Cohn-Leavitt path algebras of bi-separated graphs

TL;DR

This paper introduces Cohn-Leavitt path algebras for bi-separated graphs, generalizing Leavitt path algebras, and explores their algebraic properties and K-theory, especially for hypergraph cases.

Contribution

It defines a new class of algebras for bi-separated graphs, studies their categorical properties, and analyzes their K-theory and IBN property for hypergraph cases.

Findings

Normal forms of the algebras are computed.

The lattice of order-ideals is characterized by graph data.

A matrix criterion for IBN property is established.

Abstract

The purpose of this paper is to provide a common framework for studying various generalizations of Leavitt algebras and Leavitt path algebras. This paper consists of two parts. In part I we define Cohn-Leavitt path algebras of a new class of graphs with an additional structure called bi-separated graphs, which generalize the constructions of Leavitt path algebras of various types of graphs. We define and study the category \textbf{BSG} of bi-separated graphs with appropriate morphisms so that the functor which associates a bi-separated graph to its Cohn-Leavitt path algebra is continuous. We also characterize a full subcategory of \textbf{BSG} whose objects are direct limits of finite complete subobjects. We compute normal forms of these algebras and apply them to study some algebraic theoretic properties in terms of bi-separated graph-theoretic properties. In part II we specialize our attention to Cohn-Leavitt path algebras of a special class of bi-separated graphs called B-hypergraphs. We investigate their non-stable K-theory and show that the lattice of order-ideals of V-monoids of these algebras is determined by bi-separated graph-theoretic data. Using this information we study representations of Leavitt path algebras of regular hypergraphs and also find a matrix criterion for Leavitt path algebras of finite hypergraphs to have IBN property.

Equations

$$\sum _{j=1}^n{x}_{lj}{x}_{jk}^{\ast }={\delta }_{lk}\text{and}\sum _{i=1}^m{x}_{li}^{\ast }{x}_{ik}={\delta }_{lk}.$$

$$\varphi :F=(F^0,F^1,r_F,s_F)\to E=(E^0,E^1,r_E,s_E)$$

$$l(\mu )=\{\begin{array}{ll}0,& \text{if}\mu =v\in E^0,\\ n,& \text{if}\mu =e_1{e}_2\dots e_n.\end{array}$$

$$\text{if}v\in E^0,s(v):=v\text{and}r(v):=v,$$

$$\text{if}\mu =e_1{e}_2\dots e_n,s(\mu ):=s(e_1)\text{and}r(\mu ):=r(e_n).$$

$$\mu \cdot \nu :=\{\begin{array}{ll}0,& \text{if}\text{either}\mu =0\text{or}\nu =0,\\ 0,& \text{if}r(\mu )\ne s(\nu ),\\ \mu \nu ,& \text{if}r(\mu )=s(\nu ).\end{array}$$

$$XY=YX=\{\begin{array}{ll}e,& \text{if}X\cap Y=\{e\},\\ 0,& \text{otherwise}.\end{array}$$

$$\sum _{Y\in D}(XY)(Y{X}^{\prime }{)}^{\ast }={\delta }_{X{X}^{\prime }}s(X),$$

$$\sum _{X\in C}(YX{)}^{\ast }(X{Y}^{\prime })={\delta }_{Y{Y}^{\prime }}r(Y).$$

$$A(v,w)=\{\begin{array}{ll}e,& \text{if}e\in E^1\text{such that}s(e)=v,r(e)=w,\\ 0,& \text{otherwise}.\end{array}$$

$$B(v,w)=\{\begin{array}{ll}e,& \text{if}e\in E^1\text{such that}s(e)=v,r(e)=w,\\ 0,& \text{otherwise}.\end{array}$$

$$(L1):A{A}^{\ast }$$

$$(L2):B^{\ast }B$$

$$A=\left(\begin{array}{cccc}{e}_1& 0& \dots & 0\\ 0& e_2& \dots & 0\\ ⋮& ⋮& \dots & ⋮\\ 0& 0& \dots & e_{n-1}\end{array}\right),$$

$$e_i{e}_i^{\ast }=v_i\text{and}{e}_i^{\ast }{e}_i=v_{i+1},$$

$$L_K({\dot{\Sigma }}_n)\cong L_K({\Sigma }_n)\cong M_n(K).$$

$$A=\left(\begin{array}{cccccccc}\ast & \ast & 0& 0& 0& \dots & 0& 0\\ \ast & \ast & \ast & 0& 0& \dots & 0& 0\\ 0& \ast & \ast & \ast & 0& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& 0& 0& \dots & \ast & \ast \end{array}\right),$$

$$L_K(E,\tilde{C},\tilde{D})\cong L_K(E_w,C).$$

$$\sum _{Y\in D}(XY)(Y{X}^{\prime }{)}^{\ast }={\delta }_{X{X}^{\prime }}s(X),$$

$$\sum _{X\in C}(YX{)}^{\ast }(X{Y}^{\prime })={\delta }_{Y{Y}^{\prime }}r(Y),$$

$$1_{{\mathcal{A}}_K(\dot{E})}=\sum _{v\in E^0}v.$$

$$C^{\prime }$$

$$S^{\prime }$$

$$D^{\prime }$$

$$T^{\prime }$$

$$S_1$$

$$S_2$$

$$T_1$$

$$T_2$$

$$\{\left(\dot{{\mathcal{E}}_A}=(\mathcal{E},(\mathcal{C},\mathcal{S}),(\mathcal{D},\mathcal{T}){)}_A,\hookrightarrow \right)\mid A\text{is a finite subset of}{E}^0\bigsqcup E^1\}.$$

$$\{\left(\dot{{\mathcal{E}}_i}=({\mathcal{E}}_i,({\mathcal{C}}_i,{\mathcal{S}}_i),({\mathcal{D}}_i,{\mathcal{T}}_i)),\hookrightarrow \right)\mid i\in I\}$$

$$vw={\delta }_{v,w}v.$$

$$ve={\delta }_{v,s(e)}e,$$

$$ev={\delta }_{v,r(e)}e,$$

Full text

In part II we specialize our attention to Cohn-Leavitt path algebras of a special class of bi-separated graphs called B-hypergraphs. We investigate their non-stable K-theory and show that the lattice of order-ideals of V-monoids of these algebras is determined by bi-separated graph-theoretic data. Using this information we study representations of Leavitt path algebras of regular hypergraphs and also find a matrix criterion for Leavitt path algebras of finite hypergraphs to have IBN property.

1. Introduction

A unital ring $R$ is said to have Invariant Basis Number (IBN) property if whenever the free left modules $R^{m}$ and $R^{n}$ are isomorphic for some natural numbers $m$ and $n$, then $m=n$. In a series of papers [20, 21, 22, 23], W.G. Leavitt studied rings which do not satisfy IBN property. For natural numbers $m,n$ with $m<n$, a non-IBN ring $R$ is said to have module type $(m,n)$ if $m$ and $n$ are the least positive integers such that $R^{m}\cong R^{n}$. For any arbitrary field $K$ and natural numbers $m,n$ with $m<n$, Leavitt constructed ‘universal’ $K$-algebras $L_{K}(m,n)$ of module type $(m,n)$ (now called Leavitt algebras).

$L_{K}(m,n)$ is presented as a unital $K$-algebra with generators $x_{ij},x_{ij}^{\ast}$ where $1\leq i\leq m,1\leq j\leq n$, and relations

[TABLE]

Leavitt also proved that $L_{K}(m,n)$ is simple if and only if $m=1$, and that $L_{K}(m,n)$ is a domain for all $m>1$.

In a seemingly unrelated development, J. Cuntz constructed and studied in [11] and [12], a class of $C^{\ast}$-algebras, now known as Cuntz algebras $\mathcal{O}_{n}$, which are generated by $n$ isometries $S_{1},S_{2},\dots,S_{n}$ such that $\sum_{i=1}^{n}S_{i}S_{i}^{\ast}=1$. It turns out that the Leavitt algebra $L_{\mathbb{C}}(1,n)$ of type $(1,n)$ is a dense $\ast$-subalgebra of the Cuntz algebra $\mathcal{O}_{n}$. Brown [10], and McClanahan [24, 25], studied $C^{\ast}$-algebras $U_{(m,n)}^{nc}$ which are the $C^{\ast}$-analogs of $L_{K}(m,n)$.

Cuntz and Krieger [13] generalized the construction of $\mathcal{O}_{n}$ by considering a class of $C^{\ast}$-algebras, known as Cuntz-Krieger algebras associated to an $n\times n$ matrix with entries in $\{0,1\}$. In a subsequent development, in [19], Cuntz-Krieger algebras were realized as special cases of a broader class of $C^{\ast}$-algebras arising from directed graphs called graph $C^{\ast}$-algebras. The interested reader is referred to [30] for further information on this important class of $C^{\ast}$-algebras.

Algebraic analogs of graph $C^{\ast}$-algebras were defined and studied for row-finite graphs independently in [3] and [5], and for arbitrary graphs in [1] and [31], under the name Leavitt path algebras $L_{K}(E)$. These algebras generalize Leavitt algebras $L_{K}(1,n)$ in a similar way as the graph $C^{\ast}$-algebras generalize Cuntz algebras $\mathcal{O}_{n}$. The interested reader may consult the book [2] and the references provided there.

In [4] and [7], Ara and Goodearl initiated the study of a much larger class of algebras called Cohn-Leavitt path algebras and their $C^{\ast}$-analogs based on the concept of separated graphs $(E,C)$. The Cohn-Leavitt path algebras extend the existing version of Leavitt path algebras and graph $C^{\ast}$-algebras for a particular choice of $C$. It was shown that any free product of algebras $L_{K}(1,n)$ appears as $L_{K}(E,C)$ for a suitable separated graph $(E,C)$. Also, for any $n\geq m\geq 1$, there is a separated graph $(E,C)$ such that $L_{K}(E,C)\cong M_{m+1}(L_{K}(m,n))\cong M_{n+1}(L_{K}(m,n))$ and the corner of $L_{K}(E,C)$ corresponding to one vertex of $E$ is isomorphic to $L_{K}(m,n)$ itself. Similarly, free products of Cuntz algebra $\mathcal{O}_{n}$ and matrix $C^{\ast}$-algebras $M_{m+1}(U_{(m,n)}^{\text{nc}})$ appear as $C^{\ast}(E,C)$ for suitable separated graphs $(E,C)$. However, their motivation to study Cohn-Leavitt path algebras of separated graphs was of $K$-theoretic nature and towards developing techniques to answer realization problem for von Neumann regular rings (cf. [6]). Using Bergman’s machinery [8], they describe the $\mathcal{V}$-monoid of $L_{K}(E,C)$ and study the lattice of trace ideals of Cohn-Leavitt path algebras in terms of separated graph theoretic data.

Independently, in [14], Hazrat defined the concept of weighted Leavitt path algebras $L_{K}(E,w)$ of weighted graphs as a graph theoretic generalization of Leavitt algebras of type $(m,n)$ for any natural numbers $m,n$. The Gröbner–Shirshov bases of $L_{K}(E,w)$ (called normal forms) were found, and as an application, the characterization of $L_{K}(E,w)$ which are domains was studied in [15]. As another application of normal forms, Gelfand-Kirillov dimensions of weighted Leavitt path algebras were studied in [28]. However, Cohn-Leavitt path algebras and weighted Leavitt path algebras are not special cases of each other.

In [29], Raimund defined and studied Leavitt path algebras associated to hypergraphs. He showed that Leavitt path algebras of separated graphs and vertex weighted Leavitt path algebras of row-finite vertex-weighted graphs are examples of hypergraphs.

In this article, we initiate the study of bi-separated graphs $\dot{E}=(E,(C,S),(D,T))$ and the associated Cohn-Leavitt path algebras $\mathcal{A}_{K}(\dot{E})$. With a particular choice of $(C,S)$ and/or $(D,T)$, we show that the Leavitt path algebras, Cohn-Leavitt path algebras of separated graphs, weighted Leavitt path algebras and Leavitt path algebras of hypergraphs are special cases of $\mathcal{A}_{K}(\dot{E})$.

In section 2 we recall some preliminaries on graphs and their Leavitt path algebras. We also recall the definitions of various generalizations of Leavitt path algebras and some of their properties. The part I of this paper begins with section 3, where we define the Cohn-Leavitt path algebras of bi-separated graphs and state some very basic results that follow from the definitions. We also show how the various generalizations of Leavitt path algebras introduced in the previous section are special cases of $\mathcal{A}_{K}(\dot{E})$. In section 4, we define the category BSG of bi-separated graphs and show that every object in this category is a direct limit of countable complete sub-objects (see Proposition 4.7). However, this statement does not hold if we replace countable by finite. We then define a new sub-category of BSG, which we call “tame category tBSG” and show that this category characterizes all objects of BSG which are direct limits of finite complete sub-objects. Section 5 deals with computation of normal forms of $\mathcal{A}_{K}(\dot{E})$ using Bergman’s diamond lemma and some of their applications. In particular we find bi-separated graph theoretic conditions to study algebraic properties of Cohn-Leavitt path algebras such as simplicity, semiprimitivity, von Neumann regularity, finiteness etc and also characterize the algebras which are domains.

In part II of this paper we focus our attention to the study of B-hypergraphs. In section 6 we define B-hypergraphs $(\dot{E},\Lambda)$ and their $H$-monoids and we show that $H$-monoids are isomorphic to the $\mathcal{V}$-monoids of the corresponding Cohn-Leavitt path algebras. In section 7, we introduce the partially ordered set of admissible triples AT$(\dot{E},\Lambda)$ for each B-hypergraph $(\dot{E},\Lambda)$ and show that this poset is a lattice. We further show that the lattice of order-ideals of $H$-monoid of $(\dot{E},\Lambda)$ is isomorphic to the lattice AT$(\dot{E},\Lambda)$, which establishes that AT$(\dot{E},\Lambda)$ is isomorphic to the complete lattice of trace ideals of Cohn-Leavitt path algebra of $(\dot{E},\Lambda)$. In section 8 we study the representations of Leavitt path algebras of regular hypergraphs and show that the category of unital right modules of these algebras is a full subcategory and a retract of quiver representations of underlying graphs of the hypergraphs. Also, we give a characterization of Leavitt path algebras of regular hypergraphs having a finite dimensional representation in terms of their $H$-monoids. Finally in section 9 we provide a matrix criteria for a Leavitt path algebra of a finite hypergraph having invariant basis number.

Notation 1.1**.**

Throughout this paper, $K$ denotes a fixed field; $\mathbb{Z}$ denotes the set of integers; $\mathbb{Z}^{+}$ denotes the set of non-negative integers; $\mathbb{N}$ denotes the set of positive integers. $\delta$ is Kronecker delta (i.e $\delta_{ij}=0$ if $i\neq j$ and $\delta_{ij}=1$ if $i=j$). By a ring (resp. $K$-algebra) we mean an associative (not necessarily commutative or unital) ring (resp. $K$-algebra).

2. Preliminaries

In this section we recall some preliminary definitions and propositions and fix some conventions which will be used throughout the article.

A graph (or * quiver*) $E=(E^{0},E^{1},r,s)$ consists of two sets $E^{0},E^{1}$ called the set of vertices and the set of edges respectively, and two functions $r,s:E^{1}\rightarrow E^{0}$ called the range map and the source map respectively. We place no restriction on the cardinalities of $E^{0}$ and $E^{1}$ or on the properties of the functions $r$ and $s$. We say a graph is finite if both $E^{0}$ and $E^{1}$ are finite. A vertex $v\in E^{0}$ is called a source (resp. sink) if $r^{-1}(v)=\emptyset$ (resp. $s^{-1}(v)=\emptyset$).

A subgraph $F=(F^{0},F^{1},r_{F},s_{F})$ of $E=(E^{0},E^{1},r_{E},s_{E})$ is defined by $F^{0}\subseteq E^{0}$, $F^{1}\subseteq E^{1}$ and $r_{F}$ is the restriction of $r_{E}$ on $F^{1}$ and $s_{F}$ is the restriction of $s_{E}$ on $F^{1}$. Let $V$ be a subset of $E^{0}$. The induced subgraph on $V$ is the subgraph $E_{V}=(V,E_{V}^{1},r_{V},s_{V})$ such that $E_{V}^{1}:=s^{-1}(V)\cap r^{-1}(V)$, $r_{V}$ and $s_{V}$ are restrictions of $r_{E}$ and $s_{E}$ on $E_{V}^{1}$ respectively. A subgraph is full if it is induced on its set of vertices.

A graph morphism

[TABLE]

is a pair of maps $\phi^{0}:F^{0}\rightarrow E^{0}$ and $\phi^{1}:F^{1}\rightarrow E^{1}$ such that $r_{E}(\phi^{1}(e))=\phi^{0}(r_{F}(e))$ and $s_{E}(\phi^{1}(e))=\phi^{0}(s_{F}(e))$, for every $e\in F^{1}$.

We denote the category of graphs along with graph morphisms by Gra. Given a family of graphs $\{E_{i}\}_{i\in I}$ in Gra, we define their disjoint union $\bigsqcup\limits_{i\in I}E_{i}$ to be the graph whose vertex set is $\bigsqcup\limits_{i\in I}E_{i}^{0}$, edge set is $\bigsqcup\limits_{i\in I}E_{i}^{1}$, and the source and range maps are trivial extensions of $s_{i}$ and $r_{i}$ respectively for all $i\in I$.

A path $\mu$ in a graph $E$ is either a vertex $v\in E^{0}$ or a finite sequence of edges $\mu=e_{1}e_{2}\dots e_{n}$ such that $r(e_{i})=s(e_{i+1})$, for $i=1,\dots,n-1$. The set of all paths in $E$ is denoted by $E^{\star}$. We define the length function $l:E^{\star}\rightarrow\mathbb{Z}^{+}$ by

[TABLE]

We denote the set of all paths in $E$ of length $n$ by $E^{n}$, and hence $E^{\star}=\bigcup_{n\geq 0}E^{n}$. The source and range functions $s,r$ can be extended to $E^{\star}$ as follows:

[TABLE]

[TABLE]

The (free) path category $\mathcal{C}_{E}$ of a graph $E$ is the small category with Ob($\mathcal{C}_{E}):=E^{0}$ and for $v,w\in E^{0}$, Mor$(v,w):=\{\mu\in E^{\star}\mid s(\mu)=v,r(\mu)=w\}$. In other words, the elements of $\mathcal{C}_{E}$ are paths in $E$ and the partial multiplication is defined by path concatenation. The path [math]-semigroup $S^{0}(E)$ of a graph $E$ is the set $E^{\star}\sqcup\{0\}$ along with multiplication defined by extension of partial multiplication of $\mathcal{C}_{E}$ by [math]. That is

[TABLE]

Definition 2.1**.**

Let $E$ be a graph. The Path $K$-algebra of $E$, denoted by $K(E)$, is defined to be the quotient of the free associative $K$-algebra generated by $E^{0}\cup E^{1}$ modulo the following relations:

1. (1)

$vw=\delta_{vw}v$, for all $v,w\in E^{0}$, 2. (2)

$s(e)e=er(e)=e$, for all $e\in E^{1}$.

In other words, the path $K$-algebra of $E$ is obtained as the contracted $K$-algebra of the graph [math]-semigroup $S^{0}(E)$ (i.e, the zero of $K(E)$ and $S^{0}(E)$ are identified). The following proposition follows from an application of Bergman’s diamond lemma [9].

Proposition 2.2**.**

Let $E$ be a graph. Then $E^{\star}$ is a linear $K$-basis for $K(E)$.

We recall that an associative ring $R$ is said to have a set of local units $U$ if $U$ is a set of idempotents in $R$ having the property that, for each finite subset $r_{1},\ldots,r_{n}$ of $R$, there exists a $u\in U$ for which $ur_{i}u=r_{i}$, for $1\leq i\leq n$. Also, an associative ring $R$ is said to have enough idempotents if there exists a set of nonzero orthogonal idempotents $I$ in $R$ for which the set $F$ of finite sums of distinct elements of $I$ is a set of local units for $R$. We denote a ring with ring with enough idempotents by $(R,I)$. For any graph $E$ note that $(K(E),E^{0})$ is a $K$-algebra with enough idempotents. Moreover, $K(E)$ is unital if and only if $E^{0}$ is finite in which case $\sum_{v\in E^{0}}v$ is the unit.

By K-Alg we mean the category whose objects are $K$-algebras with enough idempotents and whose morphisms are $K$-algebra morphisms which map local units to local units. We note that $K(\_)$ is not a functor from Gra to K-Alg. This is because a graph morphism $\phi:F\rightarrow E$ can map two distinct vertices $v,w\in F^{0}$ to a same vertex in $E$, in which case $K(\phi)(vw)\neq 0$ in $K(E)$, but $vw=0$ in $K(F)$. However, if Gr denotes the category whose objects are graphs and morphisms are graph morphisms $\phi=(\phi^{0},\phi^{1})$ such that $\phi^{0}$ is injective. Then it is easy to verify that $K(\_)$ is a continuous functor from Gr to K-Alg (i.e, $K(\_)$ maps direct limits to direct limits).

Definition 2.3**.**

Given a graph $E$, the double of $E$, denoted by $\widehat{E}$, is defined to be the graph $(E^{0},E^{1}\sqcup\overline{E^{1}},\widehat{r},\widehat{s})$, where $\overline{E^{1}}=\{e^{\ast}\mid e\in E^{1}\}$, and the functions $\widehat{r}$ and $\widehat{s}$ are such that

$\widehat{r}|_{E^{1}}=r,\widehat{s}|_{E^{1}}=s$, $\widehat{r}(e^{\ast})=s(e)$ and $\widehat{s}(e^{\ast})=r(e)$, for all $e\in E^{1}$.

A path $\mu\in\widehat{E}^{\star}$ is called a generalized path. We say a graph $E$ is connected if its double $\widehat{E}$ is connected. That is, for any $v,w\in E^{0}$, there is a generalized path $\mu\in\widehat{E}^{\star}$ such that $s(\mu)=v$ and $r(\mu)=w$. The connected components of $E$ are the graphs $\{E_{j}\}_{j\in J}$ such that $E=\bigsqcup\limits_{j\in J}E_{j}$, where every $E_{j}$ is connected.

A $K$-algebra $R$ is called a $G$-graded algebra if $R=\bigoplus\limits_{g\in G}R_{g}$, where $G$ is a group and each $R_{g}$ is a $K$-subspace of $R$ and $R_{g}R_{h}\subseteq R_{gh}$ for all $g,h\in G$. The set $R^{H}=\bigcup_{g\in G}R_{g}$ is called the set of homogeneous elements of $R$. $R_{g}$ is called the $g$-component of $R$ and the nonzero elements of $R_{g}$ are called homogeneous of degree $g$. We write deg$(r)=g$ if $r\in A_{g}-\{0\}$. Note that for any graph $E$ we can define $\mathbb{Z}$-grading on $K(\widehat{E})$ by defining deg($v)=0$ for every $v\in E^{0}$, deg($e)=1$ and deg($e^{\ast})=-1$.

A $\ast$-ring is an associative unital ring $R$ with an antiautomorphism $\ast:R\rightarrow R$ that is also an involution. That is, $\ast$ satisfies the following properties: for every $x,y\in R$, $(x+y)^{\ast}=x^{\ast}+y^{\ast}$, $(xy)^{\ast}=y^{\ast}x^{\ast}$, $(x^{\ast})^{\ast}=x$ and $1^{\ast}=1$. Let $K$ be a $\ast$-field with involution $\overline{\phantom{A}}:K\rightarrow K$. A $\ast$-algebra is a $K$-algebra $R$ that is also a $\ast$-algebra such that $(kr)^{\ast}=\overline{k}r^{\ast}$ for every $k\in K$ and $r\in R$. Note that for any graph $E$ and a $\ast$-field $K$ with involution $\overline{\phantom{A}}:K\rightarrow K$, $K(\widehat{E})$ is a $\ast$-algebra with respect to the involution defined (on the generators) by $(v)\mapsto v$ for every $v\in E^{0}$, $e\mapsto e^{\ast}$ for every $e\in E^{1}$ and $e^{\ast}\mapsto e$ for each $e^{\ast}\in\overline{E^{1}}$.

