Leavitt path algebras of hypergraphs

TL;DR

This paper introduces a new class of Leavitt path algebras for hypergraphs, extending existing frameworks, and explores their algebraic properties, K-theory, and dimensions, providing new insights into their structure.

Contribution

It defines Leavitt path algebras of hypergraphs, generalizing previous models, and analyzes their algebraic and K-theoretic properties, including bases, dimensions, and regularity.

Findings

Linear bases for the algebras are found.

Gelfand-Kirillov dimension is computed.

Results on simplicity, regularity, and Noetherian properties are obtained.

Abstract

We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of finitely separated graphs and Leavitt path algebras of row-finite vertex-weighted graphs. We find linear bases for those algebras, compute their Gelfand-Kirillov dimension, obtain some results on ring-theoretic properties like simplicity, von Neumann regularity and Noetherianess and investigate their K-theory and graded K-theory. By doing so we obtain new results on the Gelfand-Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of weighted Leavitt path algebras.

Tables

	(1’)	(2’)	(3’)	(4’)	(5’)
(1’)	$uvw$	$vw{h}_{ij}$,$vw{h}_{ij}^{\ast }$	-	-	-
(2’)	$h_{ij}vw$, $h_{ij}^{\ast }vw$	$v{h}_{ij}w$, $v{h}_{ij}^{\ast }w$, $g_{ij}v{h}_{kl}$ etc.	$v{g}_{ij}{h}_{kl}$, $v{g}_{ij}{h}_{kl}^{\ast }$ etc.	$v{h}_{i1}{h}_{j1}^{\ast }$	$v{h}_{1i}^{\ast }{h}_{1j}$
(3’)	-	$g_{ij}{h}_{kl}v$, $g_{ij}^{\ast }{h}_{kl}v$ etc.	$f_{ij}{g}_{kl}{h}_{pq}$, $f_{ij}{g}_{kl}{h}_{pq}^{\ast }$ etc.	$g_{kl}{h}_{i1}{h}_{j1}^{\ast }$, $g_{kl}^{\ast }{h}_{i1}{h}_{j1}^{\ast }$	$g_{kl}{h}_{1i}^{\ast }{h}_{1j}$, $g_{kl}^{\ast }{h}_{1i}^{\ast }{h}_{1j}$
(4’)	-	$h_{i1}{h}_{j1}^{\ast }v$	$h_{i1}{h}_{j1}^{\ast }{g}_{kl}$, $h_{i1}{h}_{j1}^{\ast }{g}_{kl}^{\ast }$	-	$h_{i1}{h}_{11}^{\ast }{h}_{1j}$
(5’)	-	$h_{1i}^{\ast }{h}_{1j}v$	$h_{1i}^{\ast }{h}_{1j}{g}_{kl}$, $h_{1i}^{\ast }{h}_{1j}{g}_{kl}^{\ast }$	$h_{1i}^{\ast }{h}_{11}{h}_{j1}^{\ast }$	-

Equations

$$\{v,h_{ij},h_{ij}^{\ast }\mid v\in H^0,h\in H^1,1\le i\le |s(h)|,1\le j\le |r(h)|\}$$

$$\{v_i,w_i,h_{ij},h_{ij}^{\ast }\mid 1\le i,j\le 2\}$$

$$s_H([(h,i)]):=\{[(s_{H_i}(h{)}_1,i)],\dots ,[(s_{H_i}(h{)}_{|s_{H_i}(h)|},i)]\}$$

$$r_H([(h,i)]):=\{[(r_{H_i}(h{)}_1,i)],\dots ,[(r_{H_i}(h{)}_{|r_{H_i}(h)|},i)]\}.$$

$$h_{i1}{h}_{j1}^{\ast }(h\in H^1,1\le i,j\le |s(h)|)\text{ and }{h}_{1i}^{\ast }{h}_{1j}(h\in H^1,1\le i,j\le |r(h)|)$$

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

$$v{h}_{ij}$$

$$h_{ij}v$$

$$v{h}_{ij}^{\ast }$$

$$h_{ij}^{\ast }v$$

$$g_{ij}{h}_{kl}$$

$$g_{ij}^{\ast }{h}_{kl}$$

$$g_{ij}{h}_{kl}^{\ast }$$

$$g_{ij}^{\ast }{h}_{kl}^{\ast }$$

$$h_{i1}{h}_{j1}^{\ast }={\delta }_{ij}s(h{)}_i-\sum _{k=2}^{|r(h)|}{h}_{ik}{h}_{jk}^{\ast }.$$

$$h_{1i}^{\ast }{h}_{1j}={\delta }_{ij}r(h{)}_i-\sum _{k=2}^{|s(h)|}{h}_{ki}^{\ast }{h}_{kj}.$$

$$A\leq B\Leftrightarrow\big{[}A=B\big{]}~{}\lor~{}\big{[}l(A)<l(B)\big{]}~{}\lor\big{[}l(A)=l(B)~{}\land~{}\forall C,D\in\overline{\langle}X\rangle:m(CAD)<m(CBD)\big{]}.$$

$$h_{i1}{h}_{11}^{\ast }{h}_{1j}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$V=V^1\subseteq V^2\subseteq V^3\subseteq \dots ,A=\bigcup _{n\in \mathbb{N}}{V}^n\text{ and }{d}_V(n):=\dim V^n<\infty .$$

$$\mathrm{GKdim}A:=\underset{n\to \infty }{\mathrm{limsup}}{\log }_n{d}_V(n).$$

$$\mathrm{GKdim}(A):=\sup \{\mathrm{GKdim}(B)\mid B\text{ is a finitely generated subalgebra of }A\}.$$

$$p^{i_1}o{p}^{i_2}\dots o{p}^{i_k}$$

$$(i_1+\cdots +i_k)|p|+(k-1)|o|\le n.$$

$$\alpha =o_1{p}_1^{l_1}{q}_1{o}_2{p}_2^{l_2}{q}_2\dots o_k{p}_k^{l_k}{q}_k{o}_{k+1}$$

$$p_i\underset{\gamma :=}{\underbrace{q_i{o}_{i+1}{p}_{i+1}{q}_{i+1}\dots o_{j-1}{p}_{j-1}{q}_{j-1}{o}_j{y}_{m+1}\dots y_t}}{p}_i$$

$$\alpha =o_1{p}_1^{l_1}{q}_1{o}_2{p}_2^{l_2}{q}_2\dots o_k{p}_k^{l_k}{q}_k{o}_{k+1}$$

$$l_1|p_1|+\cdots +l_k|p_k|\le n$$

$$p_1^{l_1}{o}_1{p}_2^{l_2}\dots o_{d-1}{p}_d^{l_d}$$

$$l_1|p_1|+\cdots +l_d|p_d|+|o_1|+\cdots +|o_{d-1}|\le n.$$

$$\nu (ye)+\nu (ex)=\nu (yex)=\nu (0)=-\infty$$

$$\nu (xzy)=\nu (xezfy)=\nu (xe)+\nu (z)+\nu (fy)\ge 0.$$

$$\nu (xyx)=\nu (xfyex)=\nu (x)+\nu (fye)+\nu (x).$$

$$\max \{\nu (x),\nu (y)\}\ge \nu (x+y)=\nu (yx)=\nu (x)+\nu (y).$$

Full text

1. 1 Introduction
2. 2 Notation
3. 3 Leavitt path algebras of hypergraphs
4. 4 Direct limits
5. 5 Linear bases
6. 6 The Gelfand-Kirillov dimension
7. 7 Valuations and local valuations
8. 8 Additional applications of the linear bases
9. 9 The $V$-monoid
10. 10 The graded $V$-monoid

