i,y}=\\sum_{1\\leq i\\leq m}k_iw_i.x'+\\sum_{1\\leq i\\leq m,y\\in X_i}k_{i,y}w_{i,y}.x'.\n\\end{equation}\nClearly \n\\begin{equation}\nw_i.x'=0\n\\end{equation}\nfor any $1\\leq i\\leq m$ since $x$ and $x'$ have no common vertex. On the other hand, we have for any $1\\leq i\\leq m$ and $y\\in X_i$\n\\begin{equation}\nw_{i,y}.x'=\\begin{cases}\nw_{i,px'},\\quad &\\text{if }y=p\\text{ and }r(p)=s(x'),\\\\\nw_{i,q^*},\\quad &\\text{if }y=q^*(x')^*,\\\\\nw_{i,pq^*},\\quad &\\text{if }y=pq^*(x')^*,\\\\\nw_{i,px'_{k+1}\\dots x'_{m'}},\\quad &\\text{if }y=p(x'_k)^*\\dots (x'_1)^*\\text{ for some }1\\leq k \\leq m',\\\\\n0,\\quad &\\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere $p,q\\in E^{\\geq 1}$. Note that $y_1\\neq (x'_k)^*$ for any $1\\leq k\\leq m'$, since $x_iy_1$ is a basis path and $x$ and $x'$ do not have a common vertex. It follows from (21), (22) and (23) that\n\\begin{equation}\nk_i=0 ~(1\\leq i\\leq m)\n\\end{equation}\nand \n\\begin{equation}\n\\sum_{y\\in X_i}k_{i,y}w_{i,y}=\\sum_{y\\in X_i}k_{i,y}w_{i,y}.x'~(1\\leq i\\leq m).\n\\end{equation}\nOne can deduce from (25) that\n\\begin{equation}\nk_{i,y}=0~(1\\leq i\\leq m,y\\in X_i).\n\\end{equation} \nThe proof is essentially the same as the proof that (17) implies (19) (just replace $x$ by $x'$). It follows from (20), (24) and (26) that $\\theta(w'_{m'})=0$, a contradiction. Thus $W(F_x,\\phi_x)$ and $W(F_{x'},\\phi_{x})$ are not isomorphic.\n\\end{proof}\n\n\\section*{Acknowledgments}\nI would like to thank Daniel Gon\\c{c}alves for encouraging me to write this paper. I would also like to thank Tran Giang Nam, who asked me if I know any modules over Leavitt path algebras that are counter-examples to the converse of Schur's lemma. This question led to Section 6.\n\n\\bibliographystyle{alpha} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction.} Let $(R,\\mathfrak{n})$ be a regular local ring and $k=R\/\\mathfrak{n}$ its residue\nfield which we assume of characteristic zero.\n\nGiven an ideal $I\\subseteq \\mathfrak{n}^2$, a classical problem in Commutative\nAlgebra is to study the Poincar{\\'e} series\n$$\\mathbb{P}_A(z):=\\sum_{i\\ge 0}\\dim_kTor_i^A(k,k))z^i$$ of the\nlocal ring $(A=R\/I,\\mathfrak{m}=\\mathfrak{n}\/I).$ This is the generating function of\nthe sequence of Betti numbers of a minimal free resolution of $k$\nover $A.$\n\nDue to the classical conjecture of Serre, the main issue is concerning the rationality of this series.\nWe know by the example of Anick, see \\cite{A}, that this series can be non rational,\nbut there are relatively few classes of local rings for which the question has been settled.\nSee \\cite{Av} for a detailed study of these and other relevant related problems in local algebra.\n\n Given a Cohen-Macaulay local ring $A=R\/I,$ we say that $A$ is {\\bf stretched} if there exists\n an artinian reduction $B$ of $A$ such that the square of its maximal ideal is a principal ideal.\n Instead, if the square of the maximal ideal of an artinian reduction is minimally generated by\n two elements, we say that $A$ is {\\bf almost stretched}. See \\cite{S1}, \\cite{RV}, \\cite{EV1}\n and \\cite{EV2} for papers concerning these notions.\n\n In \\cite{S2} J. Sally computed the Poincar{\\'e} series of a stretched Cohen-Macaulay local\n rings and obtained, as a corollary, the rationality of the series. It follows that local\n Gorenstein rings of multiplicity at most five have rational Poincar{\\'e} series.\n\n In this paper we compute the Poincar{\\'e} series of an almost stretched Gorenstein local\n ring, thus exhibiting its rationality. Using this result, we can prove the rationality of\n the Poincar{\\'e} series of any Gorenstein local rings of multiplicity at most seven.\n\n We are not developing new methods for the computation of the Betti numbers of the minimal\n $A$-free resolution of $k;$ rather we show that the structure theorem we proved in \\cite{EV1}\n for Artinian almost stretched Gorenstein local rings is very much suitable to the computation of $Tor_i^A(k,k).$\n\n In the following, for a local ring $(A,\\mathfrak{m},k:=A\/\\mathfrak{m})$ of dimension $d,$ we denote by $h$ the\n embedding codimension of $A$, namely the integer $h:=\\dim_k(\\mathfrak{m}\/\\mathfrak{m}^2)-d.$ Recall, see \\cite{Ab},\n that the multiplicity $e$ of a Cohen-Macaulay local ring $A$ of embedding codimension $h$\n satisfies the inequality $e\\ge h+1.$ Further, in the extremal case $e=h+1,$ it is well known that\n $\\mathbb{P}_A(z)$ is rational.\n\n The main result of this paper is the following theorem.\n\n \\begin{theorem} Let $A=R\/I$ be an almost stretched Gorenstein local ring of dimension\n $d$ and embedding codimension $h.$ Then $$\\mathbb{P}_A(z)=\\frac{(1+z)^d}{1-hz+z^2}.$$\n \\end{theorem}\n\n\n\n\n\n\n \\bigskip\n \\section{Proof of the Theorem}\n\n The main ingredient of the proof of our result are the following classical\n ``change of rings\" theorems.\n The first one, see \\cite{T}, relates the Betti numbers of $A$ with those of\n $A\/xA$ when $x$ is a non-zero divisor in the local ring $A.$\n\n\\vskip 2mm\na) Let $x$ be a non-zero divisor in $A.$ Then\n\n $$\\mathbb{P}_A(z)=\\begin{cases}\n (1+z)\\mathbb{P}_{A\/xA}(z) & \\ \\ x\\in \\mathfrak{m}\\setminus \\mathfrak{m}^2\\\\\n (1-z^2)\\mathbb{P}_{A\/xA}(z) & \\ \\ x\\in \\mathfrak{m}^2 .\n\\end{cases}$$\n\n\n\\vskip 2mm The second one, see \\cite{GL}, relates the Betti numbers of $A$ with\nthose of $A\/xA$ when $x$ is a socle element.\n\\vskip 2mm\nb) Let $x\\in \\mathfrak{m}\\setminus \\mathfrak{m}^2$ be an element in the socle $(0:_A\\mathfrak{m})$ of $A.$\nThen $$\\mathbb{P}_A(z)=\\frac{\\mathbb{P}_{A\/xA}(z)}{1-z \\ \\mathbb{P}_{A\/xA}(z)}.$$\n\n\\vskip 2mm The third one, see \\cite{AL}, relates the Betti numbers of the\nArtinian Gorenstein local ring $A$ with those of $A$ modulo the socle.\n\\vskip 2mm\nc) If $(A,\\mathfrak{m})$ is an Artinian local Gorenstein ring, then\n$$\\mathbb{P}_A(z)=\\frac{\\mathbb{P}_{A\/(0:\\mathfrak{m})}(z)}{1+z^2 \\ \\mathbb{P}_{A\/(0:\\mathfrak{m})}(z)}.$$\n\nWe start now proving the Theorem. Let $J:=(a_1,\\dots,a_d)$ be the ideal generated\nby a minimal reduction of $\\mathfrak{m},$ such that $A\/J$ is almost stretched and Gorenstein.\n Since $\\{a_1,\\dots,a_d\\}$ is a regular sequence on $A,$ we have by a)\n $$\\mathbb{P}_A(z)=(1+z)^d \\ \\mathbb{P}_{A\/J}(z).$$\n\nHence we may assume that $(A=R\/I,\\mathfrak{m}=\\mathfrak{n}\/I)$ is an Artinian almost stretched\nGorenstein local ring of embedding dimension $h.$\nIn this case we proved in \\cite{EV1}, Proposition 4.8, that we can find\nintegers $s\\ge t+1\\ge 3$ depending on the Hilbert function of $A$, a minimal\nsystem of generators $\\{x_1,\\dots,x_h\\}$ of the maximal ideal $\\mathfrak{n}$ of $R$ and an\nelement $a\\in R$ such that $I$ is generated by the elements:\n$$ \\{x_1x_j\\}_{j=3,\\dots,h}\\ \\ \\{x_ix_j\\}_{2\\le i n$, e.g., more taxa than data points.\nThis overparameterization calls for regularization.\nEnforcing sparsity via an $\\ell_1$ penalty on the parameters $\\beta$, Lasso regression is among the most common regularization techniques for such settings.\nThe problem with enforcing sparsity in a ``linear-in-$\\ilr{}$'' model is that a zero entry in $\\beta$ does not correspond directly to a zero effect of the relative abundance of any single taxon.\nThis motivates \\emph{log-contrast} estimation \\citep{Aitchison1984} for the $p > n$ setting with a sparsity penalty \\citep{Lin2014c,Combettes2019Logcontrast}\n\\begin{equation}\\label{eq:logcontrast}\n \\min_{\\beta} \\sum_{i=1}^n \\|y_i - \\beta^T \\log(x_i) \\|_2^2 + \\lambda \\| \\beta \\|_1 \\quad \\text{subject to}\\: \\sum_{i=1}^p \\beta_i = 0 \\:.