2.1. Leavitt path algebras and their generalizations

In this subsection we recall the definitions of Leavitt path algebras and their various generalizations.

Recall that a vertex $v\in E^{0}$ is called regular if $0<|s^{-1}(v)|<\infty$. The set of all row regular vertices in $E$ is denoted by Reg$(E)$.

Definition 2.4** (Cohn-Leavitt path algebras of graphs).**

Let $E$ be a graph and $S$ be any subset of row regular vertices Reg$(E)$ of $E$. The Cohn-Leavitt path algebra $CL_{K}^{S}(E)$ of $E$ relative to $S$ is obtained from $P_{K}(\widehat{E})$ by imposing the following (Cuntz-Krieger) relations:

1. CK1:

$\quad e^{\ast}f=\delta_{e,f}r(e)$ for every $e,f\in E^{1}$, 2. CK2:

$\quad v=\sum\limits_{e\in s^{-1}(v)}ee^{\ast}$ for every $v\in S$.

In particular, $CL_{K}^{\emptyset}(E)$ is called Cohn path algebra of $E$ and denoted by $C_{K}(E)$. The algebra $CL_{K}^{\text{Reg}(E)}(E)$ is called Leavitt path algebra of $E$ and denoted by $L_{K}(E)$.

Definition 2.5** (Cohn-Leavitt path algebras of separated graphs).**

A separated graph is a pair $(E,C)$, where $E$ is a graph and $C=\bigsqcup_{v\in E^{0}}C_{v}$, $C_{v}$ being a partition of $s^{-1}(v)$, for each vertex $v\in E^{0}$. Let $C_{\text{fin}}=\{X\in C\mid|X|<\infty\}$ and $S\subset C_{\text{fin}}$. Then the Cohn-Leavitt path algebra $CL_{K}(E,C,S)$ of $(E,C)$ relative to $S$ is defined as the quotient of $P_{K}(\widehat{E})$ obtained by imposing the following relations:

1. SCK1:

$\quad e^{\ast}f=\delta_{e,f}r(e)$, for all $e,f\in X,X\in C$, 2. SCK2:

$\quad v=\sum_{e\in X}ee^{\ast}$, for every $X\in S,v\in E^{0}$.

If $S=C_{\text{fin}}$, the Cohn-Leavitt path algebra is simply called Leavitt path algebra of $(E,C)$ and denoted by $L_{K}(E,C)$.

Definition 2.6** (Weighted Leavitt path algebras).**

Let $E$ be a row-finite graph and $w:E^{1}\rightarrow\mathbb{N}$ be a (weight) function. We define the (associated) weighted graph $E_{w}$ to be a graph $(E_{w}^{0},E_{w}^{1},r_{w},s_{w})$, where $E_{w}^{0}=E^{0}$, $E_{w}^{1}=\{e_{i}\mid e\in E^{1},1\leq i\leq w(e)\}$, $r_{w}(e_{i})=r(e)$ and $s_{w}(e_{i})=s(e)$, for all $e\in E^{1},1\leq i\leq w(e)$. Set $w(v)=\max\{w(e)\mid e\in s^{-1}(v)\}$. (Note that $w(v)$ is well-defined since $E$ is row-finite). The weighted Leavitt path algebra $WL_{K}(E_{w})$ of the weighted graph $E_{w}$ is the quotient of $P_{K}(\widehat{E}_{w})$ obtained by going modulo the following relations:

1. wCK1:

$\quad\quad\sum\limits_{1\leq i\leq w(v)}e_{i}^{\ast}f_{i}=\delta_{e,f}r(e)$, for every $e,f\in E^{1}$ and $s(e)=s(f)=v\in E^{0}$, 2. wCK2:

$\quad\quad\sum\limits_{e\in s^{-1}(v)}e_{i}e_{j}^{\ast}=\delta_{i,j}v$, for each $v\in E^{0},1\leq i,j\leq w(v)$,

where we set $e_{i}$ and $e_{i}^{\ast}$ to be zero whenever $i>w(e)$.

Definition 2.7** (Leavitt path algebras of Hypergraphs).**

Let $I$ and $X$ be sets. Recall that a function $x:I\rightarrow X$, given by $i\mapsto x_{i}=x(i)$ is called a family of elements in $X$ indexed by $I$. We denote a family $x$ of elements in $X$ indexed by $I$ by $(x_{i})_{i\in I}$.

A hypergraph is a quadruple $\mathcal{H}=(\mathcal{H}^{0},\mathcal{H}^{1},s,r)$ where $\mathcal{H}^{0}$ and $\mathcal{H}^{1}$ are sets called the set of vertices and the set of hyperedges respectively. For each $h\in\mathcal{H}^{1}$ there exists a pair of non-empty indexing sets $I_{h},J_{h}$ such that $s(h):I_{h}\rightarrow\mathcal{H}^{0}$, and $r(h):J_{h}\rightarrow\mathcal{H}^{0}$ are families of vertices.

Let $\mathcal{H}$ be a hypergraph. A hyperedge $h\in\mathcal{H}^{1}$ is called source regular (resp. range regular) if $I_{h}$ is finite (resp. $J_{h}$ is finite). The set of all source regular hyperedges of $\mathcal{H}$ is denoted by $\mathcal{H}_{\text{sreg}}^{1}$ and the set of all range regular hyperedges of $\mathcal{H}$ is denoted by $\mathcal{H}_{\text{rreg}}^{1}$. The hypergraph $\mathcal{H}$ is said to be regular if $\mathcal{H}^{1}=\mathcal{H}_{\text{sreg}}^{1}=\mathcal{H}_{\text{sreg}}^{1}$.

The Leavitt path algebra $L_{K}(\mathcal{H})$ of the hypergraph $\mathcal{H}$ is the $K$-algebra presented by the generating set $\{v,h_{ij},h_{ij}^{\ast}\mid v\in\mathcal{H}^{0},h\in\mathcal{H}^{1},i\in I_{h},j\in J_{h}\}$ and the relations

1. (1)

$uv=\delta_{u,v}u$, for every $u,v\in\mathcal{H}^{0}$, 2. (2)

$s(h)_{i}h_{ij}=h_{ij}=h_{ij}r(h)_{j}$ and $r(h)_{j}h_{ij}^{\ast}=h_{ij}^{\ast}=h_{ij}^{\ast}s(h)_{i}$, for every $h\in\mathcal{H}^{1}$, $i\in I_{h}$, and $j\in J_{h}$ 3. (3)

$\sum\limits_{j\in J_{h}}h_{ij}h_{kj}^{\ast}=\delta_{ik}s(h)_{i}$, for every $h\in\mathcal{H}_{\text{sreg}}^{1}$ and $i,k\in I_{h}$, 4. (4)

$\sum\limits_{i\in I_{h}}h_{ij}^{\ast}h_{ik}=\delta_{jk}r(h)_{j}$, for every $h\in\mathcal{H}_{\text{rreg}}^{1}$ and $1\leq j,k\in J_{h}$.

Remark 2.8*.*

Let $L$ denote any one of the $K$-algebras appearing in the definitions 2.4-2.7. Then note that $L$ satisty the following properties.

1. (1)

The algebra $L$ is unital if and only if the set of vertices $V$ in the underlying graph is finite. In this case, the unit is the sum of vertices. In general $(L,V)$ is a $K$-algebra with enough idempotents. 2. (2)

If $\overline{\phantom{A}}:K\rightarrow K$ is an involution on the field $K$, then $L$ is a $\ast$-algebra (with respect to the involution $\ast:L\rightarrow L$). 3. (3)

$L$ is a graded quotient algebra of $K(\widehat{E})$ with respect to standard $\mathbb{Z}$-grading given by length of paths.

PART I

3. Cohn-Leavitt path algebras of bi-separated graphs

Definition 3.1**.**

A bi-separated graph is a triple $\dot{E}=(E,C,D)$ such that

1. (1)

$E=(E^{0},E^{1},r,s)$ is a graph, 2. (2)

$C=\bigsqcup_{v\in E^{0}}C_{v}$, where $C_{v}$ is a partition of $s^{-1}(v)$ for every non-sink $v\in E^{0}$, 3. (3)

$D=\bigsqcup_{v\in E^{0}}D_{v}$, where $D_{v}$ is a partition of $r^{-1}(v)$ for every non-source $v\in E^{0}$, 4. (4)

$|X\cap Y|\leq 1$, for every $X\in C$ and $Y\in D$,

In the above definition, $C$ is called row-separation of $E$, $D$ is called column-separation of $E$ and $(C,D)$ is called bi-separation of $E$. The elements of $C$ are called rows and the elements of $D$ are called columns. Let $C_{\text{fin}}:=\{X\in C\mid|X|<\infty\}$ and $D_{\text{fin}}:=\{Y\in D\mid|Y|<\infty\}$. A bi-separated graph $\dot{E}$ is called finitely row-separated (resp. finitely column-separated) if $C=C_{\text{fin}}$ (resp. if $D=D_{\text{fin}}$) and is called finitely bi-separated if both $C=C_{\text{fin}}$ and $D=D_{\text{fin}}$.

In the above definition we follow the convention that if $S$ is a set, by a partition $P$ of $S$ we mean a family of pairwise disjoint nonempty subsets of $S$, whose union is $S$. For any non-empty set $S$, there always exist two trivial partitions: the partition $P_{1}$ on $S$, called the discrete partition, where every element of $P_{1}$ is a singleton and the partition $P_{S}$, called the full partition, where $S$ is the only element of $P_{S}$.

Example 3.2**.**

(Standard bi-separation of a simple graph) Let $E$ be a simple graph (that is, $e,f\in E^{1}$ such that $s(e)=s(f)$ and $r(e)=r(f)$ implies $e=f$. In other words, there are no multiedges between any two vertices). We can obtain a canonical bi-separation on $E$ by considering both $C_{v}$ and $D_{v}$ to be full partitions. In other words, by defining $C_{v}=\{s^{-1}(v)\}$ for every non-sink $v\in E^{0}$ and $D_{v}=\{r^{-1}(v)\}$ for every non-source $v\in E^{0}$. This bi-separation is called standard.

In the following examples, $E$ denotes an arbitrary graph.

Example 3.3**.**

(Trivial bi-separation of a graph) By trivial bi-separation on a graph $E$, we mean both $C_{v}$ and $D_{v}$ are discrete partitions.

Example 3.4**.**

(Cuntz-Krieger bi-separation of a graph) We can obtain another canonical bi-separation on $E$ by combining full row-separation and discrete column-separation on $E$ as follows: Consider $C_{v}=\{s^{-1}(v)\}$ for every non-sink $v\in E^{0}$ and $D_{v}=\{\{e\}\mid e\in r^{-1}(v)\}$ for every non-source $v\in E^{0}$.

Example 3.5**.**

(Separated graphs) A bi-separated graph $(E,C,D)$ in which the column-separation is discrete is called a row-separated graph or simply separated graph (cf. definition 2.5). Separated graphs are denoted by $(E,C)$.

Example 3.6**.**

(Weighted graphs) Let $E$ be a row-finite graph and $w:E^{1}\rightarrow\mathbb{N}$ be a weight map on $E$. Consider the weighted graph $E_{w}=(E_{w}^{0},E_{w}^{1},r_{w},s_{w})$. We associate a bi-separation on $E_{w}$ as follows: For every non-sink $v\in E^{0}$ and $1\leq i\leq w(v)$ define $X_{v}^{i}:=\{e_{i}\mid e\in s^{-1}(v)\}$. For every $v\in E^{0}$ and $e\in s^{-1}(v)$ define $Y_{v}^{e}:=\{e_{i}\mid 1\leq i\leq w(e)\}$. Now consider $C_{v}:=\{X_{v}^{i}\mid 1\leq i\leq w(v)\}$ and $D_{v}:=\bigsqcup_{w\in E^{0}}D_{vw}$ where $D_{vw}=\{Y_{v}^{e}\mid e\in s^{-1}(v)\cap r^{-1}(w)\}$. Here $C=C_{\text{fin}}$, since $E$ is row-finite and $D=D_{\text{fin}}$, since $w$ takes natural numbers as values.

Example 3.7**.**

(Hypergraphs) We show that any hypergraph $\mathcal{H}$ can be associated to bi-separated graph $\dot{E}=(E,C,D)$ as follows: Define $E=(\mathcal{H}^{0},E^{1},s^{\prime},r^{\prime})$, where $E^{1}=\{h_{ij}\mid h\in\mathcal{H}^{1},i\in I_{h},j\in J_{h}\}$, $s^{\prime}(h_{ij})=s(h)_{i}$ and $r^{\prime}(h_{ij})=r(h)_{j}$. For an arbitrary $h\in\mathcal{H}^{1}$, if $i\in I_{h}$ then $X_{h}^{i}:=\{h_{ij}\mid j\in J_{h}\}$ and if $j\in J_{h}$ then $Y_{h}^{j}:=\{h_{ij}\mid i\in I_{h}\}$. For $v\in E^{0}$, define $C_{v}=\{X_{h}^{i}\mid h\in\mathcal{H}^{1},v\in s(h)\}$ and $D_{v}=\{Y_{h}^{j}\mid h\in\mathcal{H}^{1},v\in r(h)\}$. By construction, $C=C_{\text{fin}}$ and $D=D_{\text{fin}}$.

Notation 3.8**.**

Given a bi-separated graph $\dot{E}=(E,C,D)$, the maps $s$ and $r$ can be extended to $C$ and $D$ respectively in well-defined manner as follows: For $X\in C$, define $s(X):=s(e)$ where $e\in X$ and for $Y\in D$, define $r(Y):=r(e)$ where $e\in Y$.

Also, for each $X\in C$ and $Y\in D$ we set

[TABLE]

We interchangeably use $XY$ and $X\cap Y$, wherever there is no cause for confusion.

Definition 3.9**.**

Let $\dot{E}=(E,C,D)$ be a bi-separated graph. The Leavitt path algebra of $\dot{E}$ with coefficients over $K$, denoted by $L_{K}(\dot{E})$, is the quotient of $K(\widehat{E})$ obtained by imposing the following relations:

1. $L1$:

for every $X,X^{\prime}\in C_{\text{fin}}$,

[TABLE] 2. $L2$:

for every $Y,Y^{\prime}\in D_{\text{fin}}$,

[TABLE]

Example 3.10**.**

(Leavitt path algebra of a standard bi-separated simple graph)

Let $E$ be a simple graph and consider the standard bi-separation $(C,D)$ on $E$. Let the set of all non-sinks of $E$ be denoted by $E_{\alpha}$ and the set of all non-sources be denoted by $E_{\omega}$. Then $|C|=|E_{\alpha}|$ and $|D|=|E_{\omega}|$. Recall that a vertex $v\in E^{0}$ is called row-regular if $0<|s^{-1}(v)|<\infty$ and $w\in E^{0}$ is called column-regular if $0<|r^{-1}(w)|<\infty$. The set of all row-regular vertices is denoted by RReg($E$) and the set of all column regular vertices is denoted by CReg($E$). Note that $|C_{\text{fin}}|=|\text{RReg}(E)|$ and $|D_{\text{fin}}|=|\text{CReg}(E)|$.

Let $A$ be a $|C_{\text{fin}}|\times|D|$ matrix over $K(\widehat{E})$ with entries

[TABLE]

where $v\in\text{RReg}(E)$ and $w\in E_{\omega}$. Let $A^{\ast}$ denote the ‘adjoint transpose’ of $A$. Similiarly, let $B$ be a $|C|\times|D_{\text{fin}}|$ matrix over $K(\widehat{E})$ with entries

[TABLE]

where $v\in E_{\alpha}$ and $w\in\text{CReg}(E)$. Let $B^{\ast}$ denote the ‘adjoint transpose’ of $B$.

Then the defining relations L1 and L2 of Leavitt path algebras are obtained by imposing the following matrix relations:

[TABLE]

where $V$ is the $|\text{RReg}(E)|\times|\text{RReg}(E)|$ diagonal matrix with diagonal entries $V(v,v)=s(v)$ and and $U$ is the $|\text{CReg}(E)|\times|\text{CReg}(E)|$ diagonal matrix with diagonal entries $U(w,w)=r(w)$.

In particular, if $E$ is finite simple graph, then the matrix $A$ (resp. the matrix $B$) is obtained from the adjacency matrix of $E$ by removing the zero rows (resp. zero columns) and replacing $1$’s with corresponding edges. We illustrate this with a few examples below.

1. (1)

For $n\geq 1$, let $\Sigma_{n}$ be the following line graph with $n$ vertices and $n-1$ edges:

\dots$$v_{1}$$v_{2}$$v_{3}$$v_{n-1}$$v_{n}$$e_{1}$$e_{2}$$e_{3}$$e_{n-1}$$\Sigma_{n}=

Then

[TABLE]

and the relations obtained are

[TABLE]

where $1\leq i\leq n-1$. In this case, it is easy to see that

[TABLE] 2. (2)

For $n\geq 3$, let $\Gamma_{n}$ denote the following graph with $n$ vertices:

\dots$$v_{1}$$v_{2}$$v_{3}$$v_{n-1}$$v_{n}$$\Gamma_{n}=

Then the adjacency matrix of $\Gamma_{n}$ has at least two entries and at most three entries in both rows and columns, i.e.,

[TABLE]

where $\ast$’s are nonzero entries filled by the corresponding edges. In this case, though, the explicit description of the Leavitt path algebra $L_{K}(\dot{\Gamma_{n}})$ is not known.

Example 3.11**.**

(Groupoid algebra of a free groupoid) Let $E$ be a graph and consider the trivial bi-separation $(C,D)$ on $E$. The defining relations of Leavitt path algebra of $(E,C,D)$ turns the free path category of $\widehat{E}$ into a free groupoid and hence $L_{K}(E,C,D)$ is the groupoid algebra of this free groupoid. In particular if $E$ has only one vertex then with respect to trivial bi-separation, the Leavitt path algebra is the group algebra of the free group with generators $E^{1}$ (Here we identified the vertex with the group identity).

Example 3.12**.**

(Leavitt path algebra of a graph) Let $E$ be any graph and $\dot{E}$ be the associated bi-separated graph with respect to Cuntz-Kreiger bi-separation on $E$. Then we have $L_{K}(\dot{E})\cong L_{K}(E)$.

Example 3.13**.**

(Leavitt path algebra of a separated graph) Let $\dot{E}=(E,C)$ be a (row) separated graph. Then it is direct that $L_{K}(\dot{E})\cong L_{K}(E,C)$.

Example 3.14**.**

(Weighted Leavitt path algebra of a weighted graph) Let $E$ be a row-finite graph and $w:E^{1}\rightarrow\mathbb{N}$ be a weight map. Consider $\dot{E}=(E,C,D)$ where $(C,D)$ is the weighted bi-separation on $E$ as in example 3.6. Then it is immediate that $L_{K}(\dot{E})\cong WL_{K}(E_{w})$.

It has been noted in [7, page 171] that neither weighted Leavitt path algebras nor Leavitt path algebras of separated graphs are particular cases of the each other. One can mix the above two examples and construct new algebras as follows:

Example 3.15** (Weighted Cohn-Leavitt path algebras of finitely separated graphs).**

Let $(E,C)$ be a finitely row-separated graph (i.e. a separated graph in which $C=C_{\text{fin}}$). Let $w:E^{1}\rightarrow\mathbb{N}$ be a function and $E_{w}$ be the associated weighted graph. For $X\in C$, set $w(X)=\max\{w(e)\mid e\in X\}$. The weighted Cohn-Leavitt path algebra $CL_{K}(E_{w},C)$ of $(E_{w},C)$ can be defined as the quotient of $P_{K}(\widehat{E}_{w})$ by factoring out the following relations:

1. wSCK1:

$\quad\sum\limits_{1\leq i\leq w(X)}e_{i}^{\ast}f_{i}=\delta_{e,f}r(e),$ for every $e,f\in X$ and $X\in C$, 2. wSCK2:

$\quad\sum\limits_{e\in X}e_{i}e_{j}^{\ast}=\delta_{i,j}s(e),$ for each $X\in C,1\leq i,j\leq w(X)$,

where we set $e_{i}$ and $e_{i}^{\ast}$ to be zero whenever $i>w(e)$.

Given a weighted finitely separated graph $(E_{w},C)$, we get a canonical bi-separated graph as follows: For $X\in C$ and $1\leq i\leq w(X)$, define $\widetilde{X}^{i}=\{e_{i}\mid e\in X\}$ and set $\widetilde{C}_{v}=\{\widetilde{X}^{i}\mid X\in C_{v}~{}\text{and}~{}1\leq i\leq w(X)\}$. Here $\widetilde{C}=\widetilde{C}_{\text{fin}}$, since $E$ is finitely separated. Now, for $e\in X$, define $\widetilde{Y}_{X}^{e}=\{e_{i}\mid 1\leq i\leq w(e)\}$ and $\widetilde{D}_{vw}=\{\widetilde{Y}_{X}^{e}\mid e\in X\}$. Observe that $\widetilde{D}=\widetilde{D}_{\text{fin}}$, since $w$ is natural number valued. Now setting $\dot{E}=(E,\widetilde{C},\widetilde{D})$, we immediately get

[TABLE]

Example 3.16** (Leavitt path algebra of a hypergraph).**

Given any hypergraph $\mathcal{H}$, consider the associated bi-separated graph $\dot{E_{H}}$ as in example 3.7. Then we have $L_{K}(\dot{E})\cong L_{K}(\mathcal{H})$.

Definition 3.17**.**

Let $\dot{E}=(E,C,D)$ be bi-separated graph. Let $S\subseteq C_{\text{fin}}$ and $T\subseteq D_{\text{fin}}$ be two distinguished sets. The Cohn-Leavitt path algebra of $\dot{E}$ with coefficients over $K$ relative to $(S,T)$, denoted by $\mathcal{A}_{K}(E,(C,S),(D,T))$, is the quotient of $K(\widehat{E})$ obtained by imposing the following relations:

1. $\mathcal{A}1$:

for every $X,X^{\prime}\in S$,

[TABLE] 2. $\mathcal{A}2$:

for every $Y,Y^{\prime}\in T$,

[TABLE]

For notational convenience we denote the bi-separated graph with given distinguished subsets as in the above definition as a 5-tuple $\dot{E}=(E,(C,S),(D,T))$ and again call it bi-separated graph if there is no confusion and denote the Cohn-Leavitt path algebra also as $\mathcal{A}_{K}(\dot{E})$. Whenever we want to distinguish the case that $S=C_{\text{fin}}$ and $T=D_{\text{fin}}$ we simply call the Cohn-Leavitt path algebra as Leavitt path algebra.

Proposition 3.18** (Universal property of $\mathcal{A}_{K}(\dot{E})$).**

Let $\dot{E}=(E,(C,S),(D,T))$ be a bi-separated graph. Suppose $\mathfrak{A}$ is a $K$-algebra which contains a set of pairwise orthogonal idempotents $\{A_{v}\mid v\in E^{0}\}$, two sets $\{A_{e}\mid e\in E^{1}\}$, $\{B_{e}\mid e\in E^{1}\}$ for which the following hold.