1. Introduction

In a series of papers [13, 14, 15, 16] William Leavitt studied algebras that are now denoted by $L_{K}(n,n+k)$ and have been coined Leavitt algebras. Let $X=(x_{ij})$ and $Y=(y_{ji})$ be $(n+k)\times n$ and $n\times(n+k)$ matrices consisting of symbols $x_{ij}$ and $y_{ji}$, respectively. Then for a field $K$, $L_{K}(n,n+k)$ is the associative, unital $K$-algebra generated by all $x_{ij}$ and $y_{ji}$ subject to the relations $XY=I_{n+k}$ and $YX=I_{n}$. Leavitt established that the algebra $L_{K}(n,n+k)$ has module type $(n,k)$. Furthermore he showed that the algebras $L_{K}(1,k+1)$ are simple and the algebras $L_{K}(n,n+k)$, $n\geq 2$ domains. Recall that a ring $R$ has module type $(n,k)$ if $n$ and $k$ are the least positive integers such that $R^{n}\cong R^{n+k}$ as left $R$-modules.

Leavitt path algebras were introduced by G. Abrams and G. Aranda Pino in 2005 [1] and independently by P. Ara, M. Moreno and E. Pardo in 2007 [5] as $K$-algebras associated to directed graphs. For the directed graph with one vertex and $k+1$ loops one recovers the Leavitt algebra $L_{K}(1,k+1)$. The definition and the development of the theory were inspired on the one hand by Leavitt’s construction of $L_{K}(1,k+1)$ and on the other hand by the Cuntz algebras $\mathcal{O}_{n}$ [9] and the Cuntz-Krieger algebras in $C^{*}$-algebra theory [20]. The Cuntz algebras and later Cuntz-Krieger type $C^{*}$-algebras revolutionised $C^{*}$-theory, leading ultimately to the astounding Kirchberg-Phillips classification theorem [17]. The Leavitt path algebras have created the same type of stir in the algebraic community. The development of Leavitt path algebras and its interaction with graph $C^{*}$-algebras have been well-documented in several publications and we refer the reader to [2] and the references therein.

Since their introductions, there have been several attempts to introduce a generalisation of the Leavitt path algebras which would cover the algebras $L_{K}(n,n+k)$ for any $n\geq 1$, as well. P. Ara and K. Goodearl introduced Leavitt path algebras of separated graphs in 2012 [4]. They showed that any Leavitt algebra $L_{K}(n,n+k)$ is a corner ring of a Leavitt path algebra of a finitely separated graph. Weighted Leavitt path algebras were introduced by R. Hazrat in 2013 [10]. For the weighted graph with one vertex and $n+k$ loops of weight $n$ one recovers the Leavitt algebra $L_{K}(n,n+k)$. If the weights of all the edges are $1$, then the weighted Leavitt path algebras reduce to the usual Leavitt path algebras.

In this paper we define and investigate Leavitt path algebras of hypergraphs. A (directed) hypergraph can be thought of as a directed graph where the edges are allowed to have multiple sources and ranges. Usually the source and the range of an edge in a hypergraph (a “hyperedge”) are required to be nonempty, finite subsets of the vertex set. In this paper we use a slightly modified definition of a hypergraph. We only require that the source and the range of a hyperedge are multisets over the vertex set with finite, nonempty support. Hence a vertex may appear in the source (resp. range) of a hyperedge more than once.

The rest of the paper is organised as follows.

In Section 2 we recall some standard notation which used throughout the paper.

In Section 3 we define the Leavitt path algebra $L_{K}(H)$ of a hypergraph $H$. We show that these algebras generalise the Leavitt path algebras of finitely separated graphs and row-finite vertex-weighted graphs. Moreover, we prove that a Leavitt path algebra $L_{K}(H)$ of a hypergraph $H$ is an involutary $\mathbb{Z}^{n}$-graded algebra with local units where $n$ is the supremum over all the cardinalities of sources of hyperedges in $H$ (the cardinality $|M|$ of a multiset $M$ with finite, nonempty support is defined in the first paragraph of Section 3).

In Section 4 we show that the category $\operatorname{\mathcal{H}}$ of hypergraphs admits arbitrary direct limits. The main result of the section is Proposition 14 which says that for any hypergraph $H$ its Leavitt path algebra $L_{K}(H)$ is a direct limit of a direct system of Leavitt path algebras of finite hypergraphs.

In Section 5 we find linear bases for the Leavitt path algebras of hypergraphs. These bases play an important role in Sections 6, 7 and 8.

In Section 6 we compute the Gelfand-Kirillov dimension of a Leavitt path algebra of a hypergraph. The GK dimension of a Leavitt path algebra of a row-finite weighted graph was already known, see [18]. But the result for the GK dimension of a Leavitt path algebra of a finitely separated graph seems to be new.

In Section 7 we show that the Leavitt path algebra $L_{K}(H)$ has a “local valuation” provided the hypergraph $H$ satisfies Condition (LV), i.e. for any hyperedge $h$ the cardinalities of $s(h)$ and $r(h)$ are at least $2$. We deduce that if $H$ is a connected hypergraph which satisfies Condition (LV) and has at least one hyperedge, then $L_{K}(H)$ is prime, nonsingular, not von Neumann regular and semiprimitive. Furthermore we classify the hypergraphs $H$ such that $L_{K}(H)$ is a domain.

In Section 8 we show that a Leavitt path algebra $L_{K}(H)$ that is finite-dimensional as a $K$-vector space or simple or left Artinian or right Artininan or von Neumann regular is isomorphic to a Leavitt path algebra of a finitely separated graph. We also prove a result on the Noetherianess of $L_{K}(H)$.

In Section 9 we compute the monoid $V(L_{K}(H))$ for any hypergraph $H$. Recall that for a ring $R$ with local units, $V(R)$ is the set of all isomorphism classes of unital finitely generated projective left $R$-modules which becomes an abelian monoid by defining $[P]+[Q]:=[P\oplus Q]$. The Grothendieck group $K_{0}(R)$ is the group completion of $V(R)$.

In Section 10 we compute the monoid $V^{\operatorname{gr}}(L_{K}(H))$ with respect to any grading on $L_{K}(H)$ induced by an “admissible weight map” (we show that for example the standard $\mathbb{Z}^{n}$-grading on $L_{K}(H)$ defined in Section 3 is induced by an admissible weight map). To the best knowledge of the author, the graded $V$-monoid of a Leavitt path algebra of a finitely separated graph or a row-finite vertex-weighted graph had not been computed before. Recall that for a graded ring $R$ with graded local units, $V^{\operatorname{gr}}(R)$ is the set of all isomorphism classes of graded unital finitely generated projective left $R$-modules which becomes an abelian monoid by defining $[P]+[Q]:=[P\oplus Q]$. The graded Grothendieck group $K^{gr}_{0}(R)$ is the group completion of $V^{\operatorname{gr}}(R)$.