\n\\end{equation}\nThis estimation respects the compositional nature of $x$ while retaining the association between the entry $\\beta_i$ and the relative abundance of the individual taxon $x_i$.\nDue to the additional constraint, individual components of $\\beta$ are still not entirely disentangled.\n\n\\xhdr{Microbiome data}\nFor concreteness, we focus on microbiome measurements as a running example, but our analysis applies more broadly to any type of high-dimensional compositional data.\nTypical bacterial microbiome measurements consist of counts of certain taxa (e.g., on the species, genus, or family level) derived from high throughput amplicon sequencing of 16S ribosomal RNA (rRNA) \\citep{johnson2019evaluation}.\nFor a total of $p$ possible taxa, we denote the relative abundances by $x \\in \\ensuremath \\mathbb{S}^{p-1}$.\nIn practice, there are many more possible taxa in the gut microbiome (up to tens of thousands) than occur in any given sample, leading to a high degree of sparsity and typically $p > n$.\nWe have already introduced multiple log-based transformations of relative abundances, which are undefined for zero entries.\nA simple and widely used strategy to avoid zero entries is to add a small constant to all absolute counts, so called \\emph{pseudo-counts}, which we also use in this work.\nThe additive constant is typically chosen in an ad-hoc fashion, for example $0.5$ \\citep{kaul2017analysis,lin2020analysis}.\n\n\\xhdr{Diversity}\nOne of the key measures to describe microbial populations that circumvents interpretability issues around compositions is \\emph{diversity}.\nDiversity is a scalar summary statistic measuring the variation of taxa within a given composition and is in this context often called $\\alpha$-diversity.\nThere is no unique definition of $\\alpha$-diversity.\nAmong the most common measures in the literature are\\footnote{We write $\\|x \\|_{0}$ for the number of non-zero entries of $x$, which is not a norm.}\n\\begin{equation*}\n\\text{richness}\\; \\|x\\|_{0}, \\quad\n \\text{Shannon diversity}\\; -\\sum_{j=1}^p x_j \\log(x_j), \\quad\n \\text{and Simpson diversity}\\; -\\sum_{j=1}^p x_j^2 .\n\\end{equation*}\nBeyond these, there exist entire families of diversity measures taking into account species, functional, or phylogenetic similarities between taxa and tracing out continuous parametric profiles for varying sensitivity to highly-abundant taxa.\nSee for example \\citep{Cobbold2012,chao2014unifying,daly2018ecological} for an overview of the possibilities and choices of estimating $\\alpha$-diversity in a specific application.\nGiven the popularity of $\\alpha$-diversity for assessing the impact and health of microbial compositions\n\\citep{Bello2018}, it is natural to formulate our causal query in terms of $\\alpha$-diversity.\nWe will return to this problem in section~\\ref{sec:diversity}.\n\n\\begin{wrapfigure}{r}{5cm}\n \\vspace{-1cm}\n \\centering\n \\begin{tikzpicture}\n \\node[obs,label={[align=center]below:{$\\ensuremath \\mathbb{R}^q$ \\\\{\\scriptsize \\color{gray}STAT}}}] (Z) at (0, 0) {$Z$};\n \\node[obs,label={[align=center]below:{$\\ensuremath \\mathbb{R}^p \\to \\ensuremath \\mathbb{S}^{p-1}$ \\\\{\\scriptsize \\color{gray} diversity $\\to$ microbiome}}}] (X) at (2, 0) {$X$};\n \\node[obs,label={[align=center]below:{$\\ensuremath \\mathbb{R}$ \\\\\\scriptsize {\\color{gray}bodyweight}}}] (Y) at (4, 0) {$Y$};\n \\node[latent,label={\\scriptsize {\\color{gray}unobserved confounder}}] (U) at (3, 1.3) {$U$};\n \\edge[Orange]{Z}{X}\n \\edge[MidnightBlue]{X}{Y}\n \\edge{U}{X,Y}\n \\node[right,align=left] (desc) at (-0.4, -1.8) {\n {\\color{Orange} \\hspace{0.8mm}1${}^{\\text{st}}$: OLS $\\to$ Compositional model} \\\\\n {\\color{MidnightBlue} 2${}^{\\text{nd}}$: OLS $\\to$ Log-contrast}\n };\n \\end{tikzpicture}%\n \\caption{Estimation of the direct causal effect $X \\to Y$ via an instrumental variable $Z$ for compositional $X$ with $X$ being represented by a summary statistic or by the entire composition.}\n \\label{fig:setup}\n\\end{wrapfigure}\n\n\\subsection{Instrumental variables}\n\nWe briefly recap the assumptions of the instrumental variable setting as depicted in Figure~\\ref{fig:setup}.\nFor an outcome $Y$, a cause $X$, and potential unobserved confounders $U$, we assume access to a discrete or continuous instrument $Z \\in \\ensuremath \\mathbb{R}^q$ satisfying\n(i) $Z \\indep U$ (the confounder is independent of the instrument),\n(ii) $Z \\dep X$ (``the instrument influences the cause''), and\n(iii) $Z \\indep Y \\,|\\, \\{X, U\\}$ (``the instrument influences the outcome only through the cause'').\nOur goal is to estimate the direct causal effect of $X$ on $Y$, written as $\\E[Y\\,|\\, do(x)]$ in the do-calculus notation \\citep{pearl2009causality} or as $\\E[Y(x)]$ in the potential outcome framework \\citep{imbens2015causal}, where $Y(x)$ denotes the potential outcome of $Y$ for the treatment value $x$.\nThe functional dependencies are $X = g(Z,U)$, $Y = f(X,U)$.\nWhile $Z, X, Y$ denote random variables, we will also consider a dataset of $n$ i.i.d.\\ samples $\\ensuremath \\mathcal{D} = \\{(z_i, x_i, y_i)\\}_{i=1}^n$ from their joint distribution.\nWe collect these datapoints into matrices\/vectors $\\ensuremath \\B{X} \\in \\ensuremath \\mathbb{R}^{n\\times p}$ or $\\ensuremath \\B{X} \\in (\\ensuremath \\mathbb{S}^{p-1})^n$, $\\ensuremath \\B{Z} \\in \\ensuremath \\mathbb{R}^{n \\times q}$, $\\ensuremath \\B{y} \\in \\ensuremath \\mathbb{R}^n$.\n\nWithout further restrictions on $f$ and $g$, the causal effect is not identified \\citep{pearl1995testability,bonet2001instrumentality,gunsilius2018testability}.\nThe most common assumption leading to identification is that of \\emph{additive noise}, namely $Y = f(X) + U$ with $\\E[U] = 0$ and $X \\dep U$.\nHere, we overload the symbols $f$ and $g$ for simplicity.\nThe implied Fredholm integral equation of first kind $\\E[Y \\,|\\, Z] = \\int f(x)\\, \\mathrm{d}P(X\\,|\\, Z)$ is generally ill-posed.\nUnder certain regularity conditions it can be solved consistently even for non-linear $f$, see e.g., \\citep{newey2003instrumental,blundell2007semi} and more recently \\citep{singh2019kernel,muandet2019dual,zhang2020maximum}.\n\n\\xhdr{Linear case}\nWhen $\\ensuremath \\mathcal{X}=\\ensuremath \\mathbb{R}^p$ and $f, g$ are linear, the standard \\emph{instrumental variable estimator} is\n\\begin{equation}\\label{eq:betaiv}\n \\ensuremath \\hat{\\beta}_{\\mathrm{iv}} = (\\ensuremath \\B{X}^T \\B{P}_{Z} \\ensuremath \\B{X})^{-1} \\ensuremath \\B{X}^T \\B{P}_{Z}\\, \\ensuremath \\B{y} \\quad \\text{with} \\quad \\B{P}_{Z} = \\ensuremath \\B{Z} (\\ensuremath \\B{Z}^T \\ensuremath \\B{Z})^{-1} \\ensuremath \\B{Z}^T \\:.\n\\end{equation}\nFor the \\emph{just-identified} case $q=p$ as well as the over-identified case $q > p$, this estimator is consistent and asymptotically unbiased, albeit not unbiased.\nIn the \\emph{under-identified} case $q < p$, where there are fewer instruments than treatments, the orthogonality of $Z$ and $U$ does not determine a unique solution.\nThe estimator $\\ensuremath \\hat{\\beta}_{\\mathrm{iv}}$ can also be interpreted as the outcome of a \\emph{two-stage least squares} (2SLS) procedure consisting of\n(1) regressing $\\ensuremath \\B{X}$ on $\\ensuremath \\B{Z}$ via OLS $\\hat{\\delta} = (\\ensuremath \\B{Z}^T \\ensuremath \\B{Z})^{-1} \\ensuremath \\B{Z}^T \\ensuremath \\B{X}$, and\n(2) regressing $\\ensuremath \\B{y}$ on the predicted values $\\hat{\\ensuremath \\B{X}} = \\ensuremath \\B{Z} \\hat{\\delta}$ via OLS, again resulting in $\\ensuremath \\hat{\\beta}_{\\mathrm{iv}}$.