1. (1)

$A_{s(e)}A_{e}=A_{e}A_{r(e)}=A_{e}$, and $A_{r(e)}B_{e}=B_{e}A_{s(e)}=B_{e}$ for all $e\in E^{1}$. 2. (2)

for every $X,X^{\prime}\in S$, $\sum\limits_{Y\in D}A_{XY}B_{YX^{\prime}}=\delta_{XX^{\prime}}A_{s(X)}$, 3. (3)

for every $Y,Y^{\prime}\in T$, $\sum\limits_{X\in C}B_{YX}A_{XY^{\prime}}=\delta_{YY^{\prime}}A_{r(X)}$

then there exists a unique map $\psi:\mathcal{A}_{K}(\dot{E})\rightarrow\mathfrak{A}$ such that $\psi(v)=A_{v}$, $\psi(e)=A_{e}$, and $\psi(e^{\ast})=B_{e}$ for all $v\in E^{0}$ and $e\in E^{1}$.

Example 3.19** (Cohn-Leavitt path algebra of a graph).**

Let $E$ be a graph and let $S\subseteq\text{RReg}(E)$. Then the Cohn-Leavitt path algebra $CL_{K}^{S}(E)$ of $E$ can be realized as Cohn-Leavitt path algebra $\mathcal{A}_{K}(\dot{E})$ of the bi-separated graph $\dot{E}=(E,(C,S),(D,T))$, where $(C,D)$ is the Cuntz-Krieger bi-separation on $E$, RReg$(E)=C_{\text{fin}}$ and $T=D$.

Example 3.20**.**

(Cohn-Leavitt path algebra of a separated graph) Let $(E,C)$ be a separated graph and $S\subseteq C_{\text{fin}}$. Set $\dot{E}=(E,(C,S),(D,T))$, where $T=D_{\text{fin}}=D$. Then from definition it is clear that $\mathcal{A}_{K}(\dot{E})\cong CL_{K}(E,C,S)$.

We say a bi-separated graph $\dot{E}=(E,(C,S),(D,T))$ is connected if the underlying graph $E$ is connected. Because of the following proposition, we assume henceforth that every bi-separated graph is connected.

Proposition 3.21**.**

Let $\dot{E}$ be a bi-separated graph. Suppose $\dot{E}=\bigsqcup\limits_{j\in J}E_{j}$ is a decomposition of $\dot{E}$ into its connected components. Then $\mathcal{A}_{K}(\dot{E})\cong\bigoplus\limits_{j\in J}\mathcal{A}_{K}(\dot{E_{j}})$, where $\dot{E_{j}}$ is the bi-separated graph structure on $E_{j}$ induced by the bi-separated graph structure on $E$.

Proof.

Follows from universal property of $\mathcal{A}_{K}(\dot{E})$. ∎

Lemma 3.22**.**

Let $\dot{E}$ be a bi-separated graph.

1. (1)

The algebra $\mathcal{A}_{K}(\dot{E})$ is unital if and only if $E^{0}$ is finite. In this case,

[TABLE] 2. (2)

For each $\alpha\in\mathcal{A}_{K}(\dot{E})$, there exists a finite set of distinct vertices $V(\alpha)$ for which $\alpha=f\alpha f$, where $f=\sum\limits_{v\in V(\alpha)}v$. Moreover, the algebra $(\mathcal{A}_{K}(\dot{E}),E^{0})$ is a ring with enough idempotents. 3. (3)

Let $\overline{\phantom{A}}:K\rightarrow K$ be an involution on the field $K$. Then with respect to the involution $\ast:\mathcal{A}_{K}(\dot{E})\rightarrow\mathcal{A}_{K}(\dot{E})$, $\mathcal{A}_{K}(\dot{E})$ is a $\ast$-algebra. 4. (4)

$\mathcal{A}_{K}(\dot{E})$* is a graded quotient algebra of $K(\widehat{E})$ with respect to standard $\mathbb{Z}$-grading given by length of paths.*

Proof.

The proof follows on similiar lines of [2, Lemma 1.2.12]. ∎

Now, the linear extension of $\ast$ in paths induces a grade reversing involution. Hence for any $\dot{E}$, $\mathcal{A}_{K}(\dot{E})$ are $\mathbb{Z}$-graded $\ast$-algebras and the (graded) categories of left modules and right modules for any of Cohn-Leavitt path algebras of bi-separated graphs are equivalent.

Let $\Gamma$ be any group with identity $\varepsilon$ and we can consider a $\Gamma$-grading on $\mathcal{A}_{K}(\dot{E})$ with $E^{0}\sqcup E^{1}$ being homogeneous. Since $v^{2}=v$ for each $v\in E^{0}$, we are forced to define deg$(v)=\varepsilon$. Since the relations $\mathcal{A}1$ and $\mathcal{A}2$ are homogeneous we can conclude that all $e^{\ast}$ in $\overline{E^{1}}$ are also homogeneous with deg$(e^{\ast})$=(deg$(e))^{-1}$. As a result, any function from $E^{1}$ to $\Gamma$ defines a unique $\Gamma$-grading on $\mathcal{A}_{K}(\dot{E})$ with deg$(v)=\varepsilon$ for all $v\in E^{0}$ and deg$(e^{\ast})=$(deg$(e))^{-1}$. A refinement or a morphism from a $\Gamma$-grading to $\Gamma^{\prime}$-grading on an algebra $A$ is given by a group homomorphism $\phi:\Gamma\rightarrow\Gamma^{\prime}$ such that for all $\gamma^{\prime}\in\Gamma^{\prime}$, $A_{\gamma^{\prime}}=\bigoplus\limits_{\phi(\gamma)=\gamma^{\prime}}A_{\gamma}$ where $A_{\gamma}:=\{a\in A\mid\text{deg}_{\Gamma}(a)=\gamma\}\cup\{0\}$. There is a universal $\Gamma$-grading on $\mathcal{A}_{K}(\dot{E})$ which is a refinement of all others:

Proposition 3.23**.**

Let $\Gamma:=FE^{1}$, the free group on $E^{1}$ with identity $\varepsilon$. The $\Gamma$-grading defined by deg${}_{\Gamma}(v)=\varepsilon$ and deg${}_{\Gamma}(e)=e$ is an initial universal object in the category of group gradings of $\mathcal{A}_{K}(\dot{E})$ with $E^{0}\sqcup E^{1}$ being homogeneous.

Proof.

For any $\Gamma^{\prime}$-grading let $\phi:\Gamma\rightarrow\Gamma^{\prime}$ be the group homomorphism given by $\phi(e)=\text{deg}_{\Gamma^{\prime}}(e)$. ∎

4. The categories BSG and tBSG

In this section we introduce two categories BSG of bi-separated graphs and the category tBSG of tame bi-separated graphs. We show that the functor $\mathcal{A}_{K}(\underline{\hskip 7.0pt})$ from BSG to K-Alg is continuous. We also show that each object of tBSG is a direct limit of sub-objects based on finite graphs, from which we obtain every Cohn-Leavitt path algebra of tame bi-separated graph as a direct limit of unital Cohn-Leavitt path algebras.

Definition 4.1**.**

We define a category BSG of bi-separated graphs as follows: The objects of BSG are bi-separated graphs (with distinguished subsets) $\dot{E}=(E,(C,S),(D,T))$. A morphism $\phi:\dot{E}\rightarrow\dot{\widetilde{E}}$ in BSG is a triple $\phi=(\phi_{0},\phi_{1},\phi_{2})$ satisfying the following conditions:

1. (1)

$\phi_{0}:E\rightarrow\widetilde{E}$ is graph morphism such that $\phi_{0}^{0},\phi_{0}^{1}$ are injective. 2. (2)

$\phi_{1}:C\rightarrow\widetilde{C}$ is a map such that $X\in C_{v}\implies\phi_{1}(X)\in C_{\phi_{0}^{0}(v)}$. 3. (3)

$\phi_{1}(S)\subset\widetilde{S}$. Moreover $\left.\phi_{0}^{1}\right|_{X}:X\rightarrow\widetilde{X}$ is a bijection, for every $X\in S$. 4. (4)

$\phi_{2}:D\rightarrow\widetilde{D}$ is a map such that $Y\in D_{v}\implies\phi_{1}(Y)\in D_{\phi_{0}^{0}(v)}$. 5. (5)

$\phi_{2}(T)\subset\widetilde{T}$. Moreover $\left.\phi_{0}^{1}\right|_{Y}:Y\rightarrow\widetilde{Y}$ is a bijection, for every $Y\in T$. 6. (6)

If $X\in S,Y\in T$ then $X\cap Y=\emptyset\implies\widetilde{X}\cap\widetilde{Y}=\emptyset$.

Proposition 4.2**.**

The category BSG admits arbitrary direct limits.

Proof.

The proof is similar to [7, Proposition 3.3]. The only addition is that we have to define $D$ and $T$ analogous to the way we define $C$ and $S$.

∎

Recall that a functor is continuous if it preserves direct limits.

Proposition 4.3**.**

The assignment $\dot{E}\rightsquigarrow\mathcal{A}_{K}(\dot{E})$ extends to a continuous covariant functor $\mathcal{A}_{K}(\underline{\hskip 7.0pt}))$ from BSG to K-Alg.

Proof.

The proof is similar to [7, Proposition 3.6]. ∎

Definition 4.4**.**

We say a morphism $\phi:\dot{E}\rightarrow\dot{\widetilde{E}}$ in BSG is complete if $\phi_{1}^{-1}(\widetilde{S})=S$ and $\phi_{2}^{-1}(\widetilde{T})=T$.

Definition 4.5**.**

Let $\dot{E}$ be an object in BSG. A sub-object of $\dot{E}$ is an object $\dot{E^{\prime}}=(E^{\prime},(C^{\prime},S^{\prime}),(D^{\prime},T^{\prime}))$ such that $E^{\prime}$ is a sub-graph of $E$ and the following conditions hold:

[TABLE]

Definition 4.6**.**

Let $\dot{E}$ be an object in BSG. A complete sub-object of $\dot{E}$ is a sub-object $\dot{E^{\prime}}$ such that the inclusion is a complete morphism.

Proposition 4.7**.**

Any object in BSG is a direct limit of countable complete sub-objects.

Proof.

Let $\dot{E}$ be an object in BSG. For a finite subset $A\subset E^{0}\sqcup E^{1}$, let $E_{A}$ be the graph generated by $A$, i.e.,

$E_{A}^{1}=A\cap E^{1}$ and $E_{A}^{0}=(A\cap E^{0})\cup s_{E}^{-1}(E_{A}^{1})\cup r_{E}^{-1}(E_{A}^{1})$.

Take $v\in E_{A}^{0}$ and set

$\mathcal{E}_{0v}=s_{E}^{-1}(v)\cup\bigcup\limits_{\begin{subarray}{c}X\in S\cap C_{v}\\ X\cap A\neq\emptyset\end{subarray}}X\cup\bigcup\limits_{\begin{subarray}{c}Y\in T\cap D_{v}\\ Y\cap A\neq\emptyset\end{subarray}}Y$.

If the graph $\mathcal{E}_{0}$ generated by $E_{A}^{0}\sqcup\bigsqcup\limits_{v\in E_{A}^{0}}\mathcal{E}_{0v}$ is a complete sub-object of the given object, we are done. If not, define

$\mathcal{E}_{1v}$ to be $\mathcal{E}_{0v}\cup\bigcup\limits_{\begin{subarray}{c}X\in S\cap C_{v}\\ X\cap\mathcal{E}_{0v}\neq\emptyset\end{subarray}}X\cup\bigcup\limits_{\begin{subarray}{c}Y\in T\cap D_{v}\\ Y\cap\mathcal{E}_{0v}\neq\emptyset\end{subarray}}Y$.

If the graph $\mathcal{E}_{1}$ generated by $\mathcal{E}_{0}^{0}\sqcup\bigsqcup\limits_{v\in\mathcal{E}_{0}^{0}}\mathcal{E}_{1v}$ is a complete sub-object of the given object we are done. If not define $\mathcal{E}_{2v}$ similarly and continue this process.

Hence we have a chain $\mathcal{E}_{0}\rightarrow\mathcal{E}_{1}\rightarrow\mathcal{E}_{2}\rightarrow\dots$.

Let $\mathcal{E}$ be the direct limit of this chain, i.e., let $\dot{\mathcal{E}}=(\mathcal{E},(\mathcal{C},\mathcal{S}),(\mathcal{D},\mathcal{T}))$ be the direct limit of the directed system $\{\dot{\mathcal{E}_{i}}=(\mathcal{E}_{i},(\mathcal{C}_{i},\mathcal{S}_{i}),(\mathcal{D}_{i},\mathcal{T}_{i}))\}$, where $i\in\mathbb{N}\cup\{0\}$.

Observe that for $i\geq 0,~{}\mathcal{S}_{i}=\{X\in S\mid X\cap\mathcal{E}_{i-1}^{1}\neq\emptyset\}$ and $\mathcal{C}_{i}=\{X\in\mathcal{E}_{i}^{1}\mid X\in C\setminus S,X\cap\mathcal{E}_{i}^{1}\neq\emptyset\}\sqcup\mathcal{S}_{i}$, where $\mathcal{E}_{-1}$ means $A$. Similar statements hold for $\mathcal{D}_{i}$ and $\mathcal{T}_{i}$.

Since the vertex set and edge set of $\mathcal{E}$ are countable union of finite sets, it is a countable sub-graph of $E$. We claim that $\dot{\mathcal{E}}$ is a complete sub-object of $\dot{E}$. For, let $X\in S$ and $X\cap\mathcal{E}^{1}\neq\emptyset$. Then $X\cap\mathcal{E}_{i}^{1}\neq\emptyset$, for some $i\in\mathbb{N}\cup\{0\}$. This implies $X\in\mathcal{S}_{i+1}$, which in turn means $X\in\mathcal{S}$. Similarly one can start with $X\in C\setminus S$ and prove that whenever $X\cap\mathcal{E}^{1}\neq\emptyset$, the set $X\cap\mathcal{E}^{1}\in\mathcal{C}\setminus\mathcal{S}$. Repeating the analogous arguments for $T$, we can conclude that for each finite subset $A\subset E^{0}\sqcup E^{1}$, there exists a countable complete sub-object $\dot{\mathcal{E}}$ of $\dot{E}$. Now by keeping the set of all finite subsets of $E^{0}\sqcup E^{1}$ as the indexing set, we get a directed system of countable complete sub-objects whose direct limit is $\dot{E}$.

∎

We emphasize that a general object in BSG cannot be written as a direct limit of finite complete sub-objects as the following example illustrates:

Example 4.8**.**

Consider the following simple graph $\Gamma_{\infty}$ on countably infinite vertices.

\dots$$\dots$$v_{-1}$$v_{0}$$v_{1}$$\Gamma_{\infty}=

Observe that $\Gamma_{\infty}$ is a simple graph. Consider the standard bi-separation $(C,D)$ on $\Gamma_{\infty}$ and let $S=C_{\text{fin}}=C$ and $T=D_{\text{fin}}=D$. Then $C_{\infty}$ cannot be written as a direct limit of finite complete sub-objects. For, if there is a complete sub-object of $\Gamma_{\infty}$, then by definition we are forced to include all the edges and so, it will no more be finite.

4.1. The category of tame bi-separated graphs

Let $\dot{E}$ be an object in BSG. Set

[TABLE]

We define a relation $\sim_{T}$ on $S_{1}$ as follows: For $X,X^{\prime}\in S_{1}$, define $X\sim_{T}X^{\prime}$ if either $X=X^{\prime}$ or there exists a finite sequence $X_{0},Y_{1},X_{1},Y_{2},X_{2},$ $\ldots,Y_{n-1},X_{n-1},Y_{n},X_{n}$ such that for each $0\leq i\leq n$, $X_{i}\in S,Y_{i}\in T$, and $X_{0}=X$, $X_{n}=X^{\prime}$, $X_{i}\cap Y_{i+1}\neq\emptyset$ and $Y_{i}\cap X_{i}\neq\emptyset$. It is not hard to see that $\sim_{T}$ is an equivalence relation on $S_{1}$. Let $S_{1}=\bigsqcup\limits_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$ be the partition of $S_{1}$ induced by $\sim_{T}$.

Define $\sim_{S}$ on $T_{1}$ similarly and let $T_{1}=\bigsqcup\limits_{\lambda^{\prime}\in\Lambda^{\prime}}\mathcal{Y}_{\lambda^{\prime}}$ be the partition induced by $\sim_{S}$.

We claim that the indexing sets $\Lambda$ and $\Lambda^{\prime}$ are in bijection. To see this, start with $\lambda\in\Lambda$. Let $X\in\mathcal{X}_{\lambda}$ be an arbitrarily fixed element. This means, there exists a $\lambda^{\prime}\in\Lambda^{\prime}$ and $Y\in\mathcal{Y}_{\lambda^{\prime}}$ such that $X\cap Y\neq\emptyset$. If $X^{\prime}\neq X$ is another element of $\mathcal{X}_{\lambda}$, and if there is a $Y^{\prime}\in T$ such that $X^{\prime}\cap Y^{\prime}\neq\emptyset$, then $Y^{\prime}\sim_{S}Y$ because $X^{\prime}\sim_{T}X$. So $Y^{\prime}$ belongs to the same $\mathcal{Y}_{\lambda^{\prime}}$ as $Y$. Also if there is another element $Y_{1}\in T$ such that $X\cap Y_{1}\neq\emptyset$, then clearly $Y_{1}\sim_{S}Y$ and so $Y_{1}$ also lies in same $\mathcal{Y}_{\lambda^{\prime}}$. This implies that the map $\Lambda\rightarrow\Lambda^{\prime}$ defined by $\lambda\mapsto\lambda^{\prime}$ is well-defined. Similarly one can define a map $\Lambda^{\prime}\rightarrow\Lambda$. It is not hard to see that these maps are inverses of each other which proves the claim. Therefore, we have the following proposition:

Proposition 4.9**.**

Let $\dot{E}$ be an object in BSG and let $S_{1},T_{1}$ be as defined in equations 1 and 3 above. Then there exist canonical partitions $S_{1}=\bigsqcup\limits_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$ and $T_{1}=\bigsqcup\limits_{\lambda^{\prime}\in\Lambda^{\prime}}\mathcal{Y}_{\lambda^{\prime}}$ of $S_{1}$ and $T_{1}$ respectively such that the indexing sets $\Lambda$ and $\Lambda^{\prime}$ are bijective.

Remark 4.10*.*

Because of the above proposition, we will denote the indexing sets of the canonical partitions of both $S_{1}$ and $T_{1}$ by $\Lambda$.

Definition 4.11**.**

A bi-separated graph $\dot{E}$ is called tame if $|\mathcal{X}_{\lambda}|<\infty$ and $|\mathcal{Y}_{\lambda}|<\infty$, for each $\lambda\in\Lambda$. The tame bi-separated graphs along with complete morphisms form a category which we call a tame (sub)category of bi-separated graphs. It will be denoted by tBSG.

Note that any finite bi-separated graph is tame. Also, the class of bi-separated graphs in examples 3.4, 3.5, 3.6, and 3.7 are all tame.

Proposition 4.12**.**

Let $\dot{E}$ be a tame bi-separated graph such that $S=C_{\emph{fin}}=C$, $T=D_{\emph{fin}}=D$ and $|E^{0}|=1$. For $\lambda\in\Lambda$, let $E_{\lambda}$ be the subgraph of $E$ with edge set $\bigcup\limits_{\begin{subarray}{c}e\in X\\ X\in\mathcal{X}_{\lambda}\end{subarray}}\{e\}$ and consider the bi-separation $C_{\lambda}=\{X\in\mathcal{X}_{\lambda}\}$, and $D_{\lambda}=\{Y\in\mathcal{Y}_{\lambda}\}$ . Then $\mathcal{A}_{K}(\dot{E})$ is isomorphic to the free-product of algebras $\mathcal{A}_{K}(\dot{E_{\lambda}})$, where $\lambda$ varies over the indexing set $\Lambda$.

Corollary 4.13** ([7] Proposition 2.10).**

Let $(E,C)$ be a separated graph with $|E^{0}|=1$. Then $L_{K}(E,C)$ is isomorphic to the free-product of algebras $L_{K}(1,|X|)$, where $L_{K}(1,|X|)$ is the Leavitt algebra of type $(1,|X|)$ and $X$ varies over $C$.

Theorem 4.14**.**

Every object $\dot{E}$ in tBSG is a direct limit of finite (complete) sub-objects. Conversely, if $\dot{E}$ in BSG is a direct limit of finite complete sub-objects then it belongs to tBSG.

Proof.

Let $\dot{E}$ be an object in tBSG. By exactly same arguments as in Proposition 4.7, $\dot{E}$ is a direct limit of the directed system

[TABLE]

It follows from the definition of tame bi-separated graphs that $\dot{\mathcal{E}_{A}}$ is finite, for each finite subset $A$ of $E^{0}\sqcup E^{1}$.

Conversely, let $\dot{E}$ be a direct limit of the directed system

[TABLE]

of finite complete sub-objects. We know that $S$ and $T$ can be partitioned as $S_{1}\sqcup S_{2}$ and $T_{1}\sqcup T_{2}$ respectively (see equations 1 to 4 before the proposition 4.9). If $S_{1}=\emptyset$, then $T_{1}=\emptyset$ and clearly $\dot{E}$ is tame.

Suppose $S_{1}\neq\emptyset$. Then we have $S_{1}=\bigsqcup\limits_{\lambda\in\Lambda}\mathcal{X}_{\lambda}$ and $T_{1}=\bigsqcup\limits_{\lambda\in\Lambda}\mathcal{Y}_{\lambda}$. That each $|\mathcal{X}_{\lambda}|$ and $|\mathcal{Y}_{\lambda}|$ is finite follows from the fact that the directed system consists of finite objects and the morphisms involved are complete. ∎

Corollary 4.15**.**

Let $\dot{E}$ be an object in tBSG. Then the Cohn-Leavitt path algebra $\mathcal{A}_{K}(\dot{E})$ is the direct limit of the directed system of unital algebras $\{\mathcal{A}_{K}(\dot{E}_{i})\}_{i\in I}$ such that whenever $j\geq i$, the map $\mathcal{A}_{K}(\dot{E}_{i})\rightarrow\mathcal{A}_{K}(\dot{E}_{j})$ is a monomorphism, where $\{\dot{E}_{i}\}_{i\in I}$ is a directed system of finite complete sub-objects of $\dot{E}$ whose direct limit is $\dot{E}$.

Proof.

By the previous theorem, $\dot{E}$ is a direct limit of a directed system $\{\dot{E}_{i}\}_{i\in I}$ consisting of its finite complete sub-objects. Therefore, by proposition 4.3 and Theorem 5.3, $\mathcal{A}_{K}(\dot{E})$ is the direct limit of the directed system of algebras $\{\mathcal{A}_{K}(\dot{E}_{i})\}_{i\in I}$. ∎

By corollary 4.15 (Cohn-)Leavitt path algebras of the classes of bi-separated graphs in examples 3.4-3.7 are direct limits of unital sub-(Cohn-)Leavitt path algebras of same type.

5. Normal forms and their applications

Definition 5.1**.**

For each pair $X,X^{\prime}\in S$, if there exists $Y\in D$ such that $X\cap Y\neq\emptyset$ and $X^{\prime}\cap Y\neq\emptyset$, then we choose and fix one such $Y$ (this may vary with $X,X^{\prime}$) and call $(XY)(YX^{\prime})^{\ast}$ a forbidden word of type I.

Similarly for each pair $Y,Y^{\prime}\in T$, if there exists $X\in D$ such that $Y\cap X\neq\emptyset$ and $Y^{\prime}\cap X\neq\emptyset$, then we choose and fix one such $X$ and call $(YX)^{\ast}(XY^{\prime})$ a forbidden word of type II.

Definition 5.2**.**

A generalized path $\mu\in\widehat{E}^{\star}$ is called normal if it does not contain any forbidden subword of types I or II. An element of $K(\widehat{E})$ is called normal if it lies in the $K$-linear span of generalized normal paths.