2. Notation

Throughout the paper $K$ denotes a field. By a ring resp. $K$-algebra we mean an associative (but not necessarily commutative or unital) ring resp. $K$-algebra. By an ideal of a ring we mean a twosided ideal. $\mathbb{N}$ denotes the set of positive integers, $\mathbb{N}_{0}$ the set of nonnegative integers, $\mathbb{Z}$ the set of integers and $\mathbb{R}_{+}$ the set of positive real numbers.

3. Leavitt path algebras of hypergraphs

Recall that a multiset is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. Formally a multiset $M$ over a set $X$ is a function $M:X\rightarrow\mathbb{N}_{0}$. If $x\in X$, then $M(x)$ is called the multiplicity of $x$. The set $M_{\operatorname{supp}}:=\{x\in X\mid M(x)>0\}$ is called the support of $M$. We denote by $\operatorname{Mult}(X)$ the set of all multisets over $X$ that have finite, nonempty support. For an element $M\in\operatorname{Mult}(X)$ we set $|M|:=\sum\limits_{x\in M_{\operatorname{supp}}}M(x)$.

Definition 1**.**

A (directed) hypergraph is a quadruple $H=(H^{0},H^{1},s,r)$ where $H^{0}$ and $H^{1}$ are sets and $s,r:H^{1}\rightarrow\operatorname{Mult}(H^{0})$ are maps. The elements of $H^{0}$ are called vertices and the elements of $H^{1}$ hyperedges. $H$ is called finite if $H^{0}$ and $H^{1}$ are finite sets and empty if $H^{0}=H^{1}=\emptyset$. In this article all hypergraphs are assumed to be nonempty.

Definition 2**.**

Let $H$ be a hypergraph. For any $h\in H^{1}$ write $s(h)=\{s(h)_{1},\dots,s(h)_{|s(h)|}\}$ and $r(h)=\{r(h)_{1},\dots,r(h)_{|r(h)|}\}$. The $K$-algebra presented by the generating set

[TABLE]

and the relations

1. (i)

$uv=\delta_{uv}u\quad(u,v\in H^{0})$, 2. (ii)

$s(h)_{i}h_{ij}=h_{ij}=h_{ij}r(h)_{j},~{}r(h)_{j}h_{ij}^{*}=h_{ij}^{*}=h_{ij}^{*}s(h)_{i}\quad(h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|)$, 3. (iii)

$\sum\limits_{k=1}^{|r(h)|}h_{ik}h_{jk}^{*}=\delta_{ij}s(h)_{i}\quad(h\in H^{1},~{}1\leq i,j\leq|s(h)|)$ and 4. (iv)

$\sum\limits_{k=1}^{|s(h)|}h_{ki}^{*}h_{kj}=\delta_{ij}r(h)_{i}\quad(h\in H^{1},~{}1\leq i,j\leq|r(h)|)$

is called the Leavitt path algebra of $H$ and is denoted by $L_{K}(H)$.

Remark 3**.**

1. (a)

One checks easily that the isomorphism class of the algebra $L_{K}(H)$ does not depend on the chosen ordering of the elements of the multisets $s(h)$ and $r(h)$ ($h\in H^{1}$). 2. (b)

Let $H$ be a hypergraph. Define a directed graph $E=(E^{0},E^{1},s^{\prime},r^{\prime})$ by $E^{0}=H^{0}$, $E^{1}=\{h_{ij}\mid h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$, $s^{\prime}(h_{ij})=s(h)_{i}$ and $r^{\prime}(h_{ij})=r(h)_{j}$. The graph $E$ is called the directed graph associated to $H$. Let $\hat{E}$ be the double graph of $E$ and $K\hat{E}$ the path $K$-algebra of $\hat{E}$ (see for example [2, Remark 1.2.4]). Then $L_{K}(H)$ is isomorphic to the quotient of $K\hat{E}$ by the ideal of $K\hat{E}$ generated by relations (iii) and (iv) in Definition 2. 3. (c)

Let $H$ be a hypergraph and $A$ a $K$-algebra which contains a set $X:=\{a_{v},b_{h,i,j},c_{h,i,j}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$ such that

1. (i)

the $a_{v}$’s are pairwise orthogonal idempotents, 2. (ii)

$a_{s(h)_{i}}b_{h,i,j}=b_{h,i,j}=b_{h,i,j}a_{r(h)_{j}},~{}a_{r(h)_{j}}c_{h,i,j}=c_{h,i,j}=c_{h,i,j}a_{s(h)_{i}}\quad(h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|)$, 3. (iii)

$\sum\limits_{k=1}^{|r(h)|}b_{h,i,k}c_{h,j,k}=\delta_{ij}a_{s(h)_{i}}\quad(h\in H^{1},~{}1\leq i,j\leq|s(h)|)$ and 4. (iv)

$\sum\limits_{k=1}^{|s(h)|}c_{h,k,i}b_{h,k,j}=\delta_{ij}a_{r(h)_{i}}\quad(h\in H^{1},~{}1\leq i,j\leq|r(h)|)$.

We call $X$ an $H$-family in $A$. By the relations defining $L_{K}(H)$, there exists a unique $K$-algebra homomorphism $\phi:L_{K}(H)\rightarrow A$ such that $\phi(v)=a_{v}$, $\phi(h_{ij})=b_{h,i,j}$ and $\phi(h^{*}_{ij})=c_{h,i,j}$ for all $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. We will refer to this as the Universal Property of $L_{K}(H)$.

Example 4**.**

Consider the hypergraph $H=(H^{0},H^{1},s,r)$ where $H^{0}=\{v_{1},v_{2},w_{1},w_{2}\}$, $H^{1}=\{h\}$, $s(h)=\{v_{1},v_{2}\}$ and $r(h)=\{w_{1},w_{2}\}$. We visualize $H$ as follows:

$H:$ $v_{1}$ $v_{2}$ $w_{1}$ $w_{2}$ h

The Leavitt path algebra $L_{K}(H)$ of $H$ is the $K$-algebra presented by the generating set

[TABLE]

and the relations

1. (i)

$uu^{\prime}=\delta_{uu^{\prime}}u\quad(u,u^{\prime}\in\{v_{1},v_{2},w_{1},w_{2}\})$, 2. (ii)

$v_{i}h_{ij}=h_{ij}=h_{ij}w_{j},~{}w_{j}h_{ij}^{*}=h_{ij}^{*}=h_{ij}^{*}v_{i}\quad(h\in H^{1},1\leq i,j\leq 2)$, 3. (iii)

$\sum\limits_{k=1}^{2}h_{ik}h_{jk}^{*}=\delta_{ij}v_{i}\quad(1\leq i,j\leq 2)$ and 4. (iv)

$\sum\limits_{k=1}^{2}h_{ki}^{*}h_{kj}=\delta_{ij}w_{i}\quad(1\leq i,j\leq 2)$.

Example 5**.**

Let $(E,C)$ be a finitely separated graph (see [4]). Write each $X\in C$ as $X=\{e_{X,1},\dots,e_{X,n_{X}}\}$. Define a hypergraph $H=(H^{0},H^{1},s^{\prime},r^{\prime})$ by $H^{0}=E^{0}$, $H^{1}=\{h_{X}\mid X\in C\}$, $s^{\prime}(h_{X})=\{s(e_{X,1})\}$ and $r^{\prime}(h_{X})=\{r(e_{X,1}),\dots,r(e_{X,n_{X}})\}$. Then $L_{K}(E,C)\cong L_{K}(H)$.