\nPractitioners are typically discouraged from using the manual two-stage approach in practice, because the OLS standard errors of the second stage are wrong---a correction is needed \\citep{angrist2008mostly}.\n\nMoreover, the two-stage description suggests that we can treat the two stages as independent and thereby seems to invite us to mix and match different regression methods as we see fit.\n\\citet{angrist2008mostly} highlight that the asymptotic properties of $\\ensuremath \\hat{\\beta}_{\\mathrm{iv}}$ rely on the fact that for OLS the residuals of the first stage are uncorrelated with $\\ensuremath \\hat{\\beta}_{\\mathrm{iv}}$ and the instruments $\\ensuremath \\B{Z}$.\nHence, for OLS we achieve consistency \\emph{even when the first stage is mis-specified}.\nFor a non-linear first stage regression we may only hope to achieve uncorrelated residuals asymptotically when the model is correctly specified.\nReplacing the OLS first stage with a non-linear model is known as the ``forbidden regression'', a term commonly attributed to Prof.~Jerry Hausmann.\nAngrist and Pischke acknowledge that the practical relevance of the forbidden regression is not well understood.\nStarting with \\citet{kelejian1971two} there is now a rich literature on the circumstances under which ``manual 2SLS'' with non-linear first (and\/or second) stage can yield consistent causal estimators.\nPrimarily interested in high-dimensional, compositional $X$, we cannot directly use OLS for either stage.\nHence we pay great attention to potential issues due to the ``forbidden regression'' and mis-specification in our proposed methods.\nBecause we aim for interpretable causal effect estimates, where we want to control the second stage $X \\to Y$ regression, we still concentrate on two-stage methods despite their potential drawbacks.\n\n\\begin{figure}\n \\includegraphics[width=.6\\linewidth]{figs\/Legend.pdf}\n \\centering\n \\includegraphics[width=0.42\\linewidth]{figs\/ShannonDiversity_withoutlegend.pdf}\n \\hfill\n \\includegraphics[width=0.42\\linewidth]{figs\/SimpsonDiversity_withoutLegend.pdf}\n \\caption{Effect estimates of gut microbiome diversity on body weight using IV methods with different sets of assumptions for Shannon diversity (left) and Simpson diversity (right).\n All methods are broadly in agreement for each diversity measure separately.\n However, the effects have opposite sign for Shannon and Simpson diversity leading to inconclusive overall results.}\n \\label{fig:diversityresults}\n\\end{figure}\n\n\n\\section{Summary statistics as the causal driver}\n\\label{sec:diversity}\n\nOne way to avoid the intricacies of compositional data in causal estimation is to consider interpretable scalar summary statistics of the data \\emph{a priori}.\nFor instance, in microbial ecology and microbiome research, the most prevalent summary statistics are measures of species diversity, see section~\\ref{sec:compdata}.\nThis suggests to reduce the estimation with a compositional cause to asking ``what is the direct causal effect of \\emph{the diversity} of a composition $X$ on the outcome $Y$?''.\nIn fact, the wording in popular books and research articles alike seems to suggest that (bio-)diversity is indeed an important \\emph{causal driver} of ecosystem functioning and human health \\citep{Chapin2000,Blaser2014}.\nSince $\\alpha$-diversity is described by a real scalar, we are in the well-studied, just- or over-identified instrumental variable setting.\nLet us now attempt an interpretation of $\\ensuremath \\hat{\\beta}_{\\mathrm{iv}}$ in this scenario.\nOur estimate for the expected value of the outcome $Y$ under an intervention on the diversity $\\alpha$, denoted by $\\E[Y \\,|\\, do(\\alpha)]$, is then simply $\\ensuremath \\hat{\\beta}_{\\mathrm{iv}} \\alpha$ (up to an intercept).\nCritically, we cannot directly reason about the effect of changing the relative abundance of an individual species, but only about the hypothetical scenario in which we ``externally set the diversity to a certain value holding all host and environmental factors constant''.\nSince there is a non-trivial subspace of $\\ensuremath \\mathbb{S}^{p-1}$ with constant diversity, there are generally infinitely many different instantiations of such an intervention.\nWe are forced to conclude that the inferred effect on the outcome is the same for each possible composition of a given diversity.\n\\emph{Hence, the causal effect of diversity $\\E[Y \\,|\\, do(\\alpha)]$ is ambiguous and hard to interpret in practice.}\n\nIn addition to these issues with viewing diversity as a causal driver, concerns have been raised about the ambiguity in measuring $\\alpha$-diversity in the first place \\citep{Willis2019,shade2017diversity,gloor2017microbiome}.\nIndeed, Figure~\\ref{fig:diversityresults} shows that on a real dataset different ways of computing diversity may lead to opposite causal effects.\nWe now describe the experimental setup as well as the cause-effect estimation methods used before returning to these results.\n\n\\xhdr{Real data}\nWe found the dataset described by \\citet{Schulfer2019} to be a good fit for our setting.\nA total of 57 new born mice were assigned randomly to a sub-therapeutic antibiotic treatment (STAT) during their early stages of development.\nAfter 21 days, the microbiome composition and bodyweight of each mouse was recorded.\nWe are interested in the causal effect of the microbiome composition $X \\in \\ensuremath \\mathbb{S}^{p-1}$ on the bodyweight $Y \\in \\ensuremath \\mathbb{R}$.\nDue to the random assignment of the antibiotic treatment, it is independent of potential confounders such as genetic factors.\nThe sub-therapeutic dose implies that antibiotics can not be detected in the mice' blood, providing reason to assume no effect of the antibiotics on the weight other than through its effect on the gut microbiome, i.e., $Z \\indep Y \\,|\\, \\{U, X\\}$.\\footnote{We remark that this work is focused on methods rather than novel biological insights.\nWe do not claim robust scientific insights for this dataset, as more scrutiny of the IV assumptions would be necessary.\nEven if valid, sub-therapeutic antibiotic treatment serves only as a weak instrument in the real data, potentially causing large bias in the IV estimates especially given the small sample size \\citep{andrews2019weak}.} \nFinally, we observe empirically, that there are statistically significant differences of microbiome compositions between the treatment and control groups.\nThus, the sub-therapeutic antibiotic treatment is a good candidate for an instrument $Z \\in \\{0, 1\\}$ in estimating the effect $X \\to Y$.\n\n\\xhdr{Methods for one-dimensional causes}\nWe compare the following methods with gradually weakened assumptions to ensure the validity of our cause-effect estimates (see Appendix~\\ref{app:ivmethods} for details):\n\\begin{enumerate}[leftmargin=*,topsep=0pt,itemsep=0pt]\n \\item \\textbf{2SLS}: The standard estimator from eq.~\\eqref{eq:betaiv}.\n \\item \\textbf{KIV} (Kernel Instrumental Variables): \\citet{singh2019kernel} relax the linearity assumption in 2SLS by allowing for non-linear $f$ in $Y = f(X) + U$, while still maintaining the additive noise assumption.\n By replacing both stages with kernel ridge regression they consistently estimate non-linear $f$ in closed form as a linear combination of functions in the reproducing kernel Hilbert space.\n \\item \\textbf{GB}: \\citet{kilbertus2020class} further relax the additive noise assumption allowing for general non-linear effects $Y = f(X, U)$.\n Under mild assumptions on the function space for $f$, the causal effect is \\emph{partially} identifiable, and GB produces lower and upper bounds for $\\E[Y\\,|\\, do(x)]$.