We will show that any element of $\mathcal{A}_{K}(\dot{E})$ has precisely one normal representative in $K(\widehat{E})$. For this, we need to use Bergman’s diamond lemma. We refer the reader to [9, pp. 180-182] for the statement of the lemma and basic terminologies.

Theorem 5.3**.**

Let $\dot{E}$ be a bi-separated graph . Then $\mathcal{A}_{K}(\dot{E})$ has a basis consisting of normal generalized paths.

Proof.

In order to apply Bergman’s diamond lemma, we replace the defining relations by the following:

1. $1^{\prime}$:

For any $v,w\in E^{0}$,

[TABLE] 2. $2^{\prime}$:

For any $v\in E^{0},e\in E^{1}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE] 3. $3^{\prime}$:

For any $e,f\in E^{1}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE] 4. $\mathcal{A}^{\prime}1:$

For each $X,X^{\prime}\in S$ for which there exists $Y\in D$ such that $X\cap Y\neq\emptyset$ and $X^{\prime}\cap Y\neq\emptyset$,

[TABLE] 5. $\mathcal{A}^{\prime}2:$

For each $Y,Y^{\prime}\in T$ for which there exists $X\in C$ such that $X\cap Y\neq\emptyset$ and $X\cap Y^{\prime}\neq\emptyset$,

[TABLE]

(i.e. In $\mathcal{A}^{\prime}1$ and $\mathcal{A}^{\prime}2$, LHS contains forbidden words).

Denote by $\Sigma$ the reduction system consisting of all pairs $\sigma=(w_{\sigma},f_{\sigma})$, where $w_{\sigma}$ equals the LHS of an equation above and $f_{\sigma}$ the corresponding RHS. Let $\langle\bar{P}\rangle$ be the monoid consisting of all words formed by letters in $E^{0}\cup E^{1}\cup\overline{E^{1}}$ and $\langle P\rangle$ be the semi-group obtained by removing the identity element of $\langle\bar{P}\rangle$. We define a partial order on $\langle P\rangle$ as follows:

Let $A=x_{1}x_{2}\dots x_{n}\in\langle P\rangle$. Set $l(A)=n$ and

[TABLE]

Define a partial order $\leq$ on $\langle P\rangle$ by $A\leq B$ if and only if one of the following holds:

1. (1)

$A=B$, 2. (2)

$l(A)<l(B)$ or 3. (3)

$l(A)=l(B),\quad\text{and for each}\quad G,H\in\langle\bar{P}\rangle,\quad m(GAH)<m(GBH)$.

Clearly $\leq$ is a semigroup partial order on $\langle P\rangle$ compatible with $\Sigma$ and also the descending chain condition is satisfied. It remains to show that all ambiguities of $\Sigma$ are resolvable. Recall from Proposition 2.2 that $\widehat{E}^{\star}$ is a linear $K$-basis for $K(\widehat{E})$. Hence it is sufficient to show that the following ambiguities are resolvable:

[TABLE]

[TABLE]

We note that there are no inclusion ambiguities. We only show how to resolve ambiguity of type $\mathcal{A}^{\prime}1-\mathcal{A}^{\prime}2$ and the other case follows similarly.

\mathcal{A}^{\prime}1$$\mathcal{A}^{\prime}2$$=$$=$$\mathcal{A}^{\prime}2$$\mathcal{A}^{\prime}1$$=$$=$$=$$=$$(XY_{1})(Y_{1}X_{1})^{\ast}(X_{1}Y)$$\bigg{[}\delta_{X,X_{1}}s(X_{1})-\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}(XY_{2})(Y_{2}X_{1})^{\ast}\bigg{]}(X_{1}Y)$$(XY_{1})\bigg{[}\delta_{Y,Y_{1}}r(Y_{1})-\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}(Y_{1}X_{2})^{\ast}(X_{2}Y)\bigg{]}$$\delta_{X,X_{1}}(X_{1}Y)-\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}(XY_{2})(Y_{2}X_{1})^{\ast}(X_{1}Y)$$\delta_{Y,Y_{1}}(XY_{1})-\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}(XY_{1})(Y_{1}X_{2})^{\ast}(X_{2}Y)$$\delta_{X,X_{1}}(X_{1}Y)-\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}(XY_{2})\bigg{[}\delta_{Y,Y_{2}}r(Y)-\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}(Y_{2}X_{2})^{\ast}(X_{2}Y)\bigg{]}$$\delta_{Y,Y_{1}}(XY_{1})-\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}\bigg{[}\delta_{X,X_{2}}s(X)-\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}(XY_{2})(Y_{2}X_{2})^{\ast}\bigg{]}(X_{2}Y)$$\delta_{X,X_{1}}(X_{1}Y)-\delta_{Y,Y_{2}}(XY_{2})+\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}(XY_{2})(Y_{2}X_{2})^{\ast}(X_{2}Y)$$\delta_{Y,Y_{1}}(XY_{1})-\delta_{X,X_{2}}(X_{2}Y)+\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}(XY_{2})(Y_{2}X_{2})^{\ast}(X_{2}Y)$$\sum\limits_{\begin{subarray}{c}X_{2}\in C\\ X_{2}\neq X_{1}\end{subarray}}\sum\limits_{\begin{subarray}{c}Y_{2}\in D\\ Y_{2}\neq Y_{1}\end{subarray}}(XY_{2})(Y_{2}X_{2})^{\ast}(X_{2}Y)

This proves the confluence condition. The final expression written above is a finite sum as $X\in S$ and $Y\in T$. If some terms in this expression contain forbidden words, we further reduce them as above. But this process has to terminate in finitely many stages (again since $X\in S$ and $Y\in T$). Also, it should be noted that in this sequence of reductions, the same forbidden word cannot appear more than once at different stages. For, that would mean that a single relation is contributing more than one forbidden words, which is a contradiction. This proves the reduction finiteness as well. ∎

Corollary 5.4**.**

Let $\dot{E}$ be a bi-separated graph. Then the natural homomorphism from the path algebra $K(E)$ to the algebra $\mathcal{A}_{K}(\dot{E})$ is an inclusion.

Proof.

From the theorem, it follows that each path $\mu\in E^{\star}$ is a part of a basis of $\mathcal{A}_{K}(\dot{E})$ as $\mu$ does not contain any forbidden word. ∎

In the following subsections we give some applications of normal forms of Cohn-Leavitt path algebras. To be precise, we find bi-separated graph theoretic properties that correspond to algebraic properties. One needs to be careful in the sense that in the general setting of bi-separated graphs, finding conditions only on underlying graphs are not enough. We also need suitable conditions on $(S,T)$ which give the defining relations of Cohn-Leavitt path algebras. In the following subsections we recall some definitions and propositions from the theory of rings with enough idempotents. Then we give their applications to the case of Cohn-Leavitt path algebras. We note that the reasoning is very similar to that of [29]. Wherever some care is required we provide complete proofs else the reader is refered to [29] for proofs.

5.1. Local valuations and their applications

Definition 5.5**.**

Let $(R,I)$ be a ring with enough idempotents. A local valuation on $(R,I)$ is a map $\nu:R\rightarrow\mathbb{Z}^{+}\cup\{-\infty\}$ such that

1. (1)

$\nu(x)=-\infty~{}\text{if and only if}~{}x=0$, 2. (2)

$\nu(x-y)\leq\max\{\nu(x),\nu(y)\}$ for any $x,y\in R$ and 3. (3)

$\nu(xy)=\nu(x)+\nu(y)$ for any $e\in I$, $x\in Re$ and $y\in eR$.

A local valuation $\nu$ on $(R,I)$ is called trivial if $\nu(x)=0$ for each $x\in R-\{0\}$.

Let $R$ be a ring. A left ideal $\mathfrak{a}$ of $R$ is called essential if $\mathfrak{a}\cap\mathfrak{b}=0\Rightarrow\mathfrak{b}=0$ for any left ideal $\mathfrak{b}$ of $R$. For any $x\in R$ recall the left annihilator ideal of $x$ is Ann$(x):=\{y\in R\mid yx=0\}$. A ring $R$ is called left non-singular if for any $x\in R$, Ann$(x)$ is essential $\Leftrightarrow x=0$. A right non-singular ring is defined similarly. A ring is non-singular is if it is both left and right non-singular.

Proposition 5.6**.**

[29, Proposition 37]** Let $(R,I)$ be a ring with enough idempotents which admits a local valuation. Then $R$ is non-singular.

A non-zero ring $R$ is called a prime ring if $\mathfrak{ab}=0\Rightarrow\mathfrak{a}=0$ or $\mathfrak{b}=0$ for any ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$. A ring with enough idempotents $(R,I)$ is connected if $eRf\neq 0$ for any $e,f\in I$.

Proposition 5.7**.**

[29, Proposition 38]** Let $(R,I)$ be a nonzero, connected ring with enough idempotents which admits a local valuation. Then $R$ is a prime ring.

A ring $R$ is said to be von Neumann regular if for any $x\in R$ there exists $y\in R$ such that $xyx=x$.

Proposition 5.8**.**

[29, Proposition 39]** Let $(R,I)$ be a ring with enough idempotents that has a nontrivial local valuation. Then $R$ is not von Neumann regular.

Recall that the Jacobson radical of a ring $R$ is the ideal consisting of those elements in $R$ that annihilate all simple (right or left) $R$-modules. A ring is called semiprimitive if its Jacobson radical is the zero ideal.

Proposition 5.9**.**

[29, Proposition 40]** Let $(R,I)$ be a connected $K$-algebra with enough idempotents which admits a local valuation $\nu$ such that $\nu(x)=0$ if and only if $x$ is a nonzero $K$-linear combination of elements in $I$. Then $R$ is semiprimitive.

Now we find conditions on a bi-separated graph $\dot{E}$ for which the corresponding Cohn-Leavitt path algebra admits a local valuation.

Definition 5.10**.**

A bi-separated graph $\dot{E}=(E,(C,S),(D,T))$ is said to satisfy Condition LV if one of the following holds:

1. $(LV1)$ :

$|S|\leq 1$ , $|T|\leq 1$ and if $X\in S$ (resp. $Y\in T$), then $|X|>1$ (resp. $Y>1)$. 2. $(LV2)$ :

$|S|>1$ or $|T|>1$, and the following two conditions are satisfied:

1. (a)

For any $X_{1},X_{2}\in S$, either there is no $Y\in D$ such that $X_{1}\cap Y\neq\emptyset~{}\text{and}~{}X_{2}\cap Y\neq\emptyset$ or $\exists~{}Y_{1},Y_{2}\in D,Y_{1}\neq Y_{2}$ such that $X_{i}\cap Y_{j}\neq\emptyset$ for each $1\leq i,j\leq 2$. 2. (b)

For any $Y_{1},Y_{2}\in T$, either there is no $X\in C$ such that $Y_{1}\cap X\neq\emptyset$ and $Y_{2}\cap X\neq\emptyset$ or $\exists~{}X_{1},X_{2}\in C,X_{1}\neq X_{2}$ such that $X_{i}\cap Y_{j}\neq\emptyset$ for each $1\leq i,j\leq 2$.

We say $\dot{E}$ satisfies Domain condition if either $|S|\leq 1,|T|\leq 1$ or $(LV2)$ holds.

Proposition 5.11**.**

Let $\dot{E}$ be a bi-separated graph and for any $a\in\mathcal{A}_{K}(\dot{E})$, let $\emph{supp}(a)$ denote the set of all normal generalized paths occuring in $\emph{NF}(a)$ with nonzero coefficients, where $\emph{NF}(a)$ is the unique normal representative of $a$. If $\dot{E}$ satisfies condition LV, then the map $\nu:\mathcal{A}_{K}(\dot{E})\rightarrow\mathbb{Z}^{+}\cup\{-\infty\}$ defined by

[TABLE]

is a local valuation on $\mathcal{A}_{K}(\dot{E})$.

Proof.

The first two conditions of a local valuation are obvious. It remains to show $\nu(ab)\,=\,\nu(a)+\nu(b),$ for any $v\in E^{0},~{}a\in\mathcal{A}_{K}(\dot{E})v$ and $b\in v\mathcal{A}_{K}(\dot{E})$. If one of $\nu(a)$ and $\nu(b)$ is [math] or $-\infty$, then the result is clear. Suppose now $\nu(a),\nu(b)\,\geq\,1$. Since any reduction preserves or decreases the length of a generalized path, it follows that $\nu(ab)\,\leq\,\nu(a)+\nu(b)$. So it remains to show that $\nu(ab)\,\geq\,\nu(a)+\nu(b)$. Let

[TABLE]

be the elements of supp($a$) with length $\nu(a)$ and

[TABLE]

be the elements of supp($b$) with length $\nu(b)$. We assume that the $p_{k}$’s are pairwise distinct and so are $q_{l}$’s. Since NF is a linear map, we can make the following conclusions:

1. (1)

If $x_{\nu(a)}^{k}y_{1}^{l}$ is not of type I or II, then

NF$(p_{k}q_{l})=p_{k}q_{l}$. 2. (2)

If $x_{\nu(a)}^{k}y_{1}^{l}$ is of type I, then there are $X,X^{\prime}\in S$ and $Y\in D$ such that $x_{\nu(a)}^{k}y_{1}^{l}\,=\,(XY)(X^{\prime}Y)^{\ast}$ and $(XY)(X^{\prime}Y)^{\ast}$ is forbidden. So

[TABLE] 3. (3)

If $x_{\nu(a)}^{k}y_{\nu(b)}^{l}$ is of type II, then there are $Y,Y^{\prime}\in T$ and $X\in C$ such that $x_{\nu(a)}^{k}y_{\nu(b)}^{l}\,=\,(XY)^{\ast}(XY^{\prime})$ and $(XY)^{\ast}(XY^{\prime})$ is forbidden. So

[TABLE]

$\mathbf{Case~{}1:}$ Assume that $x_{\nu(a)}^{k}y_{1}^{l}$ is not of type I or II, for any $k$, $l$.

Then $p_{k}q_{l}\in\text{supp}($a$)$, for any $k$, $l$. So $\nu(ab)\,\geq\,|p_{k}q_{l}|\,=\,\nu(a)+\nu(b)$.

$\mathbf{Case~{}2:}$ Assume that there are $k$ and $l$ such that $x_{\nu(a)}^{k}y_{\nu(b)}^{l}$ is of type I.

Then there are $X,X^{\prime}\in S$ and $Y\in D$ such that $x_{\nu(a)}^{k}y_{1}^{l}\,=\,(XY)(X^{\prime}Y)^{\ast}$ and $(XY)(X^{\prime}Y)^{\ast}$ is forbidden. Since $\dot{E}$ is an LV-object, there is at least one more element $\widetilde{Y}\in D$ other than $Y$ such that $X\widetilde{Y}\neq 0$ and $X^{\prime}\widetilde{Y}\neq 0$.

$\mathbf{Case~{}2.1:}$ Assume $p_{k^{\prime}}q_{l^{\prime}}\neq x_{1}^{k}\ldots x_{\nu(a)-1}^{k}(X\widetilde{Y})(X^{\prime}\widetilde{Y})^{\ast}y_{2}^{l}\ldots y_{\nu(b)}^{l}$, for any $k^{\prime},l^{\prime}$.

Then $x_{1}^{k}\ldots x_{\nu(a)-1}^{k}(X\widetilde{Y})(X^{\prime}\widetilde{Y})^{\ast}y_{2}^{l}\ldots y_{\nu(b)}^{l}\in\text{supp}(ab)$, since it does not cancel with any other term. So we are done.

$\mathbf{Case~{}2.2:}$ Assume $p_{k^{\prime}}q_{l^{\prime}}\,=\,x_{1}^{k}\ldots x_{\nu(a)-1}^{k}(X\widetilde{Y})(X^{\prime}\widetilde{Y})^{\ast}y_{2}^{l}\ldots y_{\nu(b)}^{l}$, for some $k^{\prime},l^{\prime}$. In this case, $p_{k}q_{l^{\prime}}\,=\,x_{1}^{k}\ldots x_{\nu(a)-1}^{k}(XY)(X^{\prime}\widetilde{Y})^{\ast}y_{2}^{l}\ldots y_{\nu(b)}^{l}\in\text{supp}(ab)$ and so we are done.

$\mathbf{Case~{}3:}$ Assume that there are $k$ and $l$ such that $x_{\nu(a)}^{k}y_{1}^{l}$ is of type II.

Then there are $Y,Y^{\prime}\in T$ and $X\in C$ such that $x_{\nu(a)}^{k}y_{1}^{l}\,=\,(XY)^{\ast}(XY^{\prime})$ and $(XY)^{\ast}(XY^{\prime})$ is forbidden. Again since $\dot{E}$ is an LV-object, there is at least one more element $\widetilde{X}\in C$ other than $X$ such that $\widetilde{X}Y\neq 0$ and $\widetilde{X}Y^{\prime}\neq 0$. The proof now follows in exactly the same way as in $\mathbf{Cases~{}2.1}$ and $\mathbf{2.2}$. ∎

Corollary 5.12**.**

Let $\dot{E}$ be a bi-separated graph satifying condition LV. Then

1. (1)

$\mathcal{A}_{K}(\dot{E})$* is nonsingular.* 2. (2)

$\dot{E}$* is connected implies $\mathcal{A}_{K}(\dot{E})$ is semiprimitive.* 3. (3)

$\dot{E}$* is connected and non-empty implies $\mathcal{A}_{K}(\dot{E})$ is prime.* 4. (4)

$|E^{1}|\geq 1$* implies $\mathcal{A}_{K}(\dot{E})$ is not von Neumann regular.*

Theorem 5.13**.**

Let $\dot{E}$ be a bi-separated graph. Then $\mathcal{A}_{K}(\dot{E})$ is a domain if and only if $\dot{E}$ satisfies domain condition.

Proof.

If $\dot{E}$ satisfies domain condition, then we consider the two following cases:

$\mathbf{Case~{}1:}$ Assume that $X\in S\Rightarrow|X|>1$ and $Y\in T\Rightarrow|Y|>1$. If both $S$ and $T$ are empty, then $\mathcal{A}_{K}(\dot{E})$ is a free unital $K$-algebra and hence a domain (since $K$ is a field). Otherwise, by the proposition 5.11, there is a local valuation on $\mathcal{A}_{K}(\dot{E})$. So if $ab\,=\,0$ in $\mathcal{A}_{K}(\dot{E})$, then $\nu(ab)=-\infty$, which implies $\nu(a)+\nu(b)=-\infty$. This means that $\nu(a)=-\infty$ or $\nu(b)=-\infty$. Hence $a=0$ or $b=0$. Therefore $\mathcal{A}_{K}(\dot{E})$ is a domain.

$\mathbf{Case~{}2:}$ The only remaining cases to be considered are when $S=\{X\}$ with $X=\{e\}$ or $T=\{Y\}$ with $Y=\{f\}$. In both these cases the relations imposed on $K(\widehat{E})$ are not of the form $ab=0$.

For converse, if $|E^{0}|>1$, then obviously $\mathcal{A}_{K}(\dot{E})$ is not a domain. Otherwise, we consider the following cases separately:

$\mathbf{Case~{}1:}$ Assume that there are two distinct elements $X,X^{\prime}\in S$ which have only one common $Y\in D$ such that $XY\neq 0$, $X^{\prime}Y\neq 0$. Then $(XY)(X^{\prime}Y)^{\ast}\,=\,\delta_{XX^{\prime}}s(X)\,=\,0$. So we are done.

$\mathbf{Case~{}2:}$ Assume that there are two distinct elements $Y,Y^{\prime}\in T$ which have only one common $X\in C$ such that $XY\neq 0$, $XY^{\prime}\neq 0$. Then $(XY)^{\ast}(XY^{\prime})\,=\,\delta_{YY^{\prime}}r(Y)\,=\,0$.

This completes the proof. ∎

5.2. The Gelfand-Kirillov dimension

We first recall some basic facts on the growth of algebras from [18]. Suppose $B$ is a finitely generated $K$-algebra. Choose a finite generating set of $B$ and let $V$ be the $K$-subspace of $B$ spanned by this chosen generating set. For each natural number $n$, let $V^{n}$ denote the $K$-subspace of $B$ spanned by all words in $V$ of length less than or equal to $n$. In particular, $V^{1}\,=\,V$. Then we have an ascending chain

$K\subseteq V^{1}\subseteq V^{2}\subseteq\ldots\subseteq V^{n}\subseteq\ldots$

of finite dimensional $K$-subspaces of $B$ such that $B\,=\,\bigcup_{n\in\mathbb{N}_{0}}V^{n}$, where, by convention, $V^{0}\,=\,K$. Clearly, the sequence $\{\text{dim}_{K}(V^{n})\}$ is a montonically increasing sequence and the asymptotic behaviour (see the definition 5.14) of this sequence provides an invariant of the algebra $B$, called the Gelfand-Kirillov dimension of $B$, which is defined to be

[TABLE]

Definition 5.14**.**

Given two eventually monotonically increasing functions $\phi,\psi:\mathbb{N}\rightarrow\mathbb{R}^{+}$, we say $\phi\preceq\psi$ if there are natural numbers $a$ and $b$ such that $\phi(n)\leq a\psi(bn)$, for almost all $n\in\mathbb{N}$. We say $\phi$* is asymptotically equivalent to $\psi$*, if both $\phi\preceq\psi$ and $\psi\preceq\phi$. If $\phi$ and $\psi$ are asymptotically equivalent, we write $\phi\sim\psi$.

Coming back to GK dimension of algebras, if a $K$-algebra $B$ has two distinct finite generating sets, and if $V$ and $W$ are the finite dimensional subspaces of $B$ spanned by these sets, then setting $\phi(n)=\text{dim}_{\text{K}}(V^{n})$ and $\psi(n)\,=\,\text{dim}_{\text{K}}(W^{n})$, one can show that $\phi\sim\psi$ [18, Lemma 1.1]. In this notation, if $\phi\preceq n^{m}$ for some $m\in\mathbb{N}$, then $B$ is said to have polynomial growth and in this case GKdim($B$)$\leq m$. If on the other hand, $\phi\sim a^{n}$ for some $a\in\mathbb{R}$ such that $a>1$, then $B$ is said to have exponential growth and in this case GKdim($B$)$=\infty$.

Definition 5.15**.**

A bi-separated graph $\dot{E}$ is said to satisfy Condition $(A^{\prime})$ if

1. $(A^{\prime}1)$:

$S=T=\emptyset$ implies either $|E^{1}|>0$ or $|E^{0}|=\infty$. 2. $(A^{\prime}2)$:

$S\neq\emptyset$ or $T\neq\emptyset$ implies at least one of the following holds:

1. (a)

$\exists X_{1},X_{2}\in S,~{}X_{1}\neq X_{2}$, $s(X_{1})=s(X_{2})$ and $Y\in D$ such that for $i\in\{1,2\},$ $X_{i}\cap Y\neq\emptyset$ and $(X_{i}Y),~{}(X_{i}Y)^{\ast}$ are not part of any forbidden word. 2. (b)

$\exists Y_{1},Y_{2}\in T,~{}Y_{1}\neq Y_{2}$, $r(Y_{1})=r(Y_{2})$ and $X\in C$ such that for $i\in\{1,2\},$ $Y_{i}\cap X\neq\emptyset$ and $(Y_{i}X),~{}(Y_{i}X)^{\ast}$ are not part of any forbidden word. 3. (c)

$\exists X\in S,~{}Y\in D$ such that $X\cap Y\neq\emptyset$, $s(X)=r(Y)$ and $(XY)$,$(XY)^{\ast}$ are not part of any forbidden word. 4. (d)

$\exists Y\in T,~{}X\in C$ such that $X\cap Y\neq\emptyset$, $s(X)=r(Y)$ and $(XY)$,$(XY)^{\ast}$ are not part of any forbidden word.

Proposition 5.16**.**

If $\dot{E}$ is a finite bi-separated graph and satisfies condition $A^{\prime}$ then $\mathcal{A}_{K}(\dot{E})$ has exponential growth.