By a vertex-weighted graph we mean a weighted graph in the sense of [19] such that edges emitted by the same vertex have the same weight. By [19, Examples 5,6] the class of Leavitt path algebras of row-finite vertex-weighted graphs is big enough to contain all Leavitt path algebras of row-finite directed graphs and all Leavitt algebras $L_{K}(n,n+k)$.

Example 6**.**

Let $(E,w)$ be a row-finite vertex-weighted graph. Let $\operatorname{Reg}(E)$ denote the set of all vertices in $E$ that emit at least one edge. For each $v\in\operatorname{Reg}(E)$ write $s^{-1}(v)=\{e_{v,1},\dots,e_{v,n_{v}}\}$. Define a hypergraph $H=(H^{0},H^{1},s^{\prime},r^{\prime})$ by $H^{0}=E^{0}$, $H^{1}=\{h_{v}\mid v\in\operatorname{Reg}(E)\}$, $s^{\prime}(h_{v})=\{\underbrace{v,\dots,v}_{w(v)~{}times}\}$ and $r^{\prime}(h_{v})=\{r(e_{v,1}),\dots,r(e_{v,n_{v}})\}$. Then $L_{K}(E,w)\cong L_{K}(H)$.

Recall that a ring $R$ is said to have a set of local units $X$ in case $X$ is a set of idempotents in $R$ having the property that for each finite subset $S\subseteq R$ there exists an $x\in X$ such that $xsx=s$ for any $s\in S$.

Proposition 7**.**

Let $H$ be a hypergraph. Then:

1. (i)

$L_{K}(H)$* has a set of local units, namely the set of all finite sums of distinct elements of $H^{0}$. If $H$ is finite, then $L_{K}(H)$ is a unital ring with $\sum\limits_{v\in H^{0}}v$ as multiplicative identity.* 2. (ii)

There is an involution $*$ on $L_{K}(H)$ mapping $k\mapsto k$, $v\mapsto v$, $h_{ij}\mapsto h_{ij}^{*}$ and $h_{ij}^{*}\mapsto h_{ij}$ for any $k\in K$, $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. 3. (iii)

Set $n:=\sup\{|s(h)|\mid h\in H^{1}\}$ (note that $n$ might be infinity). One can define a $\mathbb{Z}^{n}$-grading on $L_{K}(H)$ by setting $\deg(v):=0$, $\deg(h_{ij}):=\alpha_{i}$ and $\deg(h_{ij}^{*}):=-\alpha_{i}$ for any $v\in E^{0}$, $h\in E^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. Here $\alpha_{i}$ denotes the element of $\mathbb{Z}^{n}$ whose $i$-th component is $1$ and whose other components are [math].

Proof.

1. (i)

Follows from the fact that the elements of $H^{0}$ are mutually orthogonal idempotents in $L_{K}(H)$ such that $\sum\limits_{v\in H^{0}}vL_{K}(H)=\sum\limits_{v\in H^{0}}L_{K}(H)v=L_{K}(H)$. 2. (ii)

Set $X:=\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$ and let $K\langle X\rangle$ denote the free $K$-algebra generated by $X$. Clearly there is a uniquely determined involution $*$ on $K\langle X\rangle$ mapping $k\mapsto k$, $v\mapsto v$, $h_{ij}\mapsto h_{ij}^{*}$ and $h_{ij}^{*}\mapsto h_{ij}$ for any $k\in K$, $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. Since the set of the relations (i)-(iv) in Definition 2 is clearly invariant under $*$, the involution $*$ induces an involution on $L_{K}(H)$. 3. (iii)

Let $K\langle X\rangle$ be defined as above. Clearly there is a $\mathbb{Z}^{n}$-grading on $K\langle X\rangle$ defined by $\deg(v):=0$, $\deg(h_{ij}):=\alpha_{i}$ and $\deg(h_{ij}^{*}):=-\alpha_{i}$ for any $v\in E^{0}$, $h\in E^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. Since the relations (i)-(iv) in Definition 2 are clearly homogeneous, the $\mathbb{Z}^{n}$-grading on $K\langle X\rangle$ induces a $\mathbb{Z}^{n}$-grading on $L_{K}(H)$.

∎

Remark 8**.**

In the following we will refer to the grading defined in Proposition 7 (iii) as the standard grading of $L_{K}(H)$. The isomorphisms $L_{K}(E,C)\cong L_{K}(H)$ and $L_{K}(E,w)\cong L_{K}(H)$ in Examples 5 and 6 are graded isomorphisms with respect to the standard gradings of $L_{K}(E,C)$, $L_{K}(E,w)$ and $L_{K}(H)$ (see [4, Remark 2.13] and [10, Proposition 5.7]).

4. Direct limits

If $X$ and $Y$ are sets, $f:X\rightarrow Y$ is a map and $M$ is a multiset over $X$, then we define $f(M)$ as the multiset over $Y$ such that $f(M)(y)=\sum\limits_{x\in f^{-1}(y)}M(x)$. Note that if $M\in\operatorname{Mult}(X)$, then $|M|=|f(M)|$ and $|M_{\operatorname{supp}}|\leq|f(M)_{\operatorname{supp}}|$.

Definition 9**.**

Let $H$ and $I$ be hypergraphs. A hypergraph homomorphism $\phi:H\rightarrow I$ consists of two maps $\phi^{0}:H^{0}\rightarrow I^{0}$ and $\phi^{1}:H^{1}\rightarrow I^{1}$ such that $s(\phi^{1}(h))=\phi^{0}(s(h))$ and $r(\phi^{1}(h))=\phi^{0}(r(h))$ for any $h\in H^{1}$. We denote by $\operatorname{\mathcal{H}}$ the category whose objects are all hypergraphs and whose morphisms are all hypergraph homomorphisms.

Proposition 10**.**

The category $\operatorname{\mathcal{H}}$ admits arbitrary direct limits.

Proof.

Let $\{H_{i},\phi_{ij}\mid i,j\in I,i\leq j\}$ be a direct system in $\operatorname{\mathcal{H}}$. For any $l\in\{0,1\}$ we define $H^{l}$ as the direct limit of the $H^{l}_{i}$’s in the category of sets. We identify $H^{l}$ with the set $\bigsqcup\limits_{i\in I}H_{i}^{l}/\sim_{l}$ where $(x,i)\sim_{l}(y,j)\Leftrightarrow~{}\exists k\in I:\phi_{ik}^{l}(x)=\phi_{jk}^{l}(y)$. For a $[(h,i)]\in H^{1}$ we set

[TABLE]

and

[TABLE]

One checks easily that one gets well-defined maps $s_{H},r_{H}:H^{1}\rightarrow\operatorname{Mult}(H^{0})$. Set $H:=(H^{0},H^{1},s_{H},r_{H})$. For any $i\in I$ let $\phi^{0}_{i}:H^{0}_{i}\rightarrow H^{0}$ and $\phi^{1}_{i}:H^{1}_{i}\rightarrow H^{1}$ be the canonical maps and set $\phi_{i}:=(\phi_{i}^{0},\phi_{i}^{1})$. It is routine to check that $\phi_{i}:H_{i}\rightarrow H$ is a hypergraph homomorphism for any $i\in I$ and that $\{H,\phi_{i}\mid i\in I\}$ is a direct limit of the direct system $\{H_{i},\phi_{ij}\mid i,j\in I,i\leq j\}$. ∎

Definition 11**.**

Let $H$ and $I$ be a hypergraphs. Then $I$ is called a subhypergraph of $H$ if $I^{0}\subseteq H^{0}$, $I^{1}\subseteq H^{1}$ and $s_{I}$ and $r_{I}$ are the restrictions of $s_{H}$ resp. $r_{H}$ to $I^{0}$.