\n\\end{enumerate}\n\nFigure~\\ref{fig:diversityresults} shows the results of all three methods (including the naive single-stage OLS regression $X \\to Y$) on the semi-synthetic dataset using Shannon diversity (left) and Simpson diversity (right) as the $\\alpha$-diversity measure.\nAll three methods broadly agree for each diversity measure separately, supporting our confidence in the overall trend.\nHowever, the sign of the effect reverses when switching from one diversity measure to the other.\nDepending on which measure we choose, ``diversity'' can have \\emph{opposing} causal effects.\nIt is not clear how to reconcile the findings that Shannon diversity has a positive causal effect on bodyweight, while Simpson diversity affects bodyweight negatively with a coherent notion of diversity as a meaningful causal driver.\n\nTo summarize, we have identified two main obstacles in assigning causal powers to summary statistics:\nfirst, there is no clear conceptualization of external interventions, mostly due to the `many-to-one' nature posing a severe obstacle for causal interpretation.\nSecond, we may risk observing inconsistent causal effects depending on the specifics of the summary statistic such as weighting schemes. \n\n\n\\section{Compositional data as the causal driver}\n\\label{sec:composition}\n\nTaken together, our findings challenge the common portrayal of diversity as a decisive (rather than merely descriptive) summary of compositions, and suggest that causal effects should be estimated from the composition directly to establish an \\emph{interpretable} link between $X$ and $Y$.\nSuch a connection naturally allows us to reason about interventions on the relative abundance of individual taxa.\n\n\\subsection{Methods for compositional causes}\n\\label{sec:methods}\n\nWe now describe methods to estimate causal effects of compositional data using combinations of two approaches:\n(a) standard (non)-linear regression techniques (OLS, kernel ridge regression) for the first and\/or second stage on log-transformed compositions such as $\\ilr(X)$ or $\\alr(X)$, and (b) composition specific regressions for the first stage (Dirichlet regression) and the second stage (log-contrast regression).\nThe former are limited in terms of interpretation in higher dimensions, when we additionally seek to enforce sparsity.\nFor the latter, a non-linear first stage may introduce bias due to the ``forbidden regression'' problem.\nWe assess and compare the following methods.\n\n\\xhdr{ILR+ILR}\nStandard 2SLS (OLS for both stages) with $\\ilr(X) \\in \\ensuremath \\mathbb{R}^{p-1}$ as the cause;\nsince OLS minima do not depend on the chosen basis, parameter estimates for different log-transformations of $X$ are related via fixed linear transformations.\nHence, as long as no sparsity penalty is added, $\\ilr$ and $\\alr$ regression yield equivalent results.\nWe denote estimated parameters using $\\ilr$ or $\\alr$ by $\\hat{\\beta}_{\\ilr} \\in \\ensuremath \\mathbb{R}^{p-1}$ and $\\hat{\\beta}_{\\alr} \\in \\ensuremath \\mathbb{R}^p$ respectively.\nThe isometric $\\ilr$ coordinates are particularly useful due to the consistency guarantees of 2SLS given that $ZX$ has full rank.\nHowever, for interpretation moving to $\\alr$ space is beneficial as components directly correspond to taxa in the composition.\nWe outline in Appendix~\\ref{app:compdetails} how to transform $\\hat{\\beta}_{\\ilr}$ into $\\hat{\\beta}_{\\ilr}$ and vice versa with linear transformations.\n\n\\xhdr{ILR+LC \/ ALR+LC}\nAs a natural extension, we use sparse log-contrast regression~\\citep{Lin2014c,Combettes2019Logcontrast} for the second stage, while retaining $\\ilr$ (or equivalently $\\alr$) regression for the first stage.\nLog-contrast regression in eq.~\\eqref{eq:logcontrast} allows for $\\ell_1$ regularization \\emph{and} conserves interpretability in that estimated parameters correspond directly to individual relative abundances.\n\n\\xhdr{KIV}\nAs a drop-in replacement in 2SLS, we can use kernel instrumental variable regression \\citep{singh2019kernel}, essentially replacing OLS with kernel ridge regression in both stages.\nLike 2SLS, KIV cannot enforce sparsity in an interpretable fashion.\n\n\\xhdr{DIR+LC}\nHere we circumvent log-transformations entirely and deploy regression methods naturally work with compositional data in both stages.\nFor the first stage, we use a Dirichlet distribution---a common choice for modeling compositional data---where $X\\,|\\, Z \\sim \\mathrm{Dirichlet}(\\alpha_1(Z), \\ldots, \\alpha_p(Z))$ for the density $p_{\\text{Dirichlet}}(x; \\alpha) = \\frac{1}{B(\\alpha_1, \\ldots, \\alpha_p)} \\prod_{j=1}^{p}x_j^{\\alpha_j - 1}$.\nWith the mean of the Dirichlet distribution given by $\\nicefrac{\\alpha}{\\sum_{j=1}^p \\alpha_j}$, we account for the $Z$-dependence via $\\log(\\alpha_j(Z_i)) = \\omega_{0,j} + \\omega_j Z_j$.\nWe then estimate the parameters via maximum likelihood estimation with $\\ell_1$ regularization.\nFor the second stage we again resort to log-contrast regression.\nThis approach leaves room for discussion.\nClearly, if this non-linear first stage is mis-specified the ``forbidden regression'' bias may falsify our effect estimates even in the limit of infinite data.\nWe nevertheless include this method in our comparison, because modeling compositions directly via the commonly used Dirichlet regression may result in a better fit than modeling log-transformations.\n\n\\subsection{Data generation}\n\\label{sec:datageneration}\n\n\\xhdr{Semi-synthetic data}\nFor the evaluation of our methods we require ground truth to be known.\nSince counterfactuals are never observed in practice, we simulate a dataset in which we attempt to preserve the characteristics of the observed data, while maintaining control over ground truth effects.\nWe generate data according to the following model with $\\mu = \\alpha_0 + \\alpha Z$\n\\begin{align}\n Z_j &\\sim \\text{Uniform}({Z_{\\min}}, {Z_{\\max}} ), \\quad\n &X &= g(Z, U) \\sim C \\Big(\\text{ZINegBinomial}(\\mu, \\Sigma, \\theta, \\eta ) \\Big) \\oplus (\\Omega_C \\odot U ),\n \\nonumber \\\\\n U &\\sim \\text{Uniform}({U_{\\min}}, {U_{\\max}}), \\quad\n &Y &= f(X, U) = \\beta_0 + \\beta^T \\log(X) + c_Y^T \\log(\\Omega_C \\odot U ).\n \\label{eq:semisyntheticdata}\n\\end{align}\nThe treatment $X$ is assumed to follow a zero-inflated negative binomial distribution \\citep{greene1994accounting}, commonly used for modeling microbiome compositions \\citep{xu2015assessment}.\nHere, $\\eta$ is the probability of zero entries, $\\Sigma$ is the variance, and $\\theta$ the shape parameter.\nThe confounder $U$ perturbs this base composition in the direction of another fixed composition $\\Omega_C$ scaled by the scalar $U$.\\footnote{In simplex geometry $x_0 \\oplus (U \\odot x_1)$ corresponds to a line starting at $x_0$ and moving into the direction of $x_1$ by amount $U$.}\nIt also enters $Y$ additively with a prefactor $c_Y$ controlling confounding strength. \nThe setting is `linear' in how $Z$ enters $\\mu$ and how $U$ enters $X$ and $Y$ (in simplex geometry). \nTo preserve the characteristics of observed data, we estimate $\\alpha_0, \\alpha, \\beta_0, \\beta$ and $\\eta$ from the dataset described in \\citet{Schulfer2019}.\nAll other parameter choices are given in Appendix~\\ref{app:simulationdetail}.\n\n\n\\xhdr{Synthetic data}\nWe also assess our methods in a \\emph{well-specified} synthetic setting.\nInstead of $X \\in \\ensuremath \\mathbb{S}^{p-1}$, we now model $\\ilr(X)$ with fully linear relations:\n\\begin{align}\n Z_j &\\sim \\text{Uniform}(0, 1), & \\ilr(X) &= g(Z, U) = \\alpha_0 + \\alpha Z + c_X U, \n \\nonumber \\\\\n U &\\sim \\ensuremath \\mathcal{N}(\\mu_c, 1), & Y &= f(X, U) = \\beta_0 + \\beta^T \\ilr(X) + c_Y U.