Definition 5.17**.**

[29, Definition 20,21] Let $\dot{E}$ be a bi-separated graph. A quasi-cycle is a normal generalized path $p$ in $\widehat{E}$ such that $p^{2}$ is normal and none of the sub-words of $p^{2}$ of length less than $|p|$ is normal. A quasi-cycle $p$ is called self-connected if there is a normal path $o$ in $\widehat{E}$ such that $p$ is not a prefix of $o$ and $pop$ is normal.

Theorem 5.18**.**

Let $\dot{E}$ be a finite bi-separated graph. Then $\mathcal{A}_{K}(\dot{E})$ has exponential growth if and only if there is a self-connected quasi-cycle.

Remark 5.19*.*

Let $\dot{E}$ be a tame bi-separated graph and suppose that $\{\dot{E}_{\lambda}\mid\lambda\in\Lambda\}$ is a directed system of all finite sub bi-separated graphs of $\dot{E}$. By results of [26, Section 3] we have GKdim($\mathcal{A}_{K}(\dot{E})=\sup\limits_{\lambda}\text{GKdim}(\mathcal{A}_{K}(\dot{E}_{\lambda}))$.

5.3. Additional applications of Linear bases

In this subsection we fix the following notations. Let $(R,I)$ be a $K$-algebra with enough idempotents. An element $a\in R$ is called homogeneous if $a\in vRw$ for some $v,w\in I$. Let $B$ denote a $K$-basis for $R$ which consists of homogeneous elements and contains $I$. Let $l:B\rightarrow\mathbb{Z}^{+}$ be a map such that $l(b)=0\Leftrightarrow b\in I$.

Definition 5.20**.**

An element $b\in B\cap vRw$ is called left adhesive if $ab\in B$ for any $a\in B\cap Rv$ and right adhesive if $bc\in B$ for any $c\in B\cap wR$. A left valuative basis element is a left adhesive element $b\in B\cap eR$ such that $l(ab)=l(a)+l(b)$ for any $a\in B\cap Rv$. A right valuative basis element is defined similarly. A valuative basis element is an adhesive element $b\in B\cap vRw$ such that $l(abc)=l(a)+l(b)+l(c)$ for any $a\in B\cap Rv$ and $c\in B\cap wR$.

Proposition 5.21**.**

[29, Proposition 53]** Suppose there exists a valuative basis element $b\in(B-I)\cap vRv$. Then $\dim_{K}(R)=\infty$, $R$ is not simple, neither left nor right Artinian and not von Newmann regular.

Definition 5.22**.**

A bi-separated graph $\dot{E}=(E,(C,S),(D,T))$ is said to satisfy Condition $(A)$ if

1. $(A1)$:

$S=T=\emptyset$ implies $|E^{0}|=\infty$ or $|E^{1}|>0$ 2. $(A2)$:

$S\neq\emptyset$ or $T\neq\emptyset$ implies at least one of the following holds:

1. (a)

$\exists~{}X\in S,~{}Y\in D$ such that $X\cap Y\neq\emptyset$, $(XY)(XY)^{\ast}$ and $(XY)^{\ast}(XY)$ are not forbidden words. 2. (b)

$\exists~{}Y\in T,~{}X\in C$ such that $X\cap Y\neq\emptyset$, $(XY)(XY)^{\ast}$ and $(XY)^{\ast}(XY)$ are not forbidden words.

Let $B$ denote the set of all normal generalized paths of $\mathcal{A}_{K}(\dot{E})$. Let $l:B\rightarrow\mathbb{Z}^{+}$ denote the map which maps a path to its length. If $\dot{E}$ satisfy Condition $(A2)$ then we can choose either $X\in S,~{}Y\in D$ or $Y\in T,~{}X\in C$ such that $X\cap Y\neq\emptyset$ and $(XY)(XY)^{\ast}$ is not forbidden. Set $b=(XY)(XY)^{\ast}$, then $b$ is a valuative basis element. Hence we have the following corollary.

Corollary 5.23**.**

Let $\dot{E}$ be a bi-separated graph that satisfies Condition $(A)$. Then $\dim_{K}(\mathcal{A}_{K}(\dot{E}))=\infty$, $\mathcal{A}_{K}(\dot{E})$ is not simple, neither left nor right Artinian and not von Neumann regular.

Definition 5.24**.**

Let $b\in B\cap vRw$ and $b^{\prime}\in B\cap v^{\prime}Rw^{\prime}$. We say that $b$ and $b^{\prime}$ have no common left multiple if there are no $a\in B\cap Rv,~{}a^{\prime}\in B\cap Rv^{\prime}$ such that $ab=a^{\prime}b^{\prime}$. We say that $b$ and $b^{\prime}$ have no common right multiple if there are no $c\in B\cap wR,~{}c^{\prime}\in B\cap w^{\prime}R$ such that $bc=b^{\prime}c^{\prime}$.

An element $b\in B\cap vRw$ is called right cancellative if $ab=cb\Rightarrow a=c$ for any $a,c\in B\cap Rv$ and left cancellative if $ba^{\prime}=bc^{\prime}\Rightarrow a^{\prime}=c^{\prime}$ for any $a^{\prime},c^{\prime}\in B\cap wR$.

Proposition 5.25**.**

[29, Proposition 56]** If there are elements $b,b^{\prime}\in B\cap vRv$ such that $b$ is adhesive and right cancellative, $b^{\prime}$ is left adhesive and $b$ and $b^{\prime}$ have no common left multiple, then $R$ is not left Noetherian.

If there are elements $c,c^{\prime}\in B\cap vRv$ such that $c$ is adhesive and left cancellative, $c^{\prime}$ is right adhesive and $c$ and $c^{\prime}$ have no common left multiple, then $R$ is not right Noetherian.

We have the following corollary which gives a necessary condition for $\mathcal{A}_{K}(\dot{E})$ to be a left or right Noetherian in terms of $\dot{E}$.

Corollary 5.26**.**

Let $\dot{E}$ be a bi-separated graph that satisfies Condition $(A^{\prime})$. Then $\mathcal{A}_{K}(\dot{E})$ is neither left nor right Noetherian.

Proof.

We prove the statement only for conditions $(A^{\prime}2)(a)$ and $(A^{\prime}2)(c)$ leaving the other simple cases to the reader.

Suppose there exist $X_{1},X_{2}\in S$, $X_{1}\neq X_{2}$, $s(X_{1})=s(X_{2})=v$ and $Y\in D$ such that for $i\in\{1,2\}$, $X_{i}\cap Y\neq\emptyset$ and $(X_{i}Y)$, $(X_{i}Y)^{\ast}$ are part of forbidden words. Then set $b_{1}:=(X_{1}Y)(X_{1}Y)^{\ast}$, $b_{2}:=(X_{1}Y)(X_{2}Y)^{\ast}$ and $b_{3}:=(X_{2}Y)(X_{1}Y)^{\ast}$. Then $b_{1},b_{2},b_{3}\in(B-E^{0})\cap v\mathcal{A}_{K}(\dot{E})v$. It is easy to check that $b_{1}$ is adhesive and both left and right cancellative, $b_{2}$ is left adhesive, $b_{3}$ is right adhesive, $b_{1},~{}b_{2}$ have no common left multiple and $b_{1},~{}b_{3}$ have no common right multiple. Thus $\mathcal{A}_{K}(\dot{E})$ is neither left nor right Noetherian.

Now suppose that there exist $X\in S$ and $Y\in D$ such that $X\cap Y\neq\emptyset$, $s(X)=r(Y)=v$ and $(XY),~{}(XY)^{\ast}$ are not part of any forbidden word. Then both $(XY),~{}(XY)$ are in $(B-E^{0})\cap v\mathcal{A}_{K}(\dot{E})v$, they are adhesive, both left and right cancellative and have neither left nor right common multiple. Therefore $\mathcal{A}_{K}(\dot{E})$ is neither left nor right Noetherian. ∎

PART II

In this part we specialize our attention to hypergraphs and study their Cohn-Leavitt path algebras.

6. B-hypergraphs and their $H$-monoids

We would like to recast the definition of hypergraphs in terms of bi-separated graphs (with distinguished subsets) so that we can study their Cohn-Leavitt path algebras. We note that a hyperegde $h$, for which both $I_{h}$ and $J_{h}$ are infinite, does not contribute to relations in the Leavitt path algebra. We ignore such hyperedges and all other hyperedges which do not contribute to Cohn-Leavitt path algebra relations from the definition of hypergraph. Hence we modify the definition of hypergraphs as follows.

Definition 6.1**.**

A B-hypergraph is a pair $(\dot{E},\Lambda)$, where $\dot{E}=(E,(C,S),(D,T))$ is a bi-separated graph and $\Lambda$ is a nonempty indexing set whose elements are called the B-hyperedges, such that for each $\lambda\in\Lambda$ there exists $\mathcal{X}_{\lambda}\subseteq C$ and $\mathcal{Y}_{\lambda}\subseteq D$ which further satisfy the following conditions:

1. (1)

$X\notin S$ and $Y\notin T\implies X\cap Y=\emptyset$, 2. (2)

for any $\alpha,\beta\in\Lambda$ with $\alpha\neq\beta$, $X\in\mathcal{X_{\alpha}}$ and $Y\in\mathcal{Y_{\beta}}\implies X\cap Y=\emptyset$, 3. (3)

for any $\lambda\in\Lambda$, $X\in\mathcal{X}_{\lambda}$ and $Y\in\mathcal{Y}_{\lambda}\implies X\cap Y\neq\emptyset$, 4. (4)

$\Lambda=\Lambda_{T}^{S}\sqcup\Lambda_{\text{fin}}^{S}\sqcup\Lambda_{\infty}^{S}\sqcup\Lambda_{T}^{\text{fin}}\sqcup\Lambda_{T}^{\infty}$, and

[TABLE]

Remark 6.2*.*

1. (1)

It is easy to check that B-hypergraphs are tame and that B-hypergraphs, along with complete morphisms form a category. This category will be denoted by BHG. 2. (2)

Note that given a B-hypergraph $(\dot{E},\Lambda)$ with $S=C_{\text{fin}}$ and $T=D_{\text{fin}}$ we can identify $(\dot{E},\Lambda)$ with a hypergraph $\mathcal{H}$ as follows: $\mathcal{H}^{0}=E^{0}$, $\mathcal{H}^{1}=\Lambda$, and for each $\lambda\in\Lambda$ $s(\lambda)=(s(X))_{X\in\mathcal{X}_{\lambda}}$ and $r(\lambda)=(r(Y))_{Y\in\mathcal{Y}_{\lambda}}$.

Notation 6.3**.**

For $\lambda\in\Lambda_{T}$, set

[TABLE]

For $\lambda\in\Lambda^{S}$, set

[TABLE]

Definition 6.4**.**

Given a B-hypergraph $(\dot{E},\Lambda)$, its $H$-monoid $H(\dot{E},\Lambda)$ is defined as the abelian monoid generated by $E^{0}\sqcup Q\sqcup P$ modulo the following relations:

1. (1)

For $\lambda\in\Lambda_{T}$ and $q_{Z}\in Q_{\lambda}$,

[TABLE] 2. (2)

For $\lambda\in\Lambda^{S}$ and $p_{W}\in P_{\lambda}$,

[TABLE] 3. (3)

For $\lambda\in\Lambda_{T}$ and $q_{Z_{1}},q_{Z_{2}}\in Q_{\lambda}$ with $Z_{1}\subsetneq Z_{2}$

[TABLE] 4. (4)

For $\lambda\in\Lambda^{S}$ and $p_{W_{1}},p_{W_{2}}\in P_{\lambda}$ with $W_{1}\subsetneq W_{2}$

[TABLE] 5. (5)

for $\lambda\in\Lambda_{T}^{S}$,

[TABLE]

If $(\dot{E},\Lambda)$ is B-hypergraph then $H(\dot{E},\Lambda)$ is a conical monoid. This is easy to see from the relations defining $H(\dot{E},\Lambda)$ because these relations ensure that $(x+y)\neq 0$ whenever $x\neq 0$ or $y\neq 0$, for $x,y\in H(\dot{E},\Lambda)$.

Definition 6.5**.**

Let $R$ be a ring, and let $M_{\infty}(R)$ denote the set of all $\omega\times\omega$ matrices over $R$ with finitely many nonzero entries, where $\omega$ varies over $\mathbb{N}$. For idempotents $e,f\in M_{\infty}(R)$, the Murray-von Neumann equivalence $\sim$ is defined as follows: $e\sim f$ if and only if there exists $x,y\in M_{\infty}(R)$ such that $xy=e$ and $yx=f$.

Let $\mathcal{V}(R)$ denote the set of all equivalence classes $[e]$, for idempotents $e\in M_{\infty}(R)$. Define $[e]+[f]=[e\oplus f]$, where $e\oplus f$ denotes the block diagonal matrix $\begin{pmatrix}e&0\\ 0&f\end{pmatrix}$. Under this operation, $\mathcal{V}(R)$ is an abelian monoid, and it is conical, that is, $a+b=0$ in $\mathcal{V}(R)$ implies $a=b=0$. Also $\mathcal{V}(\_):$Rings$\rightarrow$Mon is a continuous functor.

Let $R$ be a unital ring and let $\mathcal{U}(R)$ be the set of all isomorphic classes of finitely generated projective left $R$-modules, endowed with direct sum as binary operation. Then $(\mathcal{U}(R),\oplus)$ is an abelian monoid. We also have $\mathcal{U}(R)\cong\mathcal{V}(R)$.

Theorem 6.6**.**

There is an isomorphism $\Gamma:H\rightarrow\mathcal{V}\circ\mathcal{A}_{K}$ of funtors $\emph{{BHG}}\rightarrow\emph{{Mon}}$.

Proof.

We first define the map $\Gamma$ as follows: For each object $\dot{E}$ in BHG,

[TABLE]

is the monoid homomorphism sending

[TABLE]

[TABLE]

and

[TABLE]

where $v\in E^{0}$, $Z$ is any non-empty finite subset of $\mathcal{Y}_{\lambda}$, diag$(s(X))$ is the diagonal matrix of order $|\mathcal{X}_{\lambda}|$ with diagonal entries coming from the set $s(\mathcal{X}_{\lambda})$ in any order (without repetition), diag$(r(Y))$ is the diagonal matrix of order $|\mathcal{Y}_{\lambda}|$ with diagonal entries coming from the set $r(\mathcal{Y}_{\lambda})$ in any order (without repetition), $B$ is the $|\mathcal{X}_{\lambda}|\times|Z|$ matrix whose columns are precisely the ones in $Z$ and whose $i^{th}$ row consists elements of $X$ if and only if the diagonal entry in the $i^{th}$ row of diag$(s(X))$ is $s(X)$, and $N$ is the $|W|\times|\mathcal{Y}_{\lambda}|$ matrix whose rows are precisely the ones in $W$ and whose $j^{th}$ column has elements from $Y$ if and only if the diagonal entry in the $j^{th}$ column of diag$(r(Y))$ is $r(Y)$.

It is not hard to see that the above map is well defined. Also the fact that every element in BHG is a direct sum of its finite complete subobjects and the continuity of the functors involved will suggest that it is enough to prove the results for finite subobjects. For the finite case, we use induction on $|\Lambda|$. For $\Lambda=\emptyset$, the result is trivial. So, suppose that the result holds for all finite objects $(\dot{F},\Lambda_{\dot{F}})$ in BHG with $|\Lambda_{\dot{F}}|\leq(n-1)$, for some $n\geq 1$. Let $(\dot{E},\Lambda_{\dot{E}})$ be a finite object with $|\Lambda_{\dot{E}}|=n$. Fix an element $\lambda\in\Lambda_{\dot{E}}$. We can now apply induction to the object $\dot{F}$ obtained from the $\dot{E}$ by deleting all the edges in $\mathcal{X}_{\lambda}$ and leaving the remaining structure as it is, keeping $F^{0}=E^{0}$.

First suppose that $\lambda\in\Lambda_{T}^{S}$. Then $H(\dot{E},\Lambda_{\dot{E}})$ is obtained from $H(\dot{F},\Lambda_{\dot{F}})$ by going modulo the relation $\sum\limits_{X\in\mathcal{X}_{\lambda}}s(X)=\sum\limits_{Y\in\mathcal{Y}_{\lambda}}r(Y)$. Also, the algebra $\mathcal{A}_{K}(\dot{E},\Lambda_{\dot{E}})$ is the Bergman algebra obtained from $\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}})$ by adjoining a universal isomorphism between the finitely generated projective modules $\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}})s(X)$ and $\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}})r(Y)$. So by [8, Theorem 5.2], $\mathcal{V}(\mathcal{A}_{K}(\dot{E},\Lambda_{\dot{E}}))$ is the quotient of $\mathcal{V}(\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}}))$ modulo the relation

[TABLE]

Since the map

[TABLE]

is an isomorphism by induction hypothesis, the desired result follows.

Now suppose $\lambda$ does not belong to $\Lambda_{T}^{S}$. Then it is either in $\Lambda_{T}^{\infty}$ or in $\Lambda_{\infty}^{S}$. Let us first assume that $\lambda\in\Lambda_{T}^{\infty}$. In this case, $H(\dot{E},\Lambda_{\dot{E}})$ is obtained from $H(\dot{F},\Lambda_{\dot{F}})$ by adjoining a new generator $q_{\mathcal{Y}_{\lambda}}$ and going modulo the relation

[TABLE]

On the algebra side, $\mathcal{A}_{K}(\dot{E},\Lambda_{\dot{E}})$ is obtained from $\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}})$ in two steps by

1. (1)

first adjoining the mutually perpendicular idempotents diag$(s(X))-BB^{*}$ and $q_{\mathcal{Y}_{\lambda}}^{\prime}$, and going modulo the relation

[TABLE]

thereby, getting a new algebra $R$ and then 2. (2)

adjoining a universal isomorphism between the left module corresponding to $[BB^{*}]$ and the left module $\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}Rr(Y)$.

So, by [8, Theorems 5.1, 5.2], $\mathcal{V}(\mathcal{A}_{K}(\dot{E},\Lambda_{\dot{E}}))$ is obtained from $\mathcal{V}(\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}}))$ by adjoining a new generator $q_{\mathcal{Y}_{\lambda}}^{\prime\prime}$ and going modulo the relation

[TABLE]

This, along with the induction hypothesis, proves the theorem for the considered case.

Finally suppose $\lambda\in\Lambda_{\infty}^{S}$. Again $H(\dot{E},\Lambda_{\dot{E}})$ is obtained from $H(\dot{F},\Lambda_{\dot{F}})$ by adjoining a new generator $p_{\mathcal{X}_{\lambda}}$ and going modulo the relation

[TABLE]

On the other hand, analogous to the previous case, the algebra $\mathcal{A}_{K}(\dot{E},\Lambda_{\dot{E}})$ is obtained from $\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}})$ in two steps by

1. (1)

first adjoining the mutually perpendicular idempotents diag$(r(Y))-N^{*}N$ and $p_{\mathcal{X}_{\lambda}}^{\prime}$, and going modulo the relation

[TABLE]

thereby, getting a new algebra $R^{\prime}$ and then 2. (2)

adjoining a universal isomorphism between the left module corresponding to $[N^{*}N]$ and the left module $\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}R^{\prime}s(X)$.

So, by [8, Theorems 5.1, 5.2], $\mathcal{V}(\mathcal{A}_{K}(\dot{E},\Lambda_{\dot{E}}))$ is obtained from $\mathcal{V}(\mathcal{A}_{K}(\dot{F},\Lambda_{\dot{F}}))$ by adjoining a new generator $p_{\mathcal{X}_{\lambda}}^{\prime\prime}$ and going modulo the relation

[TABLE]

thereby completing the proof (using induction hypothesis).

∎

Remark 6.7*.*

We note that if $M$ is any conical abelian monoid then there exists a B-hypergraph $(\dot{E},\Lambda_{\dot{E}})$ such that $M\cong H(\dot{E},\Lambda)\cong\mathcal{V(A}_{K}(\dot{E},\Lambda_{\dot{E}}))$. For two different proofs of this fact, we refer the reader to [7, Proposition 4.4] or [29, Proposition 62].

7. Ideal lattices and Simplicity

In this section $(\dot{E},\Lambda)$ always denotes a B-hypergraph. Throughout this section we use the following notation: For $\lambda\in\Lambda,$

[TABLE]

7.1. The lattice of admissible triples in $(\dot{E},\Lambda)$

Definition 7.1**.**

A subset $V$ of $E^{0}$ is called bisaturated if for each $\lambda\in\Lambda_{T}^{S}$,

[TABLE]

The set of all bisaturated subsets of $E^{0}$ is denoted by BS$(\dot{E},\Lambda)$.

Note that empty set and $E^{0}$ are always elements of BS$(\dot{E},\Lambda)$. It is easy to check that BS$(\dot{E},\Lambda)$ is closed under arbitrary intersections.

If $V$ is a subset of $E^{0}$, the bisaturated closure of $V$, denoted $\overline{V}$, is the smallest bisaturated subset of $E^{0}$ containing $V$. Since the intersection of bisaturated subsets of $E^{0}$ is again bisaturated, $\overline{V}$ is well defined.

For $V\subseteq E^{0}$, $\overline{V}$ can be explicitly constructed as follows: Define $V_{0}=V$. If $n$ is odd positive integer, define

[TABLE]

and if $n$ is even positive integer, define

[TABLE]

Then $\overline{V}=\bigcup_{n\geq 0}V_{n}$.

Definition 7.2**.**

Let $V\subseteq E^{0}$ be bisaturated and for any $\lambda\in\Lambda$, set

[TABLE]

Then set

[TABLE]

where

[TABLE]

Let $V\subseteq E^{0}$ be a bisaturated set, $\Sigma\subseteq\Lambda_{\text{fin}}^{S}/V\sqcup\Lambda_{\infty}^{S}/V$ and $\Theta\subseteq\Lambda_{T}^{\text{fin}}/V\sqcup\Lambda_{T}^{\infty}/V$. A triple $(V,\Sigma,\Theta)$ is called an admissible triple and the set of all admissible triples in $(\dot{E},\Lambda)$ is denoted by AT$(\dot{E},\Lambda)$.

We define a relation $\leq$ in AT$(\dot{E},\Lambda)$ as follows: $(V_{1},\Sigma_{1},\Theta_{1})\leq(V_{2},\Sigma_{2},\Theta_{2})$ if

[TABLE]

[TABLE]

[TABLE]

We note that $(E^{0},\emptyset,\emptyset)$ is the maximum and $(\emptyset,\emptyset,\emptyset)$ is the minimum in AT$(\dot{E},\Lambda)$.

Definition 7.3**.**

Let $V$ be a bisaturated subset of $E^{0}$, $\Sigma\subseteq\Lambda^{S}(V)\sqcup\Lambda_{\text{fin}}^{S}/V\sqcup\Lambda_{\infty}^{S}/V$, and $\Theta\subseteq\Lambda_{T}(V)\sqcup\Lambda_{T}^{\text{fin}}/V\sqcup\Lambda_{T}^{\infty}/V$. The $(\Sigma,\Theta)$-bisaturation of $V$ is defined as the smallest bisaturated subset $\overline{V}(\Sigma,\Theta)$ of $E^{0}$ containing $H$ such that

1. (1)

If $\lambda\in\Sigma$ and $s(\lambda)\subseteq\overline{V}(\Sigma,\Theta)$, then $r(\lambda)\subseteq\overline{V}(\Sigma,\Theta)$ and 2. (2)

If $\lambda\in\Theta$ and $r(\lambda)\subseteq\overline{V}(\Sigma,\Theta)$, then $s(\lambda)\subseteq\overline{V}(\Sigma,\Theta)$.