Proposition 12**.**

Let $H$ be a hypergraph. Then $H$ is a direct limit of the direct system of all finite subhypergraphs of $H$.

Proof.

Straightforward. ∎

Definition 13**.**

In Definition 2 we associated to any hypergraph $H$ a $K$-algebra $L_{K}(H)$. If $\phi:H\rightarrow I$ is a morphism in $\operatorname{\mathcal{H}}$, then there is a unique $K$-algebra homomorphism $L_{K}(\phi):L_{K}(H)\rightarrow L_{K}(I)$ such that $L_{K}(\phi)(v)=\phi^{0}(v)$, $L_{K}(\phi)(h_{ij})=(\phi^{1}(h))_{ij}$ and $L_{K}(\phi)(h_{ij}^{*})=(\phi^{1}(h))_{ij}^{*}$ for any $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$ (follows from the Universal Property of $L_{K}(H)$, see Remark 3 (c)). Let $\operatorname{\mathcal{A}_{K}^{loc}}$ denote the category of $K$-algebras with local units. One checks easily that $L_{K}:\operatorname{\mathcal{H}}\rightarrow\operatorname{\mathcal{A}_{K}^{loc}}$ is a functor that commutes with direct limits.

Proposition 14**.**

Let $H$ be a hypergraph. Then $L_{K}(H)$ is a direct limit of a direct system of Leavitt path algebras of finite hypergraphs.

Proof.

Follows from Proposition 12 and the fact that $L_{K}$ commutes with direct limits. ∎

5. Linear bases

Throughout this section $H$ denotes a hypergraph. Our goal is to find a basis for the $K$-vector space $L_{K}(H)$. We denote the directed graph associated to $H$ by $E$ and the double graph of $E$ by $\hat{E}$ (see Remark 3 (b)). We set $X:=\hat{E}^{0}\cup\hat{E}^{1}$, $\langle X\rangle:=\{\text{nonempty words over }X\}$ and $\overline{\langle X\rangle}:=\langle X\rangle\cup\{\text{empty word}\}$. Together with juxtaposition $\langle X\rangle$ is a semigroup and $\overline{\langle X\rangle}$ a monoid. If $A,B\in\overline{\langle X\rangle}$, then we call $B$ a subword of $A$ if there are $C,D\in\overline{\langle X\rangle}$ such that $A=CBD$. We denote by $K\langle X\rangle$ the free $K$-algebra generated by $X$ (i.e. $K\langle X\rangle$ is the $K$-vector space with basis $\langle X\rangle$ which becomes a $K$-algebra by linear extending the juxtaposition of words). Note that $L_{K}(H)$ is the quotient of $K\langle X\rangle$ by the relations (i)-(iv) in Definition 2.

Definition 15**.**

Let $G=(G^{0},G^{1},s,r)$ be a directed graph. A path in $G$ is a nonempty word $p=x_{1}\dots x_{n}$ over the alphabet $G^{0}\cup G^{1}$ such that either $x_{i}\in G^{1}~{}(i=1,\dots,n)$ and $r(x_{i})=s(x_{i+1})~{}(i=1,\dots,n-1)$ or $n=1$ and $x_{1}\in G^{0}$. By definition, the length $|p|$ of $p$ is $n$ in the first case and [math] in the latter case. We set $s(p):=s(x_{1})$ and $r(p):=r(x_{n})$ (here we use the convention $s(v)=v=r(v)$ for any $v\in G^{0}$).

We call a path in $\hat{E}$ a d-path. While the d-paths form a basis for the path algebra $K\hat{E}$, a basis for the Leavitt path algebra $L_{K}(H)$ is formed by the nod-paths, which we will define next.

Definition 16**.**

The words

[TABLE]

in $\langle X\rangle$ are called forbidden. A normal d-path or nod-path is a d-path such that none of its subwords is forbidden. An element of $K\langle X\rangle$ is called normal if it lies in the linear span $K\langle X\rangle_{\operatorname{nod}}$ of all nod-paths.

Theorem 17**.**

Any element of $L_{K}(H)$ has precisely one normal representative. Moreover, the map $\operatorname{NF}:L_{K}(H)\rightarrow K\langle X\rangle_{\operatorname{nod}}$ which associates to each element of $L_{K}(H)$ its normal representative is an isomorphism of $K$-vector spaces.

Proof.

In order to be able to apply [11, Theorem 15], we replace the relations (i)-(iv) in Definition 2 by the relations (i’)-(v’) below.

1. (i’)

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

[TABLE] 2. (ii’)

For any $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$,

[TABLE] 3. (iii’)

For any $g,h\in H^{1}$, $1\leq i\leq|s(g)|$, $1\leq j\leq|r(g)|$, $1\leq k\leq|s(h)|$ and $1\leq l\leq|r(h)|$,

[TABLE] 4. (iv’)

For all $h\in H^{1}$ and $1\leq i,j\leq|s(h)|$,

[TABLE] 5. (v’)

For all $h\in H^{1}$ and $1\leq i,j\leq|r(h)|$,

[TABLE]

Clearly the relations (i’)-(v’) above generate the same ideal $I$ of $K\langle X\rangle$ as the relations (i)-(iv) in Definition 2. Denote by $S$ the reduction system for $K\langle X\rangle$ defined by the relations (1’)-(5’) (i.e. $S$ is the set of all pairs $\sigma=(W_{\sigma},f_{\sigma})$ where $W_{\sigma}$ equals the left hand side of an equation in (i’)-(v’) and $f_{\sigma}$ the corresponding right hand side).

For any $A=x_{1}\dots x_{n}\in\langle X\rangle$ set $l(A):=n$ and m(A):=\big{|}\{i\in\{1,\dots,n-1\}|x_{i}x_{i+1}\text{ is forbidden}\}\big{|}. Define a partial ordering $\leq$ on $\langle X\rangle$ by

[TABLE]

Clearly $\leq$ is a semigroup partial ordering on $\langle X\rangle$ compatible with $S$ and the descending chain condition is satisfied.

It remains to show that all ambiguities of $S$ are resolvable. In the table below we list all types of ambiguities which may occur.

Note that there are no inclusion ambiguities. The (4’)-(5’) and (5’)-(4’) ambiguities $h_{i1}h_{11}^{*}h_{1j}$ and $h_{1i}^{*}h_{11}h_{j1}^{*}$ are the ones which are most difficult to resolve. We show how to resolve the ambiguity $h_{i1}h_{11}^{*}h_{1j}$ (where $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$) and leave the other ambiguities to the reader.

[TABLE]

(note that $\sum\limits_{k=1}^{|r(h)|}\delta_{kj}h_{ik}=h_{ij}=\sum\limits_{k=1}^{|s(h)|}\delta_{ik}h_{kj}$). It follows from [11, Theorem 15], that $K\langle X\rangle_{\operatorname{irr}}$ is a set of representatives for the elements of $K\langle X\rangle/I=L_{K}(H)$. Clearly $K\langle X\rangle_{\operatorname{irr}}=K\langle X\rangle_{\operatorname{nod}}$.