\n \\label{eq:syntheticdata}\n\\end{align}\nSince this setting is fully linear in log-transformed coordinates with additive noise, it satisfies the standard 2SLS assumptions.\n\n\n\\subsection{Metrics}\n\nThe appropriate choice of evaluation metric is key to cause-effect estimation tasks.\nBeyond predictive performance on observational data, we primarily aim at capturing the average causal effect (under interventions) as well as the causal parameters under specific modeling assumptions.\nWhen the true effect is linear in $\\log(X)$, we can compare the estimated causal parameters $\\hat{\\beta}$ from ILR+ILR, ILR\/ALC+LC, and DIR+LC with the ground truth $\\beta$ directly.\nIn the well-specified and linear setting, we can thus identify the causal effect of individual relative abundances $X_j$ on the outcome $Y$.\n\nIn the general case, where a measure for perfect identification of the interventional distribution $Y \\,|\\, do(X)$ is not straight forward to evaluate, we focus on the \\emph{out of sample error} (OOS MSE), which denotes the mean squared error between the true value of $Y$ under an intervention $do(X)$ and the predicted causal effect of our model.\nBecause in real observational data we do not have access to $Y \\,|\\, do(X)$ (but only $Y \\,|\\, X$), we can only evaluate both measures on (semi-)synthetic data with sufficient control over the data generating mechanism.\nFor the true causal effect we first draw an i.i.d.\\ sample $\\{x_i\\}_{i=1}^{m}$ from the data generating distribution (that are not in the training set, i.e., out of sample) and compute $\\E_{U}[f(x_i, U)]$.\nWe use $m=250$ for all experiments.\nOOS MSE is then the mean square difference to our second-stage predictions $\\hat{f}(x_i)$ on these out of sample $x_i$.\n\n\\subsection{Results}\n\nUnless stated otherwise we run each method and setting for 50 random seeds on $n=1000$ examples.\n\n\\def4cm{4cm}\n\\begin{figure}\n \\centering\n \\textbf{strong instrument}\\hspace{5cm}\\textbf{weak instrument} \\\\[2mm]\n \\includegraphics[height=4cm]{figs\/MSE_Small_Linear3InstrumentStrength_medium.pdf}\n \\hspace{2cm}\n \\includegraphics[height=4cm]{figs\/MSE_Small_Linear3InstrumentStrength_weak.pdf}\\\\[0.3cm]\n \\includegraphics[height=4cm]{figs\/Beta_Small_Linear3InstrumentStrength_medium.pdf}\n \\hspace{2cm}\n \\includegraphics[height=4cm]{figs\/Beta_Small_Linear3InstrumentStrength_weak.pdf}\n \\caption{Synthetic data for linear~$f$ with $p=3$ and $q=2$.\n The top row shows OOS MSE (lower is better) and the bottom row shows the parameter estimates $\\hat{\\beta}$ (boxes) with ground truth values (dashed lines).\n \\textbf{Left:} For strong instruments two-stage procedures outperform only fitting the second stage with consistently low OOS MSE and reliably recovery of the true causal parameters. Only fitting the second stage consistently suffers from large bias.\n \\textbf{Right:} For weak instruments the out of sample performance of two-stage systems suffers substantially, but ALR+LC still compares favorably to only fitting the second stage. \n Moreover, best performances are still achieved by two-stage systems, whereas ONLY Second LC fails consistently (top).\n Similarly, the causal parameter estimates have higher variance, but ALR+LC on average still identifies them well (bottom).}\n \\label{fig:linear3}\n\\end{figure}\n\n\\xhdr{Low-dimensional setting}\nWe first consider the fully synthetic, linear setting in eq.~\\eqref{eq:syntheticdata} with $p=3$ and $q=2$ for both strong and weak instruments.\nInstrument strength for $p=1$ is typically measured via the first-stage F-statistic, with an F-statistic beyond 10 being considered sufficient to avoid weak instrument bias in 2SLS \\citep{andrews2019weak}.\nFor $p>1$, measuring instrument strength is not as straightforward \\citep{sanderson20162ftest}.\nBecause significant F-statistics are still a necessary requirement, we ensure large ($36.3, 87.5$) and small ($1.1, 0.7$) first-stage F-statistics for all (independent) instruments separately for the strong and weak instrument case respectively.\n\nIn Figure~\\ref{fig:linear3} we compare ALR+LC, DIR+LC and `ONLY Second LC', a simple baseline consisting of a direct one-stage log-contrast regression for $X \\to Y$.\nWithout a sparsity penalty in the second stage, ILR+ILR, ILR+LC, and ALR+LC yield equivalent estimates in this low-dimensional linear setting, which is why we only report ALR+LC as a representative.\nThe direct LC regression is affected by the spurious correlation introduced via the confounder $U$ between $X$ and $Y$ and does not recover the true causal effect regardless of instrument strength (bottom row in Figure~\\ref{fig:linear3}).\nAs expected, two-stage procedures reliably recover the true causal effect with strong instruments.\nWhile the variance in parameter estimates increases, ALR+LC (and equivalent methods) still achieve low OOS MSE, and on average recover the true causal parameters.\nThe large variance in DIR+LC may be an instantiation of ``forbidden regression'' bias due to a mis-specified first-stage combined with weak instruments.\n\n\\begin{wrapfigure}{r}{6.5cm}\n\\vspace{-3mm}\n \\centering\n \\includegraphics[width=0.38\\textwidth]{figs\/MSE_Small_NonLinear3InstrumentStrength_strong.pdf}\n \\vspace{-2mm}\n \\caption{Synthetic data with non-linear~$f$, $p=3$, $q=2$.\n Despite mis-specification, ALR+LC and KIV outperform naive regression.\n DIR+LC may fail because both stages are mis-specified.}\n \\label{fig:nonlinear3_strong}\n \\vspace{-2mm}\n\\end{wrapfigure}\n\nTo further explore the effects of mis-specification, we replace $f$ in eq.~\\eqref{eq:syntheticdata} with $f(X, U) = \\beta_0 + \\frac{1}{10} \\beta^T \\ilr(X) + \\frac{1}{20} \\B{1}^T(\\ilr(X) + 1)^3 + c_Y U$, where $\\B{1} = (1, \\ldots, 1)$, and use strong instruments (F-statistic: $148.3, 100.7$).\nFigure~\\ref{fig:nonlinear3_strong} shows that while DIR+LC suffers from the mis-specification in both stages, the remaining two-stage methods ALR+LC (and equivalent ones) as well as KIV still outperform a naive direct second stage LC.\nThe well-specified first stage properly removes the influence of the confounder to better recover the causal effect.\nOverall, two-stage approaches using log-transformed coordinates work well low-dimensional settings without sparsity penalty.\nAdditional Figures and all remaining details of the simulation can be found in Appendix \\ref{app:simulationdetail} and \\ref{app:graphdetail}.\n\n\\xhdr{Sparse higher-dimensional setting}\nWe now consider the case $p=30$ and $q=20$ for synthetic and semi-synthetic data from eqs.~\\eqref{eq:syntheticdata} and~\\eqref{eq:semisyntheticdata} (for which we solely perform 20 runs with different random seeds) with sparse ground truth $\\beta$ only having four nonzero entries $-5, -3, 3, 5$ for synthetic and eight nonzero entries $5, 5, 5, 5, -5, -5, -5, -5$ for semi-synthetic data.\nThe well-specified two-stage methods for the synthetic setting shown in Figure~\\ref{fig:linear30} (left) yield encouraging results coherent with the low-dimensional setup.\nUnlike the fully synthetic setting, the semi-synthetic one also captures sparsity in the first stage, which is commonly the case for high-dimensional compositional data.\nEven though we may thus suffer from first-stage mis-specification, the two stage models perform quite well in recovering average causal effects, see Figure~\\ref{fig:linear30} (right).\nAgain, all details about the simulation setup and additional results are in Appendix \\ref{app:simulationdetail} and \\ref{app:graphdetail}.