We can construct $(\Sigma,\Theta)$-saturation of $V$ as follows- Define $\overline{V}_{0}(\Sigma,\Theta)=V$. If $n$ is odd positive integer, define

[TABLE]

and if $n$ is even positive integer, define

[TABLE]

Then $\overline{V}(\Sigma,\Theta)=\bigcup_{n\geq 0}\overline{V}_{n}(\Sigma,\Theta)$.

Proposition 7.4**.**

$(\emph{AT}(\dot{E},\Lambda),\leq)$* is a lattice, with supremum $\vee$ and infimum $\wedge$ given by*

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Proof.

Clearly, $(\widetilde{V},\widetilde{\Sigma},\widetilde{\Theta})\in$ AT$(\dot{E},\Lambda)$ and is greater than $(V_{i},\Sigma_{i},\Theta_{i})$ for $i=1,2$. Now let $(V,\Sigma,\Theta)\in$ AT$(\dot{E},\Lambda)$ such that $(V_{i},\Sigma_{i},\Theta_{i})\leq(V,\Sigma,\Theta)$ for $i=1,2$. It is enough to prove that $\widetilde{V}\subseteq V$ for all $n\in\mathbb{Z}^{+}$. We do this inductively. Define $\widetilde{V}_{n}=\overline{(V_{1}\cup V_{2})}_{n}(\Sigma_{1}\cup\Sigma_{2},\Theta_{1}\cup\Theta_{1})$. For $n=0$ the claim is clear by assumption. Now assume that $n\geq 1$ and that $\widetilde{V}_{n-1}\subseteq V$. Let $v\in\widetilde{V}_{n}$. If $v\in s(\lambda)$ or $v\in r(\lambda)$ for $\lambda\in\Lambda_{T}^{S}$, then $v\in V$ because $V$ is bisaturated. Now suppose $v\in s(\lambda)$ for $\lambda\in\Theta_{1}\cup\Theta_{2}$. By definition and the induction hypothesis, we have $r(\lambda)\subseteq\widetilde{V}_{m}\subseteq V$, where $m$ is largest even integer less than $n$. In particular, this implies that $\lambda\notin\Theta$. Since $\lambda\in\Theta_{1}\cup\Theta_{2}\subseteq\Lambda_{T}(V)\cup\Theta$ we conclude that $v\in H$, which completes the induction step. The inclusion $(\Theta_{1}\cup\Theta_{2})-\Lambda_{T}(\widetilde{V})\subseteq\Theta$ follows. Similar arguments shows that if $v\in r(\lambda)$ for $\lambda\in\Sigma_{1}\cup\Sigma_{2}$, then $v\in V$ and $(\Theta_{1}\cup\Theta_{2})-\Lambda_{T}(\widetilde{V})\subseteq\Theta$.

It is clear that $(\widehat{V},\widehat{\Sigma},\widehat{T})\in$ AT$(\dot{E},\Lambda)$ and $(\widehat{V},\widehat{\Sigma},\widehat{T})\leq(V_{i},\Sigma_{i},\Theta_{i})$ for $i=1,2$. If $(V,\Sigma,\Theta)\in$ AT$(\dot{E},\Lambda)$ such that $(V,\Sigma,\Theta)\leq(V_{i},\Sigma_{i},\Theta_{i})$ for $i=1,2$, then clearly $V\subseteq\widehat{V}$. Consider $\lambda\in\Theta-\Lambda_{T}(\widehat{V})$. Then there exists $v\in s(\lambda)-\widehat{V}$, so $v\notin V_{j}$ for some $j\in\{1,2\}$, and $\lambda\notin\Lambda_{T}(V_{j})$. Let us fix $j=1$. Since $(V,\Sigma,\Theta)\leq(V_{1},\Sigma_{1},\Theta_{1})$, it follows that $\lambda\in\Theta_{1}$. Hence, $\mathcal{Y}_{\lambda/V_{1}}$ is nonempty, hence $\mathcal{Y}_{\lambda/\widehat{V}}$ is nonempty. On the other hand, $\lambda\in\Theta$ implies that $\mathcal{Y}_{\lambda/V}$ is finite, hence $\mathcal{Y}_{\lambda/\widehat{V}}$ is finite. Thus $\lambda\in\Lambda_{T}^{\text{fin}}/\widehat{V}\sqcup\Lambda_{T}^{\infty}/\widehat{V}$. We also have $\lambda\in\Theta_{i}\sqcup\Lambda_{T}(V_{i})$ for $i=1,2$, because $(V,\Sigma,\Theta)\leq(V_{i},\Sigma_{i},\Theta_{i})$ for $i=1,2$, and consequently $\lambda\in\widehat{\Theta}$. This shows $\Theta\subseteq\widehat{\Theta}\sqcup\Lambda_{T}(V)$. Similarly we can show that $\Sigma\subseteq\widehat{\Sigma}\sqcup\Lambda^{S}(V)$ proving that $(V,\Sigma,\Theta)\leq(\widehat{V},\widehat{\Sigma},\widehat{\Theta})$. This shows that $(\widehat{V},\widehat{\Sigma},\widehat{\Theta})$ is the infimum required.

Hence AT$(\dot{E}\Lambda)$ is a lattice. ∎

7.2. The lattice of order-ideals in $H(\dot{E},\Lambda)$

Definition 7.5**.**

An order-ideal of a monoid $M$ is a submonoid $I$ of $M$ such that $x+y\in I$ for some $x,y\in M$ implies that both $x$ and $y$ belong to $I$.

Every monoid $M$ is equipped with a pre-order $\leq$ as follows: for $x,y\in M$, $x\leq y$ if and only if there exists $z\in M$ such that $x+z=y$. Hence an equivalent definition of an order-ideal $I$ is as follows: For each $x,y\in M$, if $x\leq y$ and $y\in I$ then $x\in I$.

Let $\mathcal{L}(M)$ denote the set of all order-ideals of $M$. We note that $\mathcal{L}(M)$ is closed under arbitrary intersections. For a submonoid $J$ of $M$, let $\langle J\rangle$ consists of those elements $x\in M$ such that $x\leq y$ for some $y\in J$. Then $\langle J\rangle$ denote the order-ideal generated by $J$. Then $\mathcal{L}(M)$ can be shown to a complete lattice with respect to inclusion. For, an arbitrary family $\{I_{i}\}$ of order-ideals of $M$, the supremum is given by $\langle\sum I_{i}\rangle$.

We want to study the lattice of order-ideals of $H(\dot{E},\Lambda)$. For convenience we modify some notations in the previous section as follows.

Notation 7.6**.**

[TABLE]

[TABLE]

Note that the above sums are finite.

[TABLE]

[TABLE]

Also,

[TABLE]

[TABLE]

[TABLE]

Finally set

[TABLE]

[TABLE]

Definition 7.7**.**

Let $F$ be the free abelian monoid on $E^{0}\sqcup Q^{0}\sqcup P^{0}$. We identify $H(\dot{E},\Lambda)$ with $F/\sim$, where $\sim$ is the congruence on $F$ generated by the relations

[TABLE]

and

[TABLE]

for $Z_{1},Z_{2}\in\mathcal{Z}$ with $Z_{1}\subsetneq Z_{2}$, and $q_{Z_{1}}\sim q_{Z_{2}}+\textbf{r}(Z_{2}-Z_{1}),$ and for $W_{1},W_{2}\in\mathcal{W}$ with $W_{1}\subsetneq W_{2}$, and $p_{W_{1}}\sim p_{W_{2}}+\textbf{s}(W_{2}-W_{1}).$

Lemma 7.8**.**

If $I$ is an oder-ideal of $H(\dot{E},\Lambda)$, then the set $V=I\cap E^{0}$ is bisaturated.

Proof.

Let $\lambda\in\Lambda_{T}^{S}$ and $r(\lambda)\subseteq V$, then $\textbf{r}(\lambda)=\textbf{s}(\lambda)\in I$. Since $I$ is order-ideal, and $s(X)\leq\textbf{s}(\lambda)$ for each $X\in\mathcal{X}_{\lambda}$, we have $s(X)\in I$ for each $X\in\mathcal{X}_{\lambda}$, and hence $s(\lambda)\subseteq V$. Converse follows similarly. ∎

Definition 7.9**.**

Let $V$ be a bisaturated subset of $E^{0}$.

[TABLE]

[TABLE]

If $I$ is an order-ideal of $V(\dot{E},\Lambda)$, set $\psi(I)=(V,\Sigma,\Theta),$ where

[TABLE]

Conversely, for any $(V,\Sigma,\Theta)\in AT(\dot{E},\Lambda)$, let $I(V,\Sigma,\Theta)$ denote the submonoid of $H(\dot{E},\Lambda)$ generated by the set $V\sqcup Q(\Theta)\sqcup P(\Sigma)$, where

[TABLE]

and $\langle I(V,\Sigma,\Theta)\rangle$ be the order-ideal generated by $I(V,\Sigma,\Theta)$. Set $\phi(V,\Sigma,\Theta)=\langle I(V,\Sigma,\Theta)\rangle$.

Lemma 7.10**.**

If $I$ is any order-ideal of $H(\dot{E},\Lambda)$, then $I=\phi\psi(I)$.

Proof.

Let $\psi(I)=(V,\Sigma,\Theta)$ and $I(V,\Sigma,\Theta)=J$ so that $\phi\psi(I)=\langle J\rangle$. It is clear that $J\subseteq I$ and therefore $\langle J\rangle\subseteq I$. For converse, consider a nonzero element $x\in I$. Then $x=\sum_{i}v_{i}+\sum_{j}q_{\alpha_{j}}+\sum_{k}p_{\beta_{k}}+\sum_{l}q_{Z_{l}}+\sum_{m}p_{W_{m}}$ for some $v_{i}\in E^{0}$, $\alpha_{j}\in\Lambda_{T}^{\text{fin}}$, $\beta_{k}\in\Lambda_{\text{fin}}^{S}$, $Z_{l}\in\mathcal{Z}$, and $W_{m}\in\mathcal{W}$. Since $I$ is an order ideal, $v_{i},q_{\alpha_{j}},p_{\beta_{k}},q_{Z_{l}},p_{W_{m}}\in I$, and so to prove that $x\in\langle J\rangle$, it is enough to show that $v,q_{\alpha},p_{\beta},q_{Z},p_{W}$ for all $v\in E^{0},\alpha\in\Lambda_{T}^{\text{fin}},\beta\in\Lambda_{\text{fin}}^{S},Z\in\mathcal{Z}$ and $W\in\mathcal{W}$.

Case 1 If $v\in E^{0}\cap I$, then $v\in V$ by definition of $H$, hence $v\in J$.

Case 2 Let $\alpha\in\Lambda_{T}^{\text{fin}}$ such that $q_{\alpha}\in I$.

Subcase 2.1 If $r(\alpha)\subseteq V$, then $\textbf{r}(\alpha)\in I$ and so $\textbf{s}(\alpha)=\textbf{r}(\alpha)+q_{\alpha}\in I$. Hence $s(X)\in V$ for each $X\in\mathcal{X}_{\alpha}$, and so $\textbf{s}(\alpha)\in J$. Since $q_{\alpha}\leq\textbf{s}(\alpha)$, it follows that $q_{\alpha}\in\langle J\rangle$.

Subcase 2.2 If $r(\alpha)\not\subseteq V$, then by definition $\alpha\in\Theta\cap\Lambda_{T}^{\text{fin}}/V$. Hence $q_{\alpha}\in J$.

Case 3 Let $\lambda\in\Lambda_{T}^{\infty}$ and $Z\in\mathcal{Z}_{\lambda}$ such that $q_{Z}\in I$.

Subcase 3.1 $\mathcal{Y}_{\lambda/V}=\emptyset$. This is equivalent to $r(\lambda)\subseteq V$ and the argument follows similar to subcase 2.1.

Subcase 3.2 $0<|\mathcal{Y}_{\lambda/V}|<\infty$. In this case, we have

[TABLE]

It follows that $\textbf{r}(\mathcal{Y}_{\lambda/V}-Z)\in I$ and so $r(\mathcal{Y}_{\lambda/V}-Z)\subseteq H$. Hence $\mathcal{Y}_{\lambda/V}\subseteq Z$. Since $r(Z-\mathcal{Y}_{\lambda/V})\subseteq H$, we get $q_{\lambda/V}=\textbf{r}(Z-\mathcal{Y}_{\lambda/V})+q_{Z}\in I,$ so that $\lambda\in\Theta$ by definition, and since $q_{Z}\leq q_{\lambda/V}$, we get $q_{Z}\in\langle J\rangle$.

Subcase 3.3 $|\mathcal{Y}_{\lambda/V}|=\infty$. Then there exists $Y\in\mathcal{Y}_{\lambda/V}-Z$, and we have

[TABLE]

But this implies that $r(Y)\in I$ and so $r(Y)\in V$, which contradicts $Y\in\mathcal{Y}_{\lambda/H}$. Thus $q_{Z}\in\langle J\rangle$.

The remaining cases are proved analogously. ∎

Construction 7.11**.**

Let $(\dot{E},\Lambda)$ be a B-hypergraph and $(V,\Sigma,\Theta)\in AT(\dot{E},\Lambda)$. For $A\subseteq E^{1}$, define

[TABLE]

We define the quotient B-hypergraph $(\dot{\widetilde{E}},\widetilde{\Lambda})$ as follows: $\dot{\widetilde{E}}$ is given by

[TABLE]

$r_{\widetilde{E}}$ and $s_{\widetilde{E}}$ are restriction maps of $r_{E}$ and $s_{E}$ respectively.

For $v\in\widetilde{E}^{0}$, set

[TABLE]

Let $\widetilde{\Lambda}$ be defined as follows:

[TABLE]

We note that if $\pi:M_{1}\rightarrow M_{2}$ is a monoid homomorphism and $M_{2}$ is conical, then ker $\pi:=\pi^{-1}(0)$ is an order-ideal of $M_{1}$.

Theorem 7.12**.**

Let $(\dot{E},\Lambda)$ be a B-hypergraph, $(V,\Sigma,\Theta)\in\emph{AT}(\dot{E},\Lambda)$ and $(\dot{\widetilde{E}},\widetilde{\Lambda})$ is the corresponding quotient B-hypergraph. Suppose that $I:=\langle I(V,\Sigma,\Theta)\rangle$ is the order ideal in $M:=H(\dot{E},\Lambda)$. Then there exists a monoid homomorphism $\pi:M\rightarrow\widetilde{M}:=M(\dot{\widetilde{E}})$ such that $I=\emph{ker}~{}\pi$.

Proof.

We begin by defining $\widetilde{v},\widetilde{q}_{\alpha},\widetilde{p}_{\beta},\widetilde{q}_{Z},\widetilde{p}_{W}\in\widetilde{M}$ for $v\in E^{0}$, $\alpha\in\Lambda_{T}^{\text{fin}}$, $\beta\in\Lambda_{\text{fin}}^{S}$, $Z\in\mathcal{Z}$, and $W\in\mathcal{W}$. For $v\in E^{0}$, set

[TABLE]

For $\alpha\in\Lambda_{T}^{\text{fin}}$, we define $\widetilde{q}_{\alpha}$ as follows:

1. (1)

If $s(\alpha)\subseteq V$ or $r(\alpha)\subseteq V$, $\widetilde{q}_{\alpha}=0$ 2. (2)

If $s(\alpha)\not\subseteq V$ and $r(\alpha)\not\subseteq V$, $\widetilde{q}_{\alpha}=\begin{cases}0\quad\text{if}~{}\alpha\in\Theta~{}\text{and}\\ q_{\widetilde{\alpha}}\quad\text{if}~{}\alpha\notin\Theta.\\ \end{cases}$

For $\beta\in\Lambda_{\text{fin}}^{S}$, we define $\widetilde{p}_{\beta}$ as follows:

1. (1)

If $s(\beta)\subseteq V$ or $r(\beta)\subseteq V$, $\widetilde{p}_{\beta}=0$. 2. (2)

If $s(\beta)\not\subseteq V$ and $r(\beta)\not\subseteq V$, $\widetilde{p}_{\beta}=\begin{cases}0\quad\text{if}~{}\beta\in\Sigma~{}\text{and}\\ p_{\widetilde{\beta}}\quad\text{if}~{}\beta\notin\Sigma.\\ \end{cases}$

For $\lambda\in\Lambda_{T}^{\infty}$, and $Z\in\mathcal{Z}_{\lambda}$, we define $\widetilde{q}_{Z}$ as follows:

1. (1)

If $s(\lambda)\subseteq V$, $\widetilde{q}_{Z}=0$. 2. (2)

If $s(\lambda)\not\subseteq V$, set $\widetilde{Z}=\{Y_{s}(V)\in\mathcal{Y}_{\widetilde{\lambda}}\mid Y\in Z\}$.

1. (a)

If $\lambda\in\Theta$ $\widetilde{q}_{Z}=\textbf{r}(\mathcal{Y}_{\widetilde{\lambda}}-\widetilde{Z})$, 2. (b)

If $\lambda\notin\Theta$ and $r(Z)\subseteq H$, $\widetilde{q}_{Z}=\textbf{s}(\lambda)$ 3. (c)

If $\lambda\notin\Theta$, $r(Z)\not\subseteq V$ and $\lambda\notin\Lambda_{T}^{\infty}/V$, $\widetilde{q}_{Z}=q_{\widetilde{Z}}$ and 4. (d)

If $\lambda\notin\Theta$, $r(Z)\not\subseteq V$, and $\lambda\in\Lambda_{T}^{\infty}/V$. $\widetilde{q}_{Z}=q_{\widetilde{\lambda}}+\textbf{r}(\mathcal{Y}_{\widetilde{\lambda}}-\widetilde{Z})$.

For $\lambda\in\Lambda_{\infty}^{S}$, and $W\in\mathcal{W}_{\lambda}$, we define $\widetilde{p}_{W}$ as follows:

1. (1)

If $r(\lambda)\subseteq V$, $\widetilde{p}_{W}=0$. 2. (2)

If $r(\lambda)\not\subseteq V$ set $\widetilde{W}=\{X_{r}(V)\in\mathcal{X}_{\widetilde{\lambda}}\mid X\in W\}$.

1. (a)

If $\lambda\in\Sigma$, $\widetilde{p}_{W}=\textbf{s}(\mathcal{X}_{\widetilde{\lambda}}-\widetilde{W})$, 2. (b)

If $r(\lambda)\not\subseteq V$, $\lambda\notin\Sigma$ and $s(W)\subseteq V$, $\widetilde{p}_{W}=\textbf{r}(\lambda)$, 3. (c)

If $r(\lambda)\not\subseteq V$, $\lambda\notin\Sigma$, $s(W)\not\subseteq V$ and $\lambda\notin\Lambda_{\infty}^{S}/V$, $\widetilde{p}_{W}=p_{\widetilde{W}}$ and 4. (d)

If $r(\lambda)\not\subseteq V$, $\lambda\notin\Sigma$, $s(W)\not\subseteq V$, and $\lambda\in\Lambda_{\infty}^{S}/V$, $\widetilde{p}_{W}=p_{\widetilde{\lambda}}+\textbf{s}(\mathcal{X}_{\widetilde{\lambda}}-\widetilde{W})$.

We define $\pi:M\rightarrow\widetilde{M}$ by mapping generators $v\mapsto\widetilde{v}$, for all $v\in E^{0}$, $q_{\alpha}\mapsto\widetilde{q}_{\alpha}$, for all $\alpha\in\Lambda_{T}^{\text{fin}}$, $p_{\beta}\mapsto\widetilde{p}_{\beta}$ for all $\beta\in\Lambda_{\text{fin}}^{S}$, $q_{Z}\mapsto\widetilde{q}_{Z}$ for all $Z\in\mathcal{Z}$, and $p_{W}\mapsto\widetilde{p}_{W}$ for all $W\in\mathcal{W}$. To show that $\pi$ defines a homomorphism we need to verify that images of the generators satisfy all the defining relations. Here we only show for $\lambda\in\Lambda_{T}$ and the argument follows analogously for $\lambda\in\Lambda_{S}$.

Let $\lambda\in\Lambda_{T}$. We introduce a new notation

[TABLE]

Suppose that $\lambda\in\Lambda_{T}^{S}$. If $s(\lambda)\subseteq V$, then $r(\lambda)\subseteq V$, and we get

[TABLE]

If $s(\lambda)\not\subseteq V$ then $r(\lambda)\not\subseteq V$, and we get

[TABLE]

Suppose that $\lambda\in\Lambda_{T}^{\text{fin}}$. If $s(\lambda)\subseteq V$, then $\mathcal{Y}_{\lambda/V}=\emptyset$. If $s(\lambda)\not\subseteq V$, and $r(\lambda)\subseteq V$, then $\mathcal{X}_{\lambda/V}=\emptyset$. In both of the above cases we have

[TABLE]

So let $s(\lambda)\not\subseteq V$ and $r(\lambda)\not\subseteq V$. If $\lambda\in\Theta$ then

[TABLE]

If $\lambda\notin\Theta$ then

[TABLE]

Now suppose that $\lambda\in\Lambda_{T}^{\infty}$, and $Z\in\mathcal{Z}_{\lambda}$. If $s(\lambda)\subseteq V$, then $\mathcal{Y}_{\lambda/V}=\emptyset$, and hence $r(Z)=\emptyset$. Then we have

[TABLE]

Hence we assume that $s(\lambda)\not\subseteq V$ for rest of the step. If $\lambda\in\Theta$ then $\widetilde{\lambda}\in\widetilde{\Lambda}_{\widetilde{T}}^{\widetilde{S}}$ and we have

[TABLE]

If $\lambda\notin\Theta$ and $r(Z)\subseteq V$ then we have

[TABLE]

If $\lambda\notin\Theta$, $r(Z)\not\subseteq V$ and $\lambda\notin\Lambda_{T}^{\infty}/V$ then we have

[TABLE]

If $\lambda\notin\Theta$, $r(Z)\not\subseteq V$ and $\lambda\in\Lambda_{T}^{\infty}/V$ then we have

[TABLE]

Now assume that for $\lambda\in\Lambda_{T}^{\infty}$ let $Z_{1},Z_{2}\in\mathcal{Z}_{\lambda}$ and $Z_{1}\subsetneq Z_{2}$. If $s(\lambda)\subseteq V$ then we have

[TABLE]

So we may assume that $s(\lambda)\not\subseteq V$. If $\lambda\in\Theta$,

[TABLE]

Only remaining case is when $\lambda\notin\Theta$. If $\lambda\in\Lambda_{T}^{\infty}/V-\Theta$, then

[TABLE]

Hence, we may assume that $\lambda\notin\Lambda_{T}^{\infty}/V$. If $r(Z_{2})\subseteq V$ then we have

[TABLE]

If $r(Z_{1})\subseteq V$ but $r(Z_{2})\not\subseteq V$, we have

[TABLE]

Finally, if $r(Z_{1})\not\subseteq V$, then we have

[TABLE]

Thus we have shown that $\pi$ is a monoid homomorphism.

Now we show that $I\subseteq\text{ker}~{}\pi$. Since ker$\pi$ is order-ideal, it suffices to show that $I(V,\Sigma,\Theta)\subseteq\text{ker}\pi$. For $v\in H$ we have $\pi(v)=\widetilde{v}=0$. For $\lambda\in\Theta\cap\Lambda_{T}^{\text{fin}}$, we have $\pi(q_{\lambda})=\widetilde{q}_{\lambda}=0$. If $\lambda\in\Theta\cap\Lambda_{T}^{\infty}$, then $\pi(q_{\lambda/V})=\widetilde{q}_{\lambda/V}=0$. Similarly we can verify that if $\lambda\in\Sigma\cap\Lambda_{\text{fin}}^{S}$ then $\pi(q_{\lambda})=0$ and if $\lambda\in\Sigma\cap\Lambda_{\infty}^{S}$ then $\pi(q_{\lambda/V})=0$.

We claim that $\psi(\text{ker}~{}\pi)=(V,\Sigma,\Theta)$. For, let $\psi(\text{ker}~{}\pi)=(\widetilde{V},\widetilde{\Sigma},\widetilde{\Theta})$. It follows from definition that $\widetilde{V}=I\cap E^{0}=V$ and by the previous paragraph $\Sigma\subseteq\widetilde{\Sigma}$ and $\Theta\subseteq\widetilde{\Theta}$. Consider $\lambda\in\Lambda_{T}^{\text{fin}}/V\sqcup\Lambda_{T}^{\infty}/V$. If $\mathcal{Y}_{\lambda}$ is finite and $\lambda\notin\Theta$, then $\pi(q_{\lambda})=\widetilde{q}_{\lambda}\neq 0$. Hence $q_{\lambda}\notin\text{ker}~{}\pi$ and so $\lambda\notin\widetilde{\Theta}$. If $\mathcal{Y}_{\lambda}$ is infinite and $\lambda\notin\Theta$, then $\pi(q_{\mathcal{Y}_{\lambda/V}})=\widetilde{q}_{\mathcal{Y}_{\lambda/V}}\neq 0$. Thus $\lambda\notin\widetilde{\Theta}$. Hence $\Theta=\widetilde{\Theta}$. Similarly $\Sigma=\widetilde{\Sigma}$.

Finally, since $\psi(\text{ker}~{}\pi)=(V,\Sigma,\Theta)$ and $I=\phi\circ\psi(I)$, we have that ker $\pi=\langle I(V,\Sigma,\Theta)\rangle=I$. ∎

Corollary 7.13**.**

If $(V,\Sigma,\Theta)\in\emph{AT}(\dot{E},\Lambda)$, then $(V,\Sigma,\Theta)=\psi\circ\phi(V,\Sigma,\Theta)$.

Theorem 7.14**.**

Let $\dot{E}$ be a B-hypergraph. Then there are mutually inverse lattice isomorphisms

[TABLE]

where $\phi(V,\Sigma,\Theta)=\langle I(V,\Sigma,\Theta)\rangle$ for $(V,\Sigma,\Theta)\in\emph{AT}(\dot{E},\Lambda)$ and $\psi$ is defined as in definition 7.9.

Proof.

The maps $\psi$ and $\phi$ are well defined by definition and by Lemma 7.10 $\phi\circ\psi$ is the identity map on $\mathcal{L}(H(\dot{E},\Lambda))$ and by Corollary 7.13 $\psi\circ\phi$ is the identity map on AT$(\dot{E},\Lambda)$. We only to have to show that $\psi$ and $\phi$ are order-preserving.

Suppose $I_{1}\subseteq I_{2}$ are order-ideals of $H(\dot{E},\Lambda)$ and $(V_{j},\Sigma_{j},\Theta_{j})=\psi(I_{j})$ for $j=1,2$. Clearly $V_{1}\subseteq V_{2}$. We only show that $\Theta_{1}\subseteq\Theta_{2}\sqcup\Lambda_{T}(V_{2})$. Let $\lambda\in\Theta_{1}$. First suppose that $\lambda\in\Lambda_{T}^{\text{fin}}/V_{1}$ and $q_{\lambda}\in I_{1}$. If $\lambda\in\Lambda_{T}^{\text{fin}}/V_{2}$, then $\lambda\in\Theta_{2}$. Otherwise, $r(\lambda)\subseteq V_{2}$ and so $\textbf{s}(\lambda)\in I_{2}$, which implies $s(\lambda)\in V_{2}$ and $\lambda\in\Lambda_{T}(V_{2})$. Now suppose that $\lambda\in\Lambda_{Y}^{\infty}/V_{2}$ and $q_{\lambda/V_{1}}\in I_{1}$. If $\lambda\in\Lambda_{T}^{\infty}/V_{2}$, then $q_{\lambda/V_{2}}$ is defined and also

[TABLE]

so $\lambda\in\Theta_{2}$. Otherwise, $r(\lambda)\subseteq V_{2}$ and so $\textbf{r}(\lambda/V_{1})\in I_{2}$, hence $\textbf{s}(\lambda)\in I_{2}$, again giving $\lambda\in\Lambda_{T}(V_{2})$. $\Sigma_{1}\subseteq\Sigma_{2}\sqcup\Lambda^{S}(V_{2})$ follows on similar lines.

Finally, let $(V_{1},\Sigma_{1},\Theta_{1})$ and $(V_{2},\Sigma_{2},\Theta_{2})$ are elements of AT$(\dot{E},\Lambda)$ such that $(V_{1},\Sigma_{1},\Theta_{1})\leq(V_{2},\Sigma_{2},\Theta_{2})$. Clearly $V_{1}\subseteq I(V_{2},\Sigma_{2},\Theta_{2})$. Consider $\lambda\in\Theta_{1}\cap\Lambda_{T}^{\text{fin}}$. If $\lambda\in\Theta_{2}$, then $q_{\lambda}\in I(V_{2},\Sigma_{2},\Theta_{2})$ by definition of $I(V_{2},\Sigma_{2},\Theta_{2})$. If $\lambda\in\Lambda_{T}(V)$, then

[TABLE]

and so $q_{\lambda}\in\langle I(V_{2},\Sigma_{2},\Theta_{2})\rangle$. Now consider $\lambda\in\Theta_{1}\cap\Lambda_{T}^{\infty}$. If $\lambda\in\Theta_{2}$, then $q_{\lambda/V_{2}}\in I(V_{2},\Sigma_{2},\Theta_{2})$ and since

[TABLE]

it follows that $q_{\lambda/V_{1}}\in\langle I(V_{2},\Sigma_{2},\Theta_{2})\rangle$. If $\lambda\in\Lambda(V)$, then

[TABLE]

and again $q_{\lambda/V_{1}}\in\langle I(V_{2},\Sigma_{2},\Theta_{2})\rangle$. A similar arguments shows that $\Sigma_{1}\subseteq\Sigma_{2}\sqcup\Lambda^{S}(V)$. Therefore all the generators of $I(V_{1},\Sigma_{1},\Theta_{1}$ lie in $\phi(V_{2},\Sigma_{2},\Theta_{2})$, and we conclude that $\phi(V_{1},\Sigma_{1},\Theta_{1})\subseteq\phi(V_{2},\Sigma_{2},\Theta_{2})$. Hence $\phi$ is order-preserving.

∎

7.3. The lattice of trace-ideals in $\mathcal{A}_{K}(\dot{E},\Lambda)$

Definition 7.15**.**

Let $R$ be an arbitrary ring and Idem($M_{\infty}(R))$ denote the set of idempotents in $M_{\infty}(R))$. An ideal $I$ of $R$ is called a trace-ideal provided $I$ can be generated by the entries of the matrices in some subset of Idem($M_{\infty}(R))$. We denote by Tr$(R)$ the set of all trace ideals of $R$. Since Tr$(R)$ is closed under arbitrary sums and arbitrary intersections, it forms a complete lattice with respect to inclusion.

Proposition 7.16**.**

[7, Proposition 10.10]** For any ring $R$ there are mutually inverse lattice isomorphisms

[TABLE]

given by

[TABLE]

[TABLE]

Lemma 7.17**.**

Let $(\dot{E},\Lambda)$ be a B-hypergraph. Then the trace ideals of $A:=\mathcal{A}_{K}(\dot{E},\Lambda)$ are precisely the idempotent generated ideals and the lattice isomorphism $\Phi:\mathcal{L(V}(A))\rightarrow\emph{Tr}(A)$ is expressed as

[TABLE]

Proof.

The proof goes exactly similar to [7, Proposition 6.2], except that in the present case, the V-monoid $\mathcal{V}(\mathcal{A}_{K}(\dot{E}))$ is generated by

[TABLE]

(Here, because of Theorem 6.6, we are using the notation $[q_{Z}^{\prime}]$ and $[p_{W}^{\prime}]$ for the generators of $\mathcal{V}(\mathcal{A}_{K}(\dot{E}))$. Strictly speaking, $[q_{z}^{\prime}]$ and $[p_{W}^{\prime}]$ stand for images of $q_{Z}$ and $p_{W}$ respectively under the map $\Gamma$ defined in Theorem 6.6). ∎

Theorem 7.18**.**

Let $(\dot{E}.\Lambda)$ be a B-hypergraph and $A=\mathcal{A}_{K}(\dot{E},\Lambda)$. Then there exist mutually inverse lattice isomorphisms

[TABLE]

Proof.

Set $M:=H(\dot{E},\Lambda)$ and let $\Gamma:M\rightarrow\mathcal{V}(A)$ be the monoid isomorphism. By abuse of notation we also use $\Gamma$ to denote the induced lattice isomorphism $\mathcal{L}(M)\rightarrow\mathcal{L(V}(A))$. Due to Theorem 7.14 and Proposition 7.16, we have mutually inverse lattice isomorphisms

[TABLE]

More explicitly, if $J\in\emph{Tr}(A)$, then $\zeta(J)=(H,\Sigma,\Theta)$, where

[TABLE]

[TABLE]

[TABLE]

For converse, let $(H,\Sigma,\Theta)\in\emph{AT}(\dot{E},\Lambda)$. First define $\xi(H,\Sigma,\Theta)=\langle H\sqcup P(\Sigma)\sqcup Q(\Theta)\rangle$, where $P(\Sigma)$ and $Q(\Theta)$ are defined as in Definition 7.9. Then define $J(H,\Sigma,\Theta)$ to be the order-ideal of $\mathcal{V}(A)$ generated by the set $H^{\prime}\sqcup P^{\prime}(\Sigma)\sqcup Q^{\prime}(\Sigma)$, where

[TABLE]

[TABLE]

[TABLE]

By Lemma 7.17, it follows that

[TABLE]

It is clear that $\xi(H,\Sigma,\Theta)\subseteq\Phi\Gamma\phi(H,\Sigma,\Theta)$.

If $e$ is an idempotent in $A$ such that $[e]\in J(H,\Sigma,\Theta)$, then

[TABLE]

where $v_{i}\in H$, $\alpha_{j}\in\Sigma\cap\Lambda_{\text{fin}}^{S}$, $\beta_{k}\in\Sigma\cap\Lambda_{\infty}^{S}$, $\gamma_{l}\in\Theta\cap\Lambda_{T}^{\text{fin}}$ and $\delta_{m}\in\Theta\cap\Lambda_{T}^{\infty}$. Therefore $e$ is equivalent to some idempotent $e^{\prime}\leq\mathcal{D}$ where $\mathcal{D}$ is a diagonal matrix with entries $v_{i},p_{\alpha_{j}},p_{\beta_{k}/H},q_{\gamma_{l}}$, and $q_{\delta_{m}/H}$.Thus it follows that $e$ lies in these $v_{i},p_{\alpha_{j}},p_{\beta_{k}/H},q_{\gamma_{l}}$, and $q_{\delta_{m}/H}$. Hence $\Phi\Gamma\phi=\xi$. ∎

7.4. Simplicity

A non-zero conical monoid $M$ is simple if its only order-ideals are $\{0\}$ and $M$.

Theorem 7.19**.**

Let $(\dot{E},\Lambda)$ be a B-hypergraph. Then the following conditions are equivalent

1. (1)

The only trace ideals of $\mathcal{A}_{K}(\dot{E},\Lambda)$ are [math] and $\mathcal{A}_{K}(\dot{E},\Lambda)$. 2. (2)

$H(\dot{E},\Lambda)$* is a simple monoid.* 3. (3)

$S=C_{\emph{fin}}$, $T=D_{\emph{fin}}$ and the only bisaturated subsets of $E^{0}$ are $\emptyset$ and $E^{0}$.

Proof.

From Proposition 7.16 it follows that $(1)\Leftrightarrow(2)$.

$(2)\Rightarrow(3):$ Observe that $(\Lambda_{\text{fin}}^{S}/\emptyset)\sqcup(\Lambda_{\infty}^{S}/\emptyset)=C_{\text{fin}}-S$ and $(\Lambda_{T}^{\text{fin}}/\emptyset)\sqcup(\Lambda_{T}^{\infty}/\emptyset)=D_{\text{fin}}-T$. Similarly, $(\Lambda_{\text{fin}}^{S}/E^{0})\sqcup(\Lambda_{\infty}^{S}/E^{0})=\emptyset$ and $(\Lambda_{T}^{\text{fin}}/E^{0})\sqcup(\Lambda_{T}^{\infty}/E^{0})=\emptyset$. By Theorem 7.14, the only members of AT$(\dot{E},\Lambda)$ are $(\emptyset,\emptyset,\emptyset)$ and $(E^{0},\emptyset,\emptyset)$. If $\lambda\in\Lambda_{\text{fin}}^{S}$, then $(\emptyset,\{\lambda\},\emptyset)\in\text{AT}(\dot{E},\Lambda)$. This proves that $S=C_{\text{fin}}$. Similarly $T=D_{\text{fin}}$. If $H$ is any bisaturated subset of $E^{0}$, then $(H,\emptyset,\emptyset)\in\text{AT}(\dot{E},\Lambda)$ and hence the only bisaturated subsets of $E^{0}$ are $E^{0}$ and $\emptyset$.

$(3)\Rightarrow(2):$ In this case AT$(\dot{E},\Lambda)=\{(E^{0},\emptyset,\emptyset),(\emptyset,\emptyset,\emptyset)\}$. The result follows at once from Theorem 7.14. ∎

Although the characterization of simplicity of $\mathcal{A}_{K}(\dot{E},\Lambda)$ is quite difficult, from theorem 7.19 we can easily conclude the following necessary condition.

Corollary 7.20**.**

Let $(\dot{E},\Lambda)$ be a hypergraph and suppose $L_{K}(\dot{E},\Lambda)$ is simple then the only trace ideals of $L_{K}(\dot{E},\Lambda)$ are [math] and $L_{K}(\dot{E},\Lambda)$.

8. Representations of Leavitt path algebras of regular hypergraphs

Given a graph $E$, the category of quiver representations of $E$ is the category of functors from the path category $\mathcal{C}_{E}$ to the category of $K$-vector spaces. A morphism of quiver representations is a natural transformation between two such functors. In other words, a quiver representation $\rho$ assigns a (possibly infinite dimensional) $K$-vector space $\rho(v)$ to each $v\in E^{0}$ and a linear transformation $\rho(e):\rho(s(e))\rightarrow\rho(r(e))$ to each $e\in E^{1}$. And a morphism of quiver representations $\phi:\rho\rightarrow\rho^{\prime}$ is a family of linear transformations $\{\phi_{v}\mid\rho(v)\rightarrow\rho^{\prime}(v)\}_{v\in E^{0}}$ such that for each $e\in E^{1}$ the following diagram commutes:

[TABLE]

This section generalizes the results of [17]. Throughout this section, a hypergraph always means a regular hypergraph. We will work in the category $\mathfrak{M}_{L}$ of unital (right) modules over $L:=L_{K}(\dot{E},\Lambda)$ where $(\dot{E},\Lambda)$ is a hypergraph. The category $\mathfrak{M}_{L}$ is closed under taking quotients, submodules, extensions and arbitrary sums and hence it is an abelian category with sums. Note however that it is not closed under infinite product if $E^{0}$ is infinite.

Lemma 8.1**.**

Let $M$ be a right $L$-module. Then $\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}Ms(X)$ is isomorphic (as vector space) to $\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}Mr(Y)$ for every $\lambda\in\Lambda$.

Proof.

For each $\lambda\in\Lambda$, let $[\lambda]$ be the rectangular matrix of size $|\mathcal{Y}_{\lambda}|\times|\mathcal{X}_{\lambda}|$ whose entry in $Y^{th}$ row and $X^{th}$ column is the edge $YX$. Then $[\lambda]:\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}Ms(X)\rightarrow\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}Mr(Y)$given by

[TABLE]

is a well defined linear map, where $\mu_{(XY)}$ is right multiplication by the edge $XY$. We show that $[\lambda]$ is an isomorphism with the inverse $[\lambda]^{\ast}$, which is the adjoint transpose matrix of $[\lambda]$. Note that $[\lambda]^{\ast}:\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}Mr(Y)\rightarrow\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}Ms(X)$ is given by

[TABLE]

We check their compositions:

[TABLE]

Similarly by $L1$, we get

[TABLE]

which establishes the result. ∎

Remark 8.2*.*

If $M$ is unital, then for any $m\in M$, we have $m=\sum_{k=1}^{l}m_{k}v_{k}$ for some vertices $v_{k}\in E^{0}$. Hence $M=\sum\limits_{v\in E^{0}}Mv$. When considered as paths, the vertices of $E$ form a set of orthogonal idempotents, hence the above sum is direct. Therefore we have

[TABLE]

Theorem 8.3**.**

The category $\mathfrak{M}_{L}$ is equivalent to the full subcategory of quiver representations $\rho$ of $E$ satisfying:

[TABLE]

Proof.

Let $M$ be a right $L$-module. We define a quiver representation $\rho_{M}$ as follows: $\rho_{M}(v)=Mv$ for each $v\in E^{0}$ and $\rho_{M}(e):Ms(e)\rightarrow Mr(e)$ is given by $ms(e)\rho_{M}(e):=ms(e)e=me=mr(e)$. By Lemma 8.1, (H) is satisfied. If $\varphi:M\rightarrow N$ is an $L$-module homomorphism then $\varphi_{v}$ is the linear transformation making the following diagram commtative.

[TABLE]

Since right multiplication by an edge $e$ commutes with $\varphi$, this defines a homomorphism of quiver representations.

Given a quiver representation $\rho$, we define the correspoding module $M_{\rho}:=\bigoplus\limits_{v\in E^{0}}\rho(v)$. To get an $L$-module structure on $M_{\rho}$, we define the following projections and inclusions: For each $v\in E^{0}$, define

[TABLE]

and for each $\lambda\in\Lambda$, $X\in\mathcal{X}_{\lambda}$ and $Y\in\mathcal{Y}_{\lambda}$, define

[TABLE]

[TABLE]

Now let $mv:=mp_{v}i_{v}$, $m(XY):=mp_{s(X)}i_{X}[\rho(\lambda)]p_{Y}i_{r(Y)}$, and $m(YX)^{\ast}:=mp_{r(Y)}i_{Y}[\rho(\lambda)]^{\ast}p_{X}i_{s(X)}$. To keep track of the last defining relations, we draw the following diagram:

[\lambda]$$[\lambda]^{\ast}$$\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}\rho(s(X))$$\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}\rho(s(X))$$\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}\rho(r(Y))$$\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}\rho(r(Y))$$\bigoplus\limits_{w\in E^{0}}\rho(w)$$\bigoplus\limits_{w\in E^{0}}\rho(w)$$\rho(s(X))$$\rho(s(X))$$\rho(r(Y))$$\rho(r(Y))$$p_{s(X)}$$i_{s(X)}$$i_{r(Y)}$$p_{r(Y)}$$i_{X}$$p_{X}$$p_{Y}$$i_{Y}

Here the composition of the upper arrows correspond to right multiplication by $(XY)$ and the composition by lower arrows correspond to right multiplication by $(YX)^{\ast}$. Verifying that the above defining relation satisfy defining relations of $L$ is left to the reader.

Now we show that the above constructions give equivalance of categories. By Remark 8.2, we have $M_{\rho_{M}}=\bigoplus\limits_{v\in E^{0}}Mv=M$ and their $L$-module structures also match. Given a module homomorphism $\varphi:M\rightarrow N$, we have $\varphi=\bigoplus\limits_{v\in E^{0}}\varphi_{v}:\bigoplus\limits_{v\in E^{0}}Mv\rightarrow\bigoplus\limits_{v\in E^{0}}N$.

For the composition in the other order $\rho_{M_{\rho}}(v)=M_{\rho}v=\left(\bigoplus\limits_{w\in E^{0}}\rho(w)\right)v=\rho(v)$ and $\rho(e)=\rho_{M_{\rho}}(e):M_{\rho}s(e)\rightarrow M_{\rho}r(e)$. For, let $e=(XY)$ for some $X\in\mathcal{X}_{\lambda}$ and $Y\in\mathcal{Y}_{\lambda}$, then the following diagram commutes.

[TABLE]

Finally, for any homomorphism $\{\varphi_{v}:\rho(v)\rightarrow\sigma(v)\}_{v\in E^{0}}$ from $\rho$ to $\sigma$, the $v$-component of $\bigoplus\limits_{w\in E^{0}}\varphi_{w}$ is $\varphi_{v}:\rho_{M_{\rho}}(v)=\rho(v)\rightarrow\sigma_{M_{\sigma}}(v)=\sigma(v)$. ∎

Remark 8.4*.*

We note that the full subcategory of graded quiver representations with respect to standard $\mathbb{Z}$-grading satisfying condition (H) is equivalent to the category of graded unital $L$-modules. The proof follows on similar lines of proof of Theorem 8.3.

Theorem 8.5**.**

The composition of the forgetful functor from $\mathfrak{M}_{L}$ to $\mathfrak{M}_{E}$ with $\underline{\hskip 6.0pt}\otimes_{K(E)}L$ from $\mathfrak{M}_{E}$ to $\mathfrak{M}_{L}$ is naturally equivalent to the identity functor on $\mathfrak{M}_{L}$.

Proof.

We note that both forgetful functor and $\underline{\hskip 6.0pt}\otimes_{K(E)}L$ send unital modules to unital modules. Let the composition of forgetful functor with $\underline{\hskip 6.0pt}\otimes_{K(E)}L$ be denoted by $\mathcal{F}$ and the identity functor on $\mathfrak{M}_{L}$ be denoted by $\mathcal{I}$. If $M$ is an $L$-module, the $L$-module homomorphism $M\otimes_{K(E)}L\rightarrow M$ given by $m\otimes a\mapsto ma$ defines an natural transformation from $\mathcal{F}$ to $\mathcal{I}$. To see that this is an isomorphism, we define its inverse $M\rightarrow M\otimes_{K(E)}L$ by $m\mapsto\sum\limits_{\begin{subarray}{c}v\in E^{0}\\ mv\neq 0\end{subarray}}m\otimes v$. Observe that this sum is finite since $M$ is unital.

To check that the above inverse map defines an $L$-linear map, we need to check on generators. For every $w\in E^{0}$ and $m\in M$ we have $\sum mu\otimes v=m\otimes u=\left(\sum m\otimes v\right)u$, since $E^{0}$ is a set of orthogonal idempotents. For all $\lambda\in\Lambda$, $X\in\mathcal{X}_{\lambda}$, $Y\in\mathcal{Y}_{\lambda}$ and $m\in M$ we have

[TABLE]

Similarly $\sum m(YX)^{\ast}\otimes v=(\sum m\otimes v)(YX)^{\ast}$.

The composition $m\mapsto\sum m\otimes v\mapsto\sum mv=m$. Since elements of the form $m\otimes v$ with $m\in Mv$ generate $M\otimes L$ as an $L$-module and for such elements we have $m\otimes v\mapsto mv\mapsto mv\otimes v=m\otimes v$, the other composition is also identity. ∎

Recall that the universal localization $\Sigma^{-1}A$ of an algebra $A$ with respect to a set $\Sigma=\{\sigma:P_{\sigma}\rightarrow Q_{\sigma}\}$ of homomorphisms between finitely generated projective $A$-modules, is an initial object among algebra homomorphisms $f:A\rightarrow B$ such that $\sigma\otimes id_{B}:P_{\sigma}\otimes_{A}B\rightarrow Q_{\sigma}\otimes_{A}B$ is an isomorphism for every $\sigma\in\Sigma$.

Theorem 8.6**.**

$L$* is the universal localization of $K(E)$ with respect to*

[TABLE]

[TABLE]

Proof.

Since $v\in E^{0}$ is an idempotent, the cyclic module $vK(E)$ is projective. For each $\lambda\in\Lambda$, $\sigma_{\lambda}\otimes id_{L}$ is an isomorphism with inverse $\sigma_{\lambda}^{\ast}$, where $(a_{X})_{X\in\mathcal{X}_{\lambda}}\xmapsto{\sigma_{\lambda}^{\ast}}\left(\sum\limits_{X\in\mathcal{X}_{\lambda}}(YX)^{\ast}a_{X}\right)_{Y\in\mathcal{Y}_{\lambda}}$. If $f:K(E)\rightarrow B$ is an algebra homomorphism, then $f(v)^{2}=f(v)$ and $vK(E)\otimes_{K(E)}B\cong f(v)B$ by $a\otimes b\mapsto f(a)b$ and $b\mapsto v\otimes b$.

Let $f:K(E)\rightarrow B$ be an algebra homomorphism such that $\sigma_{\lambda}\otimes id_{B}$ is an isomorphism for all $\lambda\in\Lambda$ then the composition $f(s(X))B\cong(s(X))K(E)\otimes_{K(E)}B\xhookrightarrow{i_{s(X)}\otimes id_{B}}\left(\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}s(X)K(E)\right)\otimes_{K(E)}B\xrightarrow{\sigma_{\lambda}^{\ast}\otimes~{}id_{B}}\left(\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}r(Y)K(E)\right)\otimes_{K(E)}B\xrightarrow{p_{r(Y)}\otimes id_{B}}r(Y)K(E)\otimes_{K(E)}B\cong f(r(Y))B$ is uniquely and completely determined, which we call $f((YX)^{\ast})$. Now $\tilde{f}(v):=f(v)$ for all $v\in E^{0}$, $\tilde{f}((XY)):=f((XY))$ for all $\lambda\in\Lambda$, $X\in\mathcal{X}_{\lambda}$ and $Y\in\mathcal{Y}_{\lambda}$ defines the unique homomorphism $\tilde{f}:L\rightarrow B$ factoring $f$ through $K(E)\rightarrow L$. ∎

Proposition 8.7**.**

Let $(\dot{E},\Lambda)$ be hypergraph. If $d:E^{0}\rightarrow\mathbb{N}\cup\{\infty\}$ satisfies

[TABLE]

then there is an $L$-module $M$ with $\emph{dim}_{K}(Mv)=d(v)$.

Proof.

Define the quiver representation $\rho$ by $\rho(v)=K^{d(v)}$ if $d(v)<\infty$ and $\rho(v)=K^{(\mathbb{N})}$ otherwise. Then by definition of $d$ we can find isomorphism $\theta_{\lambda}:\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}\rho(s(X))\rightarrow\bigoplus\limits_{Y\in\mathcal{Y}-\lambda}\rho(r(Y))$ for all $\lambda\in\Lambda$. Let $\rho(XY):=i_{s(X)}\theta_{\lambda}p_{r(Y)}$ for all $\lambda\in\Lambda$, $X\in\mathcal{X}_{\lambda}$ and $Y\in\mathcal{Y}_{\lambda}$. Condition (H) is satisfied by construction and the corresponding $L$-module $M$ of Theorem 8.3 has dim${}_{K}(Mv)=\text{dim}_{K}\rho(v)=d(v)$. ∎

Definition 8.8**.**

A dimension function of a hypergraph $(\dot{E},\Lambda)$ is a function $d:E^{0}\rightarrow\mathbb{N}$ satisfying $\sum\limits_{X\in\mathcal{X}_{\lambda}}s(X)=\sum\limits_{Y\in\mathcal{Y}_{\lambda}}r(Y)$ for all $\lambda\in\Lambda$.

Remark 8.9*.*

If the $L$-module $M$ is finitary, i.e, dim$(Mv)<\infty$ for all $v\in E^{0}$ then by Lemma 8.1, $d(v):=\text{dim}(Mv)$ is a dimension function. By Proposition 8.7, the converse also holds. that is, every dimension function is realizable. Moreover, since dim($M)=\sum\limits_{v\in E^{0}}\text{dim}(Mv)$, $d(v):=\text{dim}(Mv)$ has finite support if $M$ is finite dimensional.

8.1. Support subgraphs and the hypergraph monoid

Let $\mathcal{H}=(\dot{E},\Lambda)$ be a hypergraph and $E^{\prime}$ be a full subgraph of $E$. Then there is a natural (hyper) bi-separation induced on $E^{\prime}$ from $\mathcal{H}$ as follows: For each $X\in C$, if $s(X)\in(E^{\prime})^{0}$, define $X^{\prime}=X\bigcap(E^{\prime})^{1}$ and similarly for each $Y\in D$, if $r(Y)\in(E^{\prime})^{0}$, define $Y^{\prime}=Y\bigcap(E^{\prime})^{0}$. Also for each $\lambda\in\Lambda$, define $\lambda^{\prime}$ using the following data: $\mathcal{X}_{\lambda^{\prime}}=\{X^{\prime}\mid X\in\mathcal{X}_{\lambda}\}$ and $\mathcal{Y}_{\lambda^{\prime}}=\{Y^{\prime}\mid Y\in\mathcal{Y}_{\lambda}\}$. Finally define $\Lambda^{\prime}=\{\lambda^{\prime}\mid\lambda\in\Lambda,\mathcal{X}_{\lambda}\neq\emptyset~{}\text{and}~{}\lambda\neq\emptyset\}$. We call $\mathcal{H}^{\prime}=(\dot{E^{\prime}},\Lambda^{\prime})$ full sub-hypergraph of $\mathcal{H}$ (hyper-induced from $E^{\prime}$).

Definition 8.10**.**

Let $\mathcal{H}=(\dot{E},\Lambda)$ be a hypergraph. A full sub-hypergraph $\mathcal{H}^{\prime}=(\dot{E^{\prime}},\Lambda^{\prime})$ is called co-bisaturated if the following conditions are satisfied: For every $\lambda^{\prime}\in\Lambda^{\prime}$,

1. (1)

if $s(X)\in(E^{\prime})^{0}$, then $X\cap(E^{\prime})^{1}\neq\emptyset$, where $X\in\mathcal{X}_{\lambda^{\prime}}$, 2. (2)

if $r(Y)\in(E^{\prime})^{0}$, then $Y\cap(E^{\prime})^{1}\neq\emptyset$, where $Y\in\mathcal{Y}_{\lambda^{\prime}}$.

We note that a full sub-hypergraph $\dot{E^{\prime}}$ of $\dot{E}$ is a co-bisaturated if and only if $E^{0}-(E^{\prime})^{0}$ is a bisaturated subset of $E^{0}$.

Let $\mathcal{H}=(\dot{E},\Lambda)$ be a hypergraph and $M$ be a right $L(\mathcal{H})$-module. The support subgraph of $M$, denoted by $E_{M}$, is the full subgraph of $E$ induced on $V_{M}:=\{v\in E^{0}\mid Mv\neq 0\}$. The hypergraph $\mathcal{H}_{M}=(\dot{E}_{M},\Lambda_{M})$, which is the full sub-hypergraph of $\mathcal{H}$ hyper-induced from $E_{M}$, is called the support sub-hypergraph of $M$.

Lemma 8.11**.**

Let $\mathcal{H}=(\dot{E},\Lambda)$ be a hypergraph and $\mathcal{H}^{\prime}=(\dot{E^{\prime}},\Lambda^{\prime})$ be a full sub-hypergraph of $\mathcal{H}$. Then the following are equivalent:

1. (1)

$\mathcal{H}^{\prime}=\mathcal{H}_{M}$, is the support sub-hypergraph of a unital $L(\mathcal{H})$-module $M$. 2. (2)

$\mathcal{H}^{\prime}$* is co-bisaturated.* 3. (3)

The map $\theta:L(\mathcal{H})\rightarrow L(\mathcal{H}^{\prime})$ defined (on generators) by

[TABLE]

extends to an onto algebra homomorphism.

Proof.

$(1)\Rightarrow(2)$: Let $\lambda^{\prime}\in\Lambda^{\prime}$ and $X\in\mathcal{X}_{\lambda^{\prime}}$. Assume $s(X)\in(E_{M})^{0}$, then $0\neq Ms(X)\hookrightarrow\bigoplus\limits_{X\in\mathcal{X}_{\lambda}}Ms(X)\cong\bigoplus\limits_{Y\in\mathcal{Y}_{\lambda}}Mr(Y)$ implies that there exists $Y\in\mathcal{Y}_{\lambda}$ such that $X\cap Y\neq\emptyset$ which is equivalent to $X\cap(E^{\prime})^{1}\neq\emptyset$. Similarly, if $Y\in\mathcal{Y}_{\lambda^{\prime}}$ and $r(Y)\in(E_{M})^{0}$, then $Y\cap(E^{\prime})^{1}\neq\emptyset$.

$(2)\Rightarrow(3)$: We check that $\theta$ preserves the defining relations of $L(\mathcal{H})$. It is direct that path algebra relations are satisfied. Let $\lambda^{\prime}\in\Lambda^{\prime}$, $X_{1},X_{2}\in\mathcal{X}_{\lambda^{\prime}}$ and $Y_{1},Y_{2}\in\mathcal{Y}_{\lambda^{\prime}}$. If $s(X_{i}),r(Y_{i})\in(E^{\prime})^{0}$ then $X_{i}\cap(E^{\prime})^{1}\neq\emptyset$ and $Y_{1}\cap(E^{\prime})^{1}\neq\emptyset$. Hence the image of $\sum\limits_{X\in\mathcal{X}_{\lambda}}(Y_{1}X)^{\ast}(XY_{2})=\delta_{Y_{1}Y_{2}}r(Y)$ is $\sum\limits_{X\in\mathcal{X}_{\lambda^{\prime}}}(Y_{1}X)^{\ast}(XY_{2})=\delta_{Y_{1}Y_{2}}r(Y)$. Similarly, the image of $\sum\limits_{Y\in\mathcal{Y}_{\lambda}}(X_{1}Y)(YX_{2})^{\ast}=\delta_{X_{1}X_{2}}s(X)$ is $\sum\limits_{Y\in\mathcal{Y}_{\lambda^{\prime}}}(X_{1}Y)(YX_{2})^{\ast}=\delta_{X_{1}X_{2}}s(X)$.

$(3)\Rightarrow(1)$: Let $M:=L(\mathcal{H}^{\prime})\cong L(\mathcal{H})/\text{Ker}\theta$. Now $v\in(E^{\prime})^{0}$ if and only if $\theta(v)\neq 0$ and $Mv=L(\mathcal{H}^{\prime})v\neq 0$. Hence the vertex set of $E_{M}$ is $(E^{\prime})^{0}$. It is routine to check that $\mathcal{H}^{\prime}$ is full sub-hypergraph and hence $\mathcal{H}_{M}=\mathcal{H}^{\prime}$. ∎

Proposition 8.12**.**

If $M$ is a unital $L(\mathcal{H})$-module then $M$ also has the structure of a unital $L(\mathcal{H}_{M})$-module induced through the epimorphism $\theta:L(\mathcal{H})\rightarrow L(\mathcal{H}_{M})$. Moreover, $\emph{Ker}\theta$ is generated by $E^{0}-V_{M}=\{v\in E^{0}\mid Mv=0\}$ and $\emph{Ker}\theta\subseteq\emph{Ann}M$.

Proof.

Let $\rho_{M}$ be the quiver representation of $E$ corresponding to $M$ as defined in the proof of Theorem 8.3. We claim that the restriction of $\rho_{M}$ to $E_{M}$ satisfies (H). That is, if $\rho^{\prime}:=\rho_{M}|_{E_{M}}$, then for all $\lambda^{\prime}\in\Lambda_{M}$,

[TABLE]

For,

[TABLE]

Let $M^{\prime}$ be the unital $L(\mathcal{H}_{M})$-module corresponding to $\rho^{\prime}$. Now $M^{\prime}$ is also an $L(\mathcal{H})$-module via $\theta:L(\mathcal{H})\rightarrow L(\mathcal{H}_{M})$. As vector spaces $M^{\prime}=\bigoplus\limits_{v\in V_{M}}Mv\cong\bigoplus\limits{v\in E^{0}}Mv=M$. We can define an $L(\mathcal{H}_{M})$-module structure on $M$ via this isomorphism. The action of the generators on $M$ and $M^{\prime}$ is compatible with this isomorphism, so $M\cong M^{\prime}$ as $L(\mathcal{H})$-modules. Thus the $L(\mathcal{H})$-module structure of $M$ is induced from the $L(\mathcal{H}_{M})$-module structure via $\theta$.

For the second part, Let $I_{M}$ be the ideal generated by $E^{0}-V_{M}$. We show that $L(\mathcal{H}_{M})\cong L(\mathcal{H})/\text{Ker}\theta$ and $L(\mathcal{H})/I_{M}$ are isomorphic. Since, $E^{0}-V_{M}\subseteq\text{Ker}\theta$, we have a surjection from $L(\mathcal{H})/I_{M}$ to $L(\mathcal{H}_{M})$. Let $\varphi:L(\mathcal{H}_{M})\rightarrow L(\mathcal{H})/I$ be defined on generators by $x\mapsto x+I$ where $x\in E^{0}\sqcup E^{1}\sqcup\overline{E^{1}}$. It is not hard to show that $\varphi$ is a homomorphism and the inverse of the above surjection. ∎

Recall that given a hypergraph $\mathcal{H}=(\dot{E},\Lambda)$, its hypergraph monoid $H(\mathcal{H})$ is defined as the additive monoid generated by $E^{0}$ modulo the following relations:

[TABLE]

Therefore, dimension functions of $\mathcal{H}$ correspond exactly to monoid homomorphisms from $H(\mathcal{H})$ to $\mathbb{N}$.

Since $H(\mathcal{H})$ is isomorphic to the monoid $\mathcal{V}(L(\mathcal{H}))$, the generator $v$ of $H(\mathcal{H})$ corresponds to the (right) projective $L(\mathcal{H})$-module $vL(\mathcal{H})$. The corresponding relations among the isomorphism classes of the cyclic projective modules was shown to hold in the proof of Theorem 8.6. We can now reinterpret the existence of a nonzero finite dimensional represenatation in terms of the nonstable K-theory of $L(\mathcal{H})$.

Theorem 8.13**.**

$L(\mathcal{H})$* has a nonzero finite dimensional representation if and only if $\dot{E}$ has a finite, full co-bisaturated sub-hypergraph $\mathcal{G}$ with a nonzero monoid homomorphism from $\mathcal{V}(L(\mathcal{G}))$ to $\mathbb{N}$.*

Proof.

By Remark 8.9, $L(\mathcal{H})$ has a nonzero finite dimensional representation if and only if $\mathcal{H}$ has a nonzero dimension function of finite support. The support of such dimension function defines a finite, full, co-bisaturated sub-hypergraph $\mathcal{G}$ and its restriction gives a nonzero dimension function on $\mathcal{G}$ and thus a nonzero monoid homomorphism from $\mathcal{V}(L(\mathcal{G}))$ to $\mathbb{N}$ as well. Conversely, since $\mathcal{G}$ is co-bisaturated, any nonzero dimension function on $\mathcal{G}$ can be extended by [math] to a dimension function on $\mathcal{H}$ and this gives a nonzero dimension function of finite support on $\mathcal{H}$. ∎

9. Some remarks on Cohn-Leavitt path algebras of B-hypergraphs with Invariant Basis Number

Let $\mathcal{H}=(\dot{E},\Lambda)$ be a finite B-hypergraph and let $H:=H(\mathcal{H})$ be the $H$-monoid of $\mathcal{H}$. Let the Cohn-Leavitt path algebra $\mathcal{A}_{K}(\mathcal{H})$ be denoted simply by $L$ and its Grothendieck group by $\mathcal{K}_{0}(L)$. Let $\mathcal{U}(L)$ denote the submonoid of the $\mathcal{V}$-monoid $\mathcal{V}(L)$ generated by the element $[L]\in\mathcal{V}(L)$. Then $L$ has IBN property if and only if $\mathcal{U}(L)$ has infinite order. Now suppose $G(\mathcal{U}(L))$ denotes the Grothendieck group of $\mathcal{U}(L)$. Then one can show that the natural map $G(\mathcal{U}(L))\rightarrow\mathcal{K}_{0}(L)$ induced by the inclusion $\mathcal{U}(L)\hookrightarrow\mathcal{V}(L)$ is an embedding (see [16, Proposition 7]). So $[L]$, treated as an element in the group $\mathcal{K}_{0}(L)$, has infinite order. This means that the element $[L]\otimes 1$ in $\mathcal{K}_{0}(L)\otimes\mathbb{Q}$ is nonzero. We know that $[L]\in\mathcal{V}(L)$ corresponds to the element $[\sum_{v\in E^{0}}v]\in H$ under the isomorphism of functors proved in Theorem 6.6. So, if $G(H)$ denotes the Grothendieck group of $H$, from the above arguments we can conclude that $L$ has IBN if and only if $\sum_{v\in E^{0}}v$ is nonzero in $G(H)\otimes\mathbb{Q}$, which is equivalent saying that $\sum_{v\in E^{0}}v$ is not in the $\mathbb{Q}-$ linear span of the elements of $R$ in $\mathbb{Q}\Omega$ (cf. [16, Theorem 13]), where $\Omega$ is the set $E^{0}\sqcup Q\sqcup P$ ($Q$ and $P$ are as in the Definition 6.4) and

[TABLE]

9.1. A Matrix criterion for Leavitt path algebra of a finite hypergraph having IBN

In this subsection we generalize the main result of [27, Section 3]. Let $(\dot{E},\Lambda)$ be a finite hypergraph such that $|\Lambda|=h$ and $E^{0}=n$. Then by theorem 6.6, the $\mathcal{V}$-monoid of $L_{K}(\dot{E})$ is generated by the set $E^{0}$ modulo $h$ relations of the form

[TABLE]

one corresponding to each element of $\Lambda$. Let $A$ and $B$ be the coefficient matrices corresponding to the LHS and RHS respectively of the $h$ relations (6). Then it is clear that both $A$ and $B$ are $h\times n$ matrices with entries as non-negative integers. Let $T$ be a free abelian monoid on the set of all vertices. For each element $x\in T$, and for each $i$ such that $1\leq i\leq h$, let $M_{i}(x)$ denote the element of $T$ which results by applying to $x$ the relation (6) corresponding to the element $\lambda_{i}\in\Lambda$. For any sequence $\sigma$ taken from the set $\{1,2,\dots,h\}$, and any $x\in T$, let $\Delta_{\sigma}(x)\in T$ be the element obtained by applying $M_{i}$ operations in the order specified by $\sigma$.

Definition 9.1**.**

Suppose for each pair $x,y\in T$, $[x]=[y]$ in the $\mathcal{V}$-monoid if and only if there are two sequences $\sigma$ and $\sigma^{\prime}$ taken from the set $\{1,2,\dots,h\}$ such that $\Delta_{\sigma}(x)=\Delta_{\sigma^{\prime}}(y)$ in $T$. Then we say that the confluence condition holds in $T$.

Theorem 9.2**.**

Let $(\dot{E},\Lambda)$ be a finite hypergraph such that $|\Lambda|=h$ and $E^{0}=n$. Suppose $A$ and $B$ are the coefficient matrices corresponding to LHS and RHS respectively of the $h$ relations of the $\mathcal{V}$-monoid of $L_{K}(\dot{E})$. Also suppose that the confluence condition holds in $T$, the free abelian monoid on $E^{0}$. Then $L_{K}(\dot{E})$ has Invariant Basis Number if and only if $\text{rank}(B^{t}-A^{t})<\text{rank}([B^{t}-A^{t}~{}c])$, where $c$ is the column matrix of order $n\times 1$ with all its entries equal to $1$.

Proof.

Suppose that $\text{rank}(B^{t}-A^{t})<\text{rank}(B^{t}-A^{t}~{}c)$. We prove that if $m$ and $p$ are positive integers such that

[TABLE]

in the $\mathcal{V}$-monoid,then $m=p$. So let us assume that the equation (7) holds for some positive integers $m$ and $p$. Since the confluence condition holds in $T$, there are two sequences $\sigma$ and $\sigma^{\prime}$ taken from $\{1,2,\dots,h\}$ such that $\Delta_{\sigma}(m\sum_{i=1}^{n}v_{i})=\Delta_{\sigma^{\prime}}(p\sum_{i=1}^{n}v_{i})=\gamma(\text{say})$ in $T$. Now suppose $M_{j}$ is invoked $k_{j}$ times in $\Delta_{\sigma}$ and $k_{j}^{\prime}$ times in $\Delta_{\sigma^{\prime}}$. Then we have

[TABLE]

Also

[TABLE]

Let $m_{i}=(k_{i}^{\prime}-k_{i})$ for $i=1,\dots,h$. From the above two equations, we have the following system of equations-

[TABLE]

So $(m_{1},\dots,m_{h})\in\mathbb{Z}^{h}$ is a solution of the linear system $(B^{t}-A^{t})x=(m-p)c$, where $x=(x_{1},\dots,x_{h})^{t}$ and $c$ is the column matrix mentioned in the statement of the theorem. This means $\text{rank}(B^{t}-A^{t})=\text{rank}(B^{t}-A^{t}~{}~{}(m-p)c)$. We know that if $m\neq p$, then $\text{rank}(B^{t}-A^{t}~{}~{}(m-p)c)=\text{rank}(B^{t}-A^{t}~{}~{}c)$. This would mean that $\text{rank}(B^{t}-A^{t})=\text{rank}(B^{t}-A^{t}~{}~{}c)$ whenever $m\neq p$, contrary to our initial assumption. This proves the first part.

Conversely, assume that $\text{rank}(B^{t}-A^{t})=\text{rank}(B^{t}-A^{t}~{}~{}c):=r$. We will prove that there exists a pair of distinct positive integers $m$ and $p$ such that $m[\sum_{i=1}^{n}v_{i}]=p[\sum_{i=1}^{n}v_{i}]$ in the $\mathcal{V}$-monoid of $\mathcal{A}_{K}(\dot{E})$.

The fact that $\text{rank}(B^{t}-A^{t}~{}~{}c)=r$ means that after finite number of elementary row operations, $(B^{t}-A^{t}~{}~{}c)$ can be brought to the form

[TABLE]

where the entries are integers, $d_{1j_{1}}d_{2j_{2}}\dots d_{rj_{r}}\neq 0$ and $\sum_{i=1}^{r}c_{i}^{2}\neq 0$. So it is clear that one particular solution for the linear system $(B^{t}-A^{t})x=c$ is the column vector $(\frac{c_{1}}{d_{1j_{1}}},\frac{c_{2}}{d_{2j_{2}}},\dots,\frac{c_{r}}{d_{rj_{r}}},0,\dots,0)^{t}$.

Now let

[TABLE]

[TABLE]

[TABLE]

From the above definitions, it is clear that $(m-p)>0$. So the $h$-tuple $(m_{1},m_{2},\dots,m_{h})$ is a solution for the linear system $(B^{t}-A^{t})x=(m-p)c$. This, from the first part of the proof, is equivalent to showing that $m[\sum_{i=1}^{n}v_{i}]=p[\sum_{i=1}^{n}v_{i}]$. This means that $\mathcal{A}_{K}(\dot{E})$ does not have Invariant Basis Number, thereby completing the proof. ∎

Acknowledgement

The authors would like to thank Ramesh Sreekantan, Roozbeh Hazrat and Pere Ara for their valuable suggestions and comments.

The first named author gratefully acknowledges the Department of Atomic Energy (DAE), India for financial support through PhD scholarship.

The second named author was a visitor of Indian Statistical Institute Bangalore Center during the preparation of this manuscript. He would like to thank the institute for providing conducive environment for research. He thanks B. Venkatesh for his kind support. He also gratefully acknowledges the Department of Atomic Energy (DAE), India for financial support through Postdoctoral Fellowship.