Clearly the map $\operatorname{NF}:L_{K}(H)\rightarrow K\langle X\rangle_{\operatorname{nod}}$ which associates to each element of $L_{K}(H)$ its normal representative is bijective. That $\operatorname{NF}$ is linear follows from [7, Lemma 1.1] (note that $\operatorname{NF}=r_{S}$). ∎

Corollary 18**.**

The images of the nod-paths in $L_{K}(H)$ form a basis of the $K$-vector space $L_{K}(H)$.

Proof.

By Theorem 17, the map $\operatorname{NF}:L_{K}(H)\rightarrow K\langle X\rangle_{\operatorname{nod}}$ is an isomorphism of $K$-vector spaces. Its inverse is the map $K\langle X\rangle_{\operatorname{nod}}\rightarrow L_{K}(H)$ induced by the inclusion map $K\langle X\rangle_{\operatorname{nod}}\hookrightarrow K\langle X\rangle$. Since the nod-paths form a $K$-basis for $K\langle X\rangle_{\operatorname{nod}}$, the assertion of the corollary follows. ∎

6. The Gelfand-Kirillov dimension

First we want to recall some general facts on the growth of algebras. Let $A\neq\{0\}$ be a finitely generated $K$-algebra. Let $V$ be a finite-dimensional generating subspace of $A$, i.e. a finite-dimensional subspace of $A$ that generates $A$ as a $K$-algebra. For $n\geq 1$ let $V^{n}$ denote the linear span of the set $\{v_{1}\dots v_{k}\mid k\leq n,v_{1},\dots,v_{k}\in V\}$. Then

[TABLE]

Given functions $f,g:\mathbb{N}\rightarrow\mathbb{R}^{+}$, we write $f\preccurlyeq g$ if there is a $c\in\mathbb{N}$ such that $f(n)\leq cg(cn)$ for all $n$. If $f\preccurlyeq g$ and $g\preccurlyeq f$, then the functions $f,g$ are called asymptotically equivalent and we write $f\sim g$. If $W$ is another finite-dimensional generating subspace of $A$, then $d_{V}\sim d_{W}$. The Gelfand-Kirillov dimension or GK dimension of $A$ is defined as

[TABLE]

The definition of the GK dimension does not depend on the choice of the finite-dimensional generating subspace $V$. If $d_{V}\preccurlyeq n^{m}$ for some $m\in\mathbb{N}$, then $A$ is said to have polynomial growth and we have $\operatorname{GKdim}A\leq m$. If $d_{V}\sim a^{n}$ for some real number $a>1$, then $A$ is said to have exponential growth and we have $\operatorname{GKdim}A=\infty$. If $A$ does not happen to be finitely generated over $K$, then the GK dimension of $A$ is defined as

[TABLE]

Definition 19**.**

Let $H$ be a hypergraph. Let $p$ and $q$ be nod-paths. If there is a nod-path $o$ such that $p$ is not a prefix of $o$ and $poq$ is a nod-path, then we write $p\overset{\operatorname{nod}}{\Longrightarrow}q$. If $pq$ is a nod-path or $p\overset{\operatorname{nod}}{\Longrightarrow}q$, then we write $p\Longrightarrow q$.

Definition 20**.**

Let $H$ be a hypergraph. A nod2-path is a nod-path $p$ such that $p^{2}$ is a nod-path. A quasi-cycle is a nod2-path $p$ such that none of the subwords of $p^{2}$ of length $<|p|$ is a nod2-path. A quasi-cycle $p$ is called selfconnected if $p\overset{\operatorname{nod}}{\Longrightarrow}p$.

Remark 21**.**

1. (a)

Let $p=x_{1}\dots x_{n}$ be a quasi-cycle. Assume that $x_{i}=x_{j}$ for some $1\leq i<j\leq n$. Then we get the contradiction that $x_{i}\dots x_{j-1}$ is a nod2-path of length $<n$. Hence $x_{i}\neq x_{j}$ for all $i\neq j$. It follows that there is only a finite number of quasi-cycles if $(E,w)$ is finite. 2. (b)

If $x_{1}\dots x_{n}$ is a nonempty word over some alphabet, then we call the words $x_{m+1}\dots x_{n}x_{1}\dots x_{m}~{}(1\leq m\leq n)$ shifts of $x_{1}\dots x_{n}$. One checks easily that any shift of a quasi-cycle $p$ is a quasi-cycle (note that if $q$ is a shift of $p$, then any subword of $q^{2}$ of length $<|q|=|p|$ is also a subword of $p^{2}$). If $p$ and $q$ are quasi-cycles, then we write $p\approx q$ iff $q$ is a shift of $p$. Clearly $\approx$ is an equivalence relation on the set of all quasi-cycles. 3. (c)

Let $p=x_{1}\dots x_{n}$ be a quasi-cycle. Then $p^{*}:=x_{n}^{*}\dots x_{1}^{*}$ is a quasi-cycle.

The following lemma shows, that quasi-cycles behave like cycles in a way (one cannot “take a shortcut”).

Lemma 22**.**

Let $H$ be a hypergraph, $p=x_{1}\dots x_{n}$ a quasi-cycle and $1\leq i,j\leq n$. Then $x_{i}x_{j}$ is a nod-path iff $i<n$ and $j=i+1$ or $i=n$ and $j=1$.

Proof.

If $i<n$ and $j=i+1$ or $i=n$ and $j=1$, then clearly $x_{i}x_{j}$ is a nod-path. Suppose now that $x_{i}x_{j}$ is a nod-path.

case 1 Suppose $i=j$. Assume that $n>1$. Then we get the contradiction that $x_{i}$ is a nod2-path which is a subword of $p^{2}$ of length $1<|p|=n$. Hence $n=1$ and we have $i=j=1=n$.

case 2 Suppose $i<j$. Then $x_{j}\dots x_{n}x_{1}\dots x_{i}$ is a nod2-path which is a subword of $p^{2}$ of length $n-j+1+i$. It follows that $j=i+1$.

case 3 Suppose $j<i$. Then $x_{j}\dots x_{i}$ is a nod2-path which is a subword of $p^{2}$ of length $i-j+1$. It follows that $j=1$ and $i=n$. ∎

Lemma 23**.**

Let $H$ be a finite hypergraph. If there is a selfconnected quasi-cycle $p$, then $L_{K}(H)$ has exponential growth.

Proof.

Let $o$ be a nod-path such that $p$ is not a prefix of $o$ and $pop$ is a nod-path. Let $n\in\mathbb{N}$. Consider the nod-paths

[TABLE]

where $k,i_{1},\dots,i_{k}\in\mathbb{N}$ satisfy

[TABLE]

Let $A=(k,i_{1},\dots,i_{k})$ and $B=(k^{\prime},i^{\prime}_{1},\dots,i^{\prime}_{k^{\prime}})$ be different solutions of (2). Assume that $A$ and $B$ define the same nod-path in (1). After cutting out the common beginning, we can assume that the nod-path defined by $A$ starts with $o$ and the nod-path defined by $B$ with $p$ or vice versa. Since $p$ is not a prefix of $o$, it follows that $|p|>|o|$. Write $p=x_{1}\dots x_{m}$. Since the next letter after an $o$ must be a $p$, we get $x_{|o|+1}=x_{1}$ which contradicts Remark 21(a). Hence different solutions of (2) define different nod-paths in (1). Let $V$ denote the finite-dimensional generating subspace of $L_{K}(H)$ spanned by the set $\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$. By Theorem 17 the nod-paths in (1) are linearly independent in $V^{n}$. The number of solutions of (2) is $\sim 2^{n}$ and hence $L_{K}(H)$ has exponential growth. ∎

In Definition 24 below, we introduce the Condition (A’) for hypergraphs. This condition will be used again in Section 9.