\n\n\n\\def4cm{4cm}\n\\begin{figure}\n \\centering\n \\hspace{1cm}\\textbf{synthetic data \\eqref{eq:syntheticdata}}\\hspace{5.5cm}\\textbf{semi-synthetic data \\eqref{eq:semisyntheticdata}} \\\\[2mm]\n \\includegraphics[height=4cm]{figs\/MSE_Small_HighLinear_30InstrumentStrength_unknown.pdf}\n \\hspace{2cm}\n \\includegraphics[height=4cm]{figs\/MSE_Small_NegBinom30InstrumentStrength_unknown_inst20_run20.pdf}\\\\[0.3cm]\n \\includegraphics[height=4cm]{figs\/Beta_Small_HighLinear30InstrumentStrength_unknown.pdf}\n \\hspace{2cm}\n \\includegraphics[height=4cm]{figs\/Beta_Small_NegBinom30InstrumentStrength_unknown_inst20_run20.pdf}\n \\caption{Higher-dimensional results for linear $f$, $p=30$, $q=20$ and sparse $\\beta$ on synthetic (left) and semi-synthetic (right) data.\n The top row shows OOS MSE (lower is better) and the bottom row shows the parameter estimates $\\hat{\\beta}$ (boxes) with ground truth values (dashed lines).\n \\textbf{Left:} The well-specified two-stage methods ALR+LC (as representative) outperform a naive second stage LC and nearly identify the sparse true causal parameters, where naive LC fails consistently. Due to their advantage of the linear methods in this well-specified setting, KIV is not competitive in this setting (see \\ref{app:graphdetail}). \n \\textbf{Right:} Again, two-stage methods outperform a naive second stage, but variance increases due to the mis-specified first stage for ALR+LC.\n While KIV compares well in terms of OOS MSE, it does not produce direct estimates for the linear causal parameters.\n Neither method recovers those well, but the naive second stage even consistently flips their order.}\n \\label{fig:linear30}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe compositional nature of many scientific datasets poses major challenges to statistical analysis and interpretation.\nSince compositional data measurements naturally lead to dependencies among the individual components, they do not allow for a straightforward interpretation.\nMoreover, because compositions derived from modern biological high-throughput measurements are often high-dimensional, the analyst is also faced with the difficulty of finding a parsimonious sub-composition of interest.\nHowever, given the potentially profound impact of the microbiome on human health and disease, developing interpretable statistical methods to obtain meaningful insights from\n(microbiome) compositional data is of vital importance.\n\nIn this work, we initiated the analysis and development of methods for cause-effect estimation with compositional causes in instrumental variable settings.\nAs we aim for informing consequential decisions such as medical treatments, we focused on interpretability with respect to potential interventions.\nFirst, we crisply formulated the limitations of replacing compositions with information theoretic summary statistics using microbial species diversity as a hall mark example.\nEven though it would allow us to tap into a collection of well-established cause-effect estimation tools, this approach can not provide actionable insights about interventions and causal drivers.\nNext, building on a number of existing methods for compositional data such as isometries to Euclidean space ($\\ilr, \\clr$), we develop a range of methods for average treatment effect \nestimation in an instrumental variable setting with compositional causes.\nWe provided an in-depth analysis of how IV assumptions interact with specifics of compositional data and carefully explain valid and invalid interpretations of the results for different modeling choices (including mis-specified first or second stages as well as weak instrument bias).\nFinally, we evaluated the efficacy and robustness of our methods on synthetic and semi-synthetic datasets motivated by microbiome amplicon sequencing data.\nOur experimental results provide promising evidence that our methods can indeed give interpretable and theoretically sound answers to causal queries involving compositional causes from purely observational data.\n\nWe remark that this work is a first step, leading to many open questions for future work on how to extend the causal inference toolbox to compositional data both as the cause as well as the effect.\nIn particular, it highlights that one must be cautious when informing consequential treatments using models learned from compositional data.\nNevertheless, we posit that our work provides a useful starting point for how to carefully think about cause-effect estimation with compositional data.\nGoing forward, this may be a vital component in fully integrating the microbiome into our understanding of human health and palette of possible treatments of disease as well as understanding the effects of compositions in microbial ecology.\n\n\n\\begin{ack}\n We thank Dr.~Chan Wang and Dr.~Huilin Li, NYU Langone Medical Center, for kindly providing the\n pre-processed murine amplicon and associated phenotype data used in this study. We thank L\\'eo\n Simpson, TU M\u00fcnchen, and Alice Sommer, LMU M\\\"unchen, for kindly and patiently providing their \n technical and scientific support. \n EA is supported by the Helmholtz Association under the joint research\n school ``Munich School for Data Science - MUDS''.\n\\end{ack}\n\n\n\\bibliographystyle{refs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\n\nLet $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of positive characteristic $p$. Let\n\\[\nV(FG)=\\Big\\{\\; x=\\sum_{g\\in G}\\alpha_gg\\in FG\\; \\mid\\; \\chi(x)=\\sum_{g\\in G}\\alpha_g=1\\; \\Big\\}\n\\]\nbe the group of normalized units of $FG$, where $\\chi(x)$ is the augmentation map (see \\cite [Chapters 2-3, p.\\,194-196]{Bovdi_survey}). In this case, the order of the group $V(FG)$ is equal to $\\order{F}^{\\order{G}-1}$, so the order of $V(FG)$ can be very large even for a small group $G$. Note that, studying the structure of the group $V(FG)$ is a rather difficult task (for more details see the survey \\cite{Bovdi_survey}).\n\nLet $\\cd$ be an involution of the algebra $FG$. We say that the involution $\\cd$ arises from the group $G$, if $\\cd$ is a linear extension of an anti-automorphism of $G$ to $FG$. An example for such kind of involution is the canonical involution that is the linear extension of the anti-automorphism of $G$ which sends each element of $G$ to its inverse. This involution is usually denoted by $*$.\n\nAn element $u \\in V(FG)$ is called {\\it $\\cd$-unitary}, if $u^{\\cd}=u^{-1}$ with respect to the involution $\\cd$ of $FG$.\nThe set $V_{\\cd}(FG)$ of all $\\cd$-unitary units forms a subgroup of $V(FG)$ which is called $\\cd$-unitary subgroup.\nInterest in the structure of unitary subgroups arose in algebraic topology and unitary $K$-theory (see Novikov's papers \\cite{Novikov} and Bovdi's paper \\cite{Bovdi_Unitarity}). Let $L$ be a finite Galois extension of $F$ with Galois group $G$, where $F$ is a finite field of characteristic two.\nSerre \\cite{serre} identified an interesting relation between the self-dual normal basis of $L$ over $F$ and the $*$-unitary subgroup of $FG$. This relationship also makes the study of the unitary subgroups timely.\n\nThe unitary subgroups have been proven to be very useful subgroups in several studies (see \\cite{Balogh_MD, Balogh_Creedon_Gildea, Balogh_Laver, Bovdi_Erdei_II, Bovdi_Erdei_I, Bovdi_Szakacs_II, Bovdi_Kovacs_I, Bovdi_Salim, Bovdi_Grichkov, Creedon_Gildea_I} and \\cite{Creedon_Gildea_II}). However, we know very little about their structure, as even finding their order is a challenging problem. The first results in this area were published in the $1980$'s. For finite abelian $p$-groups $G$ the order of $V_*(FG)$ was given in \\cite{Bovdi_Szakacs_III}.\n\\begin{proposition}(\\cite[Theorem 2]{Bovdi_Szakacs_III})\\label{szakacs}\n\tLet $G$ be a finite abelian $2$-group. If ${F}$ is a finite field of characteristic $2$, then the order $\\order{V_*({F}G)}$ is divisible by $\\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1}$, that is, \t\n \\[\n \\order{V_*({F}G)}=\\Theta \\cdot \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1},\n\t\\]\n\twhere $\\Theta=\\order{G^2\\{2\\}}$ and $\\order{S}$ denotes the size of a finite set $S$.\n\\end{proposition}\n\nIt follows that the number $\\Theta$ does not depend on the size of the field $F$. The following breakthrough result was proved for certain non-abelian $2$-groups by Bovdi and Roza.\n\\begin{proposition}(\\cite[Corollary 2]{Bovdi_Rosa_I})\\label{roza}\n\tIf $\\order{F}=2^m\\geq 2$, then:\n\t\\[\n\t\\order{V_*({F}G)}=\\Theta \\cdot \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1},\n\t\\]\n\twhere\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\Theta=1$ if $G$ is a dihedral $2$-group;\n\t\t\\item[(ii)] $\\Theta=4$ if $G$ is a generalized quaternion $2$-group.\n\t\\end{enumerate}\n\\end{proposition}\n\nIn \\cite{Balogh_IEJA}, the value of the number $\\Theta$ was given for all non-abelian groups of order $2^4$. Although, for these groups, the number $\\Theta$ is not equal to $\\order{G^2\\{2\\}}$ it does not depend on the field $F$. Wang and Liu \\cite{Wang} evaluated $\\Theta$ in the case when $G$ is a non-abelian $2$-group given by a central extension of the form\n\\[\n1 \\longrightarrow C_{2^m} \\longrightarrow G \\longrightarrow C_2 \\times \\cdots \\times C_2 \\longrightarrow 1,\n\\]\nin which $m\\geq 1$ and $|G'|=2$. At present, the question of whether the quotient $\\Theta$ depends on the field $F$ is still open.\n\nOur main results are the following.\n\\begin{theorem}\\label{main_theorem_p}\n\tLet $G$ be a finite $p$-group, where $p$ is an odd prime and let $F$ be a finite field of characteristic $p$. If $\\cd$ is an involution of $FG$ that arises from the group $G$, then\n \\[\n \\order{V_{\\cd}(FG)}=\\order{F}^{\\frac{1}{2}(\\order{G}-\\order{G_{\\cd}})},\n \\]\n where $G_{\\cd}=\\{\\,g\\; \\vert \\; g=g^{\\cd}\\,\\}$.\n\\end{theorem}\n\n\nLet $\\xi(G)$ denote the center of the group $G$ and $\\xi(G)\\{2\\}$ denote the set of elements of order two in $\\xi(G)$.\n\\begin{theorem}\\label{main_theorem}\n\tLet $G$ be a finite $2$-group. If $F$ is a finite field of characteristic two, then\n\t\\[\n\t\\order{V_*({F}G)}=\\Theta \\cdot \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1}\n\t\\]\n\tfor some integer $\\Theta$.\n\tMoreover, if the set $T_c=\\{g\\in G\\;\\vert\\;g^2=c\\}$ is commutative for some $c\\in \\xi(G)\\{2\\}$, then $\\Theta$ does not depend on the field $F$.\n\\end{theorem}\n\nBy combining Theorem \\ref{main_theorem_p} and Theorem \\ref{main_theorem}, we have the following.\n\n\\begin{corollary}\\label{C:1}\nLet $G$ be a finite $p$-group. If $F$ is a finite field of characteristic $p$, then the order of the $*$-unitary subgroup of $FG$ determines the order of $G$.\n\\end{corollary}\n\n\\section{Proofs}\n\nLet $G$ be a finite $p$-group, let $F$ be a finite field of $char(F)=p>2$ and let $\\cd$ be an involution of $FG$ which arises from $G$. An element $x\\in FG$ is called skew-symmetric under the involution $\\cd$ if $x^{\\cd}=-x$. Let $FG^-_{\\cd}$ denote the set of all skew-symmetric elements of $FG$.\n\n\\begin{proof}[Proof of Theorem \\ref{main_theorem_p}]\nLet $z\\in FG$ such that $1+z$ is invertible. Clearly, $1-z$ and $1+z$ commute, therefore $1-z$ and $(1+z)^{-1}$ also commute.\n\nLet $Q=\\{x\\in FG\\;|\\; 1+x \\;\\text{is invertible in $FG$}\\}$.\n\tLet us define the map $f:Q \\rightarrow FG$ by\n\t\\[\n\tf(x)=(1-x)(1+x)^{-1}.\n\t\\]\n\nIf $y\\in FG^-_{\\cd}$, then $\\chi(y)=0$, so \n\\[\n\\chi(1+y)=\\chi(1-y)=\\chi(1+y^{\\cd})=1\n\\]\n and $1+y, 1-y, 1+y^{\\cd}$ are normalized units. Hence\n\\[\n\\begin{split}\nf(y)f(y)^{\\cd}=&(1-y)(1+y)^{-1}(1+y^{\\cd})^{-1}(1-y^{\\cd})\\\\\n&=(1-y)(1+y)^{-1}(1-y)^{-1}(1+y)\\\\\n&=(1+y)^{-1}(1-y)(1-y)^{-1}(1+y)\\\\\n&=1.\n\\end{split}\n\\]\nConsequently, $f(y) \\in V_{\\cd}(FG)$ and $f: FG^-_{\\cd}\\rightarrow V_{\\cd}(FG)$ is a surjection.\n\t\nLet $x\\in V_{\\cd}(FG)$. Evidently, $1+x,1+x^{\\cd}$ and $1+x^{-1}$ are invertible, because $\\chi(1+x)=\\chi(1+x^{\\cd})=\\chi(1+x^{-1})=2$ is invertible in $F$. Therefore $V_{\\cd}(FG)$ is a subset of $Q$. Let $y$ denote the element $f(x)$. Then\n\t\\[\n\t\\begin{split} y^{\\cd}&=f(x)^{\\cd}=(1+x^{\\cd})^{-1}(1-x^{\\cd})=(1+x^{-1})^{-1}(1-x^{-1})\\\\\n\t&=\\big(x^{-1}(x+1) \\big)^{-1}x^{-1}(x-1)=-(1+x)^{-1}xx^{-1}(1-x)\\\\\n&=-y.\n\t\\end{split}\n\t\\]\n\tTherefore $f:V_{\\cd}(FG) \\rightarrow FG^-_{\\cd}$ is a surjection.\n\t\n\t\n\tSimilar computation shows that\n\t\\[\n\t\\begin{split}\n\t f(f(x))=&\\big(1-(1-x)(1+x)^{-1}\\big)\\big(1+(1-x)(1+x)^{-1}\\big)^{-1}\\\\ &=\\big((1+x)-(1-x)\\big)(1+x)^{-1}\\Big(\\big((1+x)+(1-x)\\big)(1+x)^{-1}\\Big)^{-1}\\\\\n\t&=\\big((1+x)-(1-x)\\big)(1+x)^{-1} (1+x) \\big((1+x)+(1-x)\\big)^{-1}\\\\\n\t&=\\big((1+x)-(1-x)\\big)\\big((1+x)+(1-x)\\big)^{-1}\\\\\n&=x\\\\\n\t\\end{split}\n\t\\]\n\tfor every $x\\in V_{\\cd}(FG)$, so $f$ is a bijection between\n\t$FG^-_{\\cd}$ and $V_{\\cd}(FG)$.\n\t\nSince $FG^-_{\\cd}$ is a linear space over $F$ with basis\n\t$\\{\\,g-g^{\\cd}\\;\\vert \\; g\\in G\\setminus G_{\\cd}\\,\\}$, \\[\n\\order{V_{\\cd}(FG)}=\\order{FG^-_{\\cd}}=\\order{F}^{\\frac{1}{2}(\\order{G}-\\order{G_{\\cd}})}.\n\\]\n\\end{proof}\n\nLet $H$ be a normal subgroup of $G$ and let $I(H):=\\gp{1+h \\;\\mid \\; h\\in H}_{FG}$ be an ideal of $FG$ generated by the set $\\{ 1+h \\mid h\\in H\\}$.\nClearly,\n\\[\nFG\/I(H)\\cong FG\/\\ker(\\Psi) \\cong F[G\/H],\n\\]\nwhere $\\Psi: FG \\to FG\/I(H)$ is the natural homomorphism.\n\nLet us denote by $V_*(F\\overline{G})$ the $*$-unitary subgroup of the factor algebra $FG\/I(H)$, where $\\overline{G}=G\/H$.\nIt is easy to check that the set\n\\[\nN^*_{\\Psi}=\\{x\\in V(FG) \\mid \\Psi(x) \\in V_*(F\\overline{G}) \\}\n\\]\nforms a subgroup in $V(FG)$.\nLet $I(H)^+=\\{1+x \\, \\vert\\, x\\in I(H) \\}$. The subgroup $I(H)^+$ is normal in $V(FG)$ and\n$S_H=\\{ xx^* \\mid x\\in N^*_{\\Psi} \\}$ is a subset of $I(H)^+$, because $xx^* \\in 1+\\ker(\\Psi)=I(H)^+$ for all $x\\in N^*_{\\Psi}$.\n\nFirst, we need the following.\n\n\\begin{lemma}\\label{lemma_main}\n\tLet $H$ be a normal subgroup of a finite $2$-group $G$. Set $\\overline{G}=G\/H$. If $\\order{F}=2^m\\geq 2$, then\n\t\\begin{equation}\n\t\\textstyle\n\t\\order{V_*(FG)}=\\order{F}^{\\order{\\overline{G}}}\\cdot \\frac{\\order{V_*(F\\overline{G})}}{\\order{S_H}}.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tLet $\\Phi: V(FG) \\to V(FG)$ be a map such that $\\Phi(x)=xx^*$ for every $x\\in V(FG)$.\n\tThe sets $\\Phi(x)$ and $\\Phi(y)$ coincide if and only if $y \\in x\\cdot V_*(FG)$. Indeed, if $y \\in x \\cdot V_*(FG)$, then $y=xv$ for some $v \\in V_*(FG)$. Therefore\n\t\\[\n\t\\Phi(y)=yy^*=xv(xv)^*=xvv^*x^*=xx^*=\\Phi(x).\n\t\\]\n\tAssume that $\\Phi(x)=\\Phi(y)$ for some $x,y\\in V(FG)$. Then $xx^*=yy^*$, or equivalently, $y^{-1}x=y^*(x^*)^{-1}$. Therefore\n\t\\[\n\t(x^{-1}y)^{-1}=y^{-1}x=y^*(x^*)^{-1}=(x^{-1}y)^*\n\t\\]\n\twhich confirms that $x^{-1}y\\in V_*(FG)$.\n\t\n\tSince $[N^*_{\\Psi}:V_*(FG)] = \\order{S_H}$,\n\t\\[\n\t\\textstyle\n\t \\order{V_*(FG)}=\\frac{\\order{N^*_{\\Psi}}}{\\order{S_H}}=\\order{I(H)^+}\\cdot \\frac{\\order{V_*(F\\overline{G})} }{\\order{S_H}}.\n\t\\]\n\tWe should note that $V_*(FG)$ is usually not a normal subgroup of $N^*_{\\Psi}$.\n\t\n\tThe ideal $I(H)$ can be considered as a vector space over $F$ with the following basis $\\{\\, u(1+h) \\mid u \\in T(G\/H),\\; h \\in H \\,\\}$, where $T(G\/H)$ is a complete set of left coset representatives of $H$ in $G$. Consequently,\n\\[\n\\order{I(H)^+}=\\order{I(H)}=\\order{F}^{\\frac{\\order{G}}{\\order{H}}}.\n\\]\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{main_theorem}]\n\tLet $G$ be a $2$-group of order $2^n$ and let $H$ be a subgroup of $G$ generated by a central element $c$ of order two.\n\tEvidently, the set $S_H=\\{\\, xx^* \\mid x\\in N^*_{\\Psi}\\}$ is a subset of\n\t$I(H)^+ \\cap V_*(FG)$\n\tand every $y\\in S_H$ is a $*$-symmetric element. Moreover, the support of $y$ does not contain elements of order two by \\cite[Lemma 2.5]{Balogh_IEJA}.\n\tThus, every $y\\in S_H$ can be written as\n\t\\[\n\ty=1+\\sum_{g\\in G\\setminus (G\\{2\\} \\cup T_c)} \\alpha_g (g+g^{-1})\\widehat{H}+ \\sum_{g\\in T_c} \\beta_g g\\widehat{H},\\qquad (\\alpha_g, \\beta_g \\in F)\n\t\\]\n\twhere $T_c=\\{\\,g\\in G\\;\\vert \\;g^2=c\\}$ and $\\widehat{H}=1+c$. This yields that\n\t\\begin{equation}\\label{ineqush1}\n\t\\order{S_H} \\leq \\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}-\\order{T_c})+\\frac{1}{2}\\order{T_c}}=\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}+\\order{T_c})}.\n\t\\end{equation}\nLet us prove that if $T_c$ is a commutative set, then\n\t\\begin{equation}\\label{ineqush2}\n\t\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}+\\order{T_c})}\\cdot 2^{-\\frac{1}{2}\\order{T_c}} \\leq \\order{S_H}.\n\t\\end{equation}\nLet $N_1$ be a group generated by the elements $1+\\alpha_g (g+g^{-1})\\widehat{H}$, in which $g^2\\not\\in H$ and $\\alpha_g \\in F$.\nEvidently, $N_1$ is an elementary abelian subgroup of $I(H)^+$.\nSince $g^2\\not\\in H$ (equivalently $g\\in G\\setminus (G\\{2\\} \\cup T_c)$), \n\\[\n1+\\alpha_g (g+g^{-1})\\widehat{H}=1+\\alpha_g g^{-1}(1+g^2)\\widehat{H}\\not=1\n\\]\nand\n\t\\[\n\t1+\\alpha_g (g+g^{-1})\\widehat{H}=(1+\\alpha_g g\\widehat{H})(1+\\alpha_g g\\widehat{H})^*\\in S_H.\n\t\\]\nIf $z\\in N_1$, then\n\t\\[\n\tz=\\prod_{g^2\\not\\in H} (1+\\alpha_g (g+g^{-1})\\widehat{H})=\\prod_{g^2\\not\\in H}(1+\\alpha_g g\\widehat{H})(1+\\alpha_g g\\widehat{H})^*,\n\t\\]\nso $(1+\\alpha_g g\\widehat{H})\\in I(H)^+$ and\n\t\\[\n\tz=\\Big(\\prod_{g^2\\not\\in H}(1+\\alpha_g g\\widehat{H})\\Big) \\cdot \\Big(\\prod_{g^2\\not\\in H}(1+\\alpha_g g\\widehat{H})\\Big)^*\\in S_H.\n\t\\]\n\tThus $N_1$ is a subgroup in the set $S_H$ and $\\order{N_1}=\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}-\\order{T_c})}$. \t \n\t\n\tThe map $\\tau: F\\to F$ defined by $\\tau(\\alpha)=\\alpha+\\alpha^2$ $(\\alpha \\in F)$ is a homomorphism on the additive group of the field $F$ with kernel $\\ker(\\tau) =\\{0,1\\}$ (see \\cite[Lemma 10]{Balogh_IEJA}). Therefore $\\order{\\imf(\\tau)}=\\frac{\\order{F}}{2}$.\n\t\nSuppose that $T_c$ is a commutative set.\nLet $N_2$ be a group generated by the elements\t $1+\\alpha_g g\\widehat{H}$, in which $g\\in T_c$ and $\\alpha_g \\in F$.\n\tSince\n\t\\[\n\t(1+\\omega g+\\omega g^2)(1+\\omega g+\\omega g^2)^*=1+(\\omega+\\omega^2) g \\widehat{H}\n\t\\]\n\twe have $1+\\alpha_g g\\widehat{H}\\in S_H$ for every $\\alpha_g \\in \\imf(\\tau)$.\n\tThe group $N_2$, being $T_c$ commutative, is a subgroup in $S_H$ and \\[\n\t \\order{N_2}=\\order{\\imf(\\tau)}^{\\frac{1}{2}\\order{T_c}}=\\order{F}^{\\frac{1}{2}\\order{T_c}}\\cdot 2^{-\\frac{1}{2}\\order{T_c}}.\n\t\\]\n\tLet $x\\in N_1$ and $y\\in N_2$. There exist $x_1\\in I(H)^+$ and $y_1\\in N^*_{\\Psi}$ such that $x_1x_1^*=x$ and $y_1y_1^*=y$. Since $I(H)^+$ is an elementary $2$-group, the element $y$ commutes with $x_1$ and\n\t\\[\n\tyx=yx_1x_1^*=x_1yx_1^*=x_1y_1y_1^*x_1^*=(x_1y_1)(x_1y_1)^*\\in S_H.\n\t\\]\n\tTherefore $N_1\\times N_2$ is a subgroup in $S_H$ and\n\t\\[\n\t\\order{N_1\\times N_2}=\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}-\\order{T_c})+\\frac{1}{2}\\order{T_c}}\\cdot 2^{-\\frac{1}{2}\\order{T_c}}.\n\t\\]\n\tConsequently,\n\t\\[\n\t\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}+\\order{T_c})}\\cdot 2^{-\\frac{1}{2}\\order{T_c}} \\leq \\order{S_H}.\n\t\\]\n\t\n\tNow, we are ready to prove the theorem. If $n=3$, then the theorem is true by Propositions \\ref{szakacs} and \\ref{roza}.\n\tSuppose that $n>3$. In the factor group $\\overline{G}=G\/H$ the element $\\overline{g}$ has order two\n\tif and only if either $g\\in G\\{2\\}$ or $g \\in T_c$. Therefore, $\\order{\\overline{G}\\{ 2 \\}} = \\frac{\\order{G\\{2\\}}+\\order{T_c}}{2}$.\n\tAccording to the inductive hypothesis\n\t\\[\\order{V_*(F\\overline{G})}=2^s\\cdot \\order{F}^{\\frac{1}{2}(\\order{\\overline{G}}+\\order{\\overline{G}\\{2\\}})-1}=2^s\\cdot \\order{F}^{\\frac{1}{4}(\\order{{G}}+\\order{{G}\\{2\\}}+\\order{T_c})-1}\n\t\\]\n\tfor some $s \\geq 0$. Using Lemma \\ref{lemma_main} and equation (\\ref{ineqush1}) we obtain that\n\t\\[\n\t\\begin{split}\n\t\\order{V_*(FG)}&=\\textstyle\\order{F}^{\\frac{\\order{G}}{2}}\\cdot \\frac{\\order{V_*(F\\overline{G})}}{\\order{S_H}}\\\\\n&\\textstyle \\geq \\order{F}^{\\frac{\\order{G}}{2}}\\cdot \\frac{2^s\\cdot \\order{F}^{\\frac{1}{4}(\\order{G}+\\order{G\\{2\\}}+\\order{T_c})-1}}\t {\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}+\\order{T_c})}}\\\\\n\t&=2^s\\cdot \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1}.\n\t\\end{split}\n\t\\]\n\tTherefore\n\t\\begin{equation}\\label{inequV}\n\t2^s\\cdot \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1} \\leq \\order{V_*(FG)}\n\t\\end{equation}\n\tand $\\order{V_*(FG)}$ is divisible by $\\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1}$.\n\t\n\t\n\tSimilarly, using Lemma \\ref{lemma_main} and (\\ref{ineqush2}), we obtain that\n\t\\[\n\t\\begin{split}\n\t\\order{V_*(FG)}&=\\textstyle\\order{F}^{\\frac{\\order{G}}{2}}\\cdot \\frac{\\order{V_*(F\\overline{G})}}{\\order{S_H}}\\\\\n &\\textstyle \\leq \\order{F}^{\\frac{\\order{G}}{2}}\\cdot \\frac{2^s\\cdot \\order{F}^{\\frac{1}{4}(\\order{G}+\\order{G\\{2\\}}+\\order{T_c})-1}}\t {\\order{F}^{\\frac{1}{4}(\\order{G}-\\order{G\\{2\\}}+\\order{T_c})}\\cdot 2^{-\\frac{1}{2}\\order{T_c}}}\\\\\n\t&=2^{s+\\frac{1}{2}\\order{T_c}}\\cdot \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1}.\n\t\\end{split}\n\t\\]\n\t\n\tThe size of the set $T_c$ does not depend on the field $F$. Since $2^s$ does not depend on the field $F$ by the inductive hypothesis, the proof is complete.\n\\end{proof}\n\nWe should remark that $2^s$ in the inequality ($\\ref{inequV}$) is usually not inherited via the factorization. For example, for the semidihedral group $D_{16}^-$ of order $16$ we have that\n\\[\n\\order{V_*(FD_{16}^-)}=2\\cdot \\order{F}^{\\frac{1}{2}(\\order{D_{16}^-}+\\order{D_{16}^-\\{2\\}})-1}\n\\]\n by \\cite[Lemma $3.4$]{Balogh_IEJA}. However, $D_{8}\\cong D_{16}^-\/H$, where $H$ is the center of $D_{16}^-$ and $D_{8}$ is the dihedral group of order $8$ and\n$\\order{V_*(FD_{8})}=\\order{F}^{\\frac{1}{2}(\\order{D_{8}}+\\order{D_{8}\\{2\\}})-1}$ by Proposition \\ref{roza}.\n\n\\begin{proof}[Proof of Corollary \\ref{C:1}]\nIf $p=2$, then Theorem \\ref{main_theorem} implies that\n\t\\[\n\t\\order{F}^{\\frac{\\order{G}}{2}-1} \\leq \\order{F}^{\\frac{1}{2}(\\order{G}+\\order{G\\{2\\}})-1} \\leq \\order{V_*(FG)} \\leq \\order{F}^{\\order{G}-1}.\n\t\\]\t\n\tIf $p$ is an odd prime, then $\\order{V_*(FG)}=\\order{F}^{\\frac{1}{2}(\\order{G}-1)}$ by Theorem \\ref{main_theorem_p}. Hence, $\\order{F}^{\\frac{\\order{G}}{2}-1} \\leq \\order{V_*(FG)} \\leq \\order{F}^{\\order{G}-1}$, which confirms that the order of $V_*(FG)$ determines the order of $G$ for every finite $p$-groups.\n\\end{proof}\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}