We call a multiset $M:X\rightarrow\mathbb{N}_{0}$ a set and identify it with $M_{\operatorname{supp}}$ if $M(x)\leq 1$ for any $x\in X$. If the multiset $M$ is not a set, then we call it a proper multiset.

Definition 24**.**

$H$ satisfies Condition (A’) if there is an $h\in H^{1}$ such that $|s(h)|,|r(h)|\geq 2$ and moreover $s(h)$ is a proper multiset or $r(h)$ is a proper multiset or $s(h)$ and $r(h)$ are sets with nonempty intersection.

Corollary 25**.**

If $H$ is a finite hypergraph that satisfies Condition (A’), then $L_{K}(H)$ has exponential growth.

Proof.

Since $H$ satisfies Condition (A’), we can choose an $h\in H^{1}$ such that $|s(h)|,|r(h)|\geq 2$ and $s(h)$ is a proper multiset or $r(h)$ is a proper multiset or $s(h)$ and $r(h)$ are sets with nonempty intersection.

First suppose that $s(h)_{\operatorname{supp}}\cap r(h)_{\operatorname{supp}}\neq\emptyset$. By Remark 3 (a) we may assume that $s(h)_{2}=r(h)_{2}$. Clearly $p:=h_{22}$ is a quasi-cycle. Further $h_{22}h_{22}^{*}h_{22}$ is a nod-path and therefore $p$ is selfconnected. Hence, by the previous lemma, $L_{K}(H)$ has exponential growth.

Now suppose that $s(h)_{\operatorname{supp}}\cap r(h)_{\operatorname{supp}}=\emptyset$ and $s(h)$ is a proper multiset. Choose $1\leq i<j\leq|s(h)|$ such that $s(h)_{i}=s(h)_{j}$. Clearly $p:=h_{j2}h_{j2}^{*}$ is a quasi-cycle. Further $h_{j2}h_{j2}^{*}h_{j2}h_{i2}^{*}h_{j2}h_{j2}^{*}$ is a nod-path and therefore $p$ is selfconnected. Hence, by the previous lemma, $L_{K}(H)$ has exponential growth. The case that $s(h)_{\operatorname{supp}}\cap r(h)_{\operatorname{supp}}=\emptyset$ and $r(h)$ is a proper multiset can be handled similarly. ∎

If $H$ is a hypergraph, then we denote by $E^{\prime}$ the subset of $\{h_{ij},h_{ij}^{*}\mid h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$ consisting of all the elements which are not a letter of a quasi-cycle. We denote by $P^{\prime}$ the set of all nod-paths which are composed from elements of $E^{\prime}$.

Lemma 26**.**

Let $H$ be a finite hypergraph. Then $|P^{\prime}|<\infty$.

Proof.

Let $p^{\prime}=x_{1}\dots x_{n}\in P^{\prime}$. Assume that there are $1\leq i<j\leq n$ such that $x_{i}=x_{j}$. Then $x_{i}\dots x_{j-1}$ is a nod2-path. Since for any nod2-path $q$ which is not a quasi-cycle there is a shorter nod2-path $q^{\prime}$ such that any letter of $q^{\prime}$ already appears in $q$, we get a contradiction. Hence the $x_{i}$’s are pairwise distinct. It follows that $|P^{\prime}|<\infty$ since $|E^{\prime}|<\infty$. ∎

Let $H$ be a hypergraph. A sequence $p_{1},\dots,p_{k}$ of quasi-cycles such that $p_{i}\not\approx p_{j}$ for any $i\neq j$ is called a chain of length $k$ if $p_{1}\Longrightarrow p_{2}\Longrightarrow\dots\Longrightarrow p_{k}$. We call a nod-path $p$ trivial if $p=v$ for some $v\in H^{0}$ and nontrivial otherwise.

Lemma 27**.**

Let $H$ be a hypergraph. If there is no selfconnected quasi-cycle, then any nontrivial nod-path $\alpha$ can be written as

[TABLE]

where $k\geq 0$, $o_{i}$ is the empty word or $o_{i}\in P^{\prime}~{}(1\leq i\leq k+1)$, $p_{1},\dots,p_{k}$ is a chain of quasi-cycles, $l_{i}$ is a nonnegative integer $(1\leq i\leq k)$, and $q_{i}\neq p_{i}$ is a prefix of $p_{i}~{}(1\leq i\leq k)$.

Proof.

Let $\alpha=x_{1}\dots x_{r}$ be a nontrivial nod-path. Then $x_{1},\dots,x_{r}\in\{e_{i},e_{i}^{*}\mid e\in E^{1},1\leq i\leq w(e)\}$. Let $r_{1}$ be minimal such that $x_{r_{1}}\not\in E^{\prime}$ (if such an $r_{1}$ does not exist, then $\alpha\in P^{\prime}$ and (3) holds with $k=0$ and $o_{1}=\alpha$). Then $o_{1}:=x_{1}\dots x_{r_{1}-1}$ is either the empty word or $o_{1}\in P^{\prime}$. Since $x_{r_{1}}\not\in E^{\prime}$, we have that $x_{r_{1}}$ is a letter of a quasi-cycle, say $p_{1}$. By Remark 21(b) we may assume that $x_{r_{1}}$ is the first letter of $p_{1}$. Let $r_{2}>r_{1}$ be minimal such that $x_{r_{2}}$ is not a letter of $p_{1}$. Then $x_{r_{1}},\dots,x_{r_{2}-1}$ are letters of $p_{1}$. It follows from Lemma 22 that $x_{r_{1}}\dots x_{r_{2}-1}=p_{1}^{l_{1}}q_{1}$ for some $l_{1}\geq 0$ and $q_{1}\neq p_{1}$ that is a prefix of $p_{1}$. Let $r_{3}\geq r_{2}$ be minimal such that $x_{r_{3}}\not\in E^{\prime}$ and set $o_{2}:=x_{r_{2}}\dots x_{r_{3}-1}$. Then $o_{2}$ is either the empty word or $o_{2}\in P^{\prime}$. Since $x_{r_{3}}\not\in E^{\prime}$, we have that $x_{r_{3}}$ is a letter of a quasi-cycle, say $p_{2}$, and we may assume that $p_{2}$ starts with $x_{r_{3}}$. Let $r_{4}>r_{3}$ be minimal such that $x_{r_{4}}$ is not a letter of $p_{2}$. Then $x_{r_{3}}\dots x_{r_{4}-1}=p_{2}^{l_{2}}q_{2}$ for some $l_{2}\geq 0$ and $q_{2}\neq p_{2}$ that is a prefix of $p_{2}$ (see above).

By repeating the procedure described in the previous paragraph one gets that $\alpha$ can be written as in Equation (3) where $k\geq 0$, $o_{i}$ is the empty word or $o_{i}\in P^{\prime}~{}(1\leq i\leq k+1)$, $p_{1},\dots,p_{k}$ are quasi-cycles, $l_{i}$ is a nonnegative integer $(1\leq i\leq k)$, and $q_{i}\neq p_{i}$ is a prefix of $p_{i}~{}(1\leq i\leq k)$. It remains to show that $p_{1},\dots,p_{k}$ is a chain of quasi-cycles.

First we show that $p_{i}\Longrightarrow p_{i+1}$ for any $1\leq i\leq k-1$. Because of (3) we know that $p_{i}\underbrace{q_{i}o_{i+1}}_{\beta:=}p_{i+1}$ is a nod-path. If $\beta$ is the empty word, then $p_{i}\Longrightarrow p_{i+1}$ by the definition of $\Longrightarrow$. Otherwise $p_{i}$ is not a prefix of $\beta$ since $q_{i}$ is the empty word or $|q_{i}|<|p_{i}|$ and $o_{i}$ is the empty word or $o_{i+1}\in P^{\prime}$. Hence we get again $p_{i}\Longrightarrow p_{i+1}$.

Now assume that $p_{i}\approx p_{j}$ for some $1\leq i<j\leq n$. Write $p_{i}=y_{1}\dots y_{t}$. Then $p_{j}=y_{m+1}\dots y_{t}y_{1}\dots y_{m}$ for some $1\leq m\leq t$. Because of (3) we have that $p_{i}q_{i}o_{i+1}p_{i+1}q_{i+1}\dots o_{j-1}p_{j-1}q_{j-1}o_{j}p_{j}$ is a nod-path. It follows that

[TABLE]

is a nod-path. By the construction in the first paragraph of this proof, the first letter in $\alpha$ after $q_{i}$ is not a letter of $p_{i}$. Hence $p_{i}$ is not a prefix of $\gamma$ and hence we get the contradiction $p_{i}\overset{\operatorname{nod}}{\Longrightarrow}p_{i}$. Therefore $p_{i}\not\approx p_{j}$ for any $i\neq j$ and thus $p_{1},\dots,p_{k}$ is a chain of quasi-cycles. ∎

Theorem 28**.**

Let $H$ be a finite hypergraph. Then:

1. (i)

$L_{K}(H)$* has polynomial growth iff there is no selfconnected quasi-cycle.* 2. (ii)

If $L_{K}(H)$ has polynomial growth, then $\operatorname{GKdim}L_{K}(H)=d$ where $d$ is the maximal length of a chain of quasi-cycles.

Proof.

If there is a selfconnected quasi-cycle, then $L_{K}(H)$ has exponential growth by Lemma 23. Suppose now that there is no selfconnected quasi-cycle. Let $V$ denote the finite-dimensional generating subspace of $L_{K}(H)$ spanned by $\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$. By Theorem 17 the nod-paths of length $\leq n$ form a basis for $V^{n}$. By Lemma 27 we can write any nontrivial nod-path $\alpha$ of length $\leq n$ as

[TABLE]

where $k\geq 0$, $o_{i}$ is the empty word or $o_{i}\in P^{\prime}~{}(1\leq i\leq k+1)$, $p_{1},\dots,p_{k}$ is a chain of quasi-cycles, $l_{i}$ is a nonnegative integer $(1\leq i\leq k)$, and $q_{i}\neq p_{i}$ is a prefix of $p_{i}~{}(1\leq i\leq k)$. Clearly

[TABLE]

since $|\alpha|\leq n$. Now fix a chain $p_{1},\dots,p_{k}$ of quasi-cycles and further $o_{i}$’s and $q_{i}$’s as above. The number of solutions $(l_{1},\dots,l_{k})$ of (5) is $\sim n^{k}$. This implies that the number of nod-paths $\alpha$ of length $n$ or less that can be written as in (4) (corresponding to the choice of the $p_{i}$’s, $o_{i}$’s and $q_{i}$’s) is $\preccurlyeq n^{k}\leq n^{d}$. Since there are only finitely many quasi-cycles and finitely many choices for the $o_{i}$’s and $q_{i}$’s (note that $|P^{\prime}|<\infty$ by Lemma 26), the number of nod-paths of length $n$ or less is $\preccurlyeq n^{d}$.

On the other hand, choose a chain $p_{1},\dots,p_{d}$ of length $d$. Then $p_{1}o_{1}p_{2}\dots o_{d-1}p_{d}$ is a nod-path for some $o_{1},\dots,o_{d-1}$ such that for any $i\in\{1,\dots,d-1\}$, $o_{i}$ is either the empty word or a nod-path such that $p_{i}$ is not a prefix of $o_{i}$. Consider the nod-paths

[TABLE]

where $l_{1},\dots,l_{d}\in\mathbb{N}$ satisfy

[TABLE]

Let $A=(l_{1},\dots,l_{d})$ and $B=(l^{\prime}_{1},\dots,l^{\prime}_{d})$ be different solutions of (7). Assume that $A$ and $B$ define the same nod-path in (6). After cutting out the common beginning, we can assume that the nod-path defined by $A$ starts with $o_{i}p_{i+1}$ for some $i\in\{1,\dots,d-1\}$ and the nod-path defined by $B$ with $p_{i}o_{i}$ or $p_{i}^{2}$. If $o_{i}$ is the empty word, then we get the contradiction $p_{i}=p_{i+1}$, since $p_{i}$ and $p_{i+1}$ are quasi-cycles. Suppose now that $o_{i}$ is not the empty word. Since $p_{i}$ is not a prefix of $o_{i}$, it follows that $|o_{i}|<|p_{i}|$. Further $|p_{i}|<|o_{i}|+|p_{i+1}|$ (otherwise $p_{i+1}$ would be a subword of $p_{i}$ of length $<|p_{i}|$). Write $p_{i}=x_{1}\dots x_{k}$ and $p_{i+1}=y_{1}\dots y_{m}$.

case 1 Assume that $|p_{i}|\leq|p_{i+1}|$. Then $o_{i}=x_{1}\dots x_{j}$ and $p_{i+1}=x_{j+1}\dots x_{k}x_{1}\dots x_{j}y_{k+1}\dots y_{m}$ for some $j\in\{1,\dots,k-1\}$. By Remark 21(b), $x_{j+1}\dots x_{k}x_{1}\dots x_{j}$ is a quasi-cycle. It follows that $k=m$. Hence we get the contradiction $p_{i}\approx p_{i+1}$.

case 2 Assume that $|p_{i}|>|p_{i+1}|$. Then $o_{i}=x_{1}\dots x_{j}$ and $p_{i+1}=x_{j+1}\dots x_{k}x_{1}\dots x_{l}$ for some $j\in\{1,\dots,k-1\}$ and $l\in\{1,\dots,j-1\}$. But this yields the contradiction that $p_{i+1}$ is a subword of $p_{i}^{2}$ of length $<|p_{i}|$.

Hence different solutions of (7) define different nod-paths in (6). The number of solutions of (7) is $\sim n^{d}$ and thus $n^{d}\preccurlyeq$ the number of nod-paths of length $n$ or less. ∎

Remark 29**.**

Let $H$ be a nonfinite hypergraph. One can use Theorem 28 to determine $\operatorname{GKdim}L_{K}(H)$ as follows. Let $\{H_{i}\mid i\in I\}$ be the direct system of all finite subhypergraphs of $H$. Then $L_{K}(E,w)=\varinjlim\limits_{i}L_{K}(H_{i})$ by Proposition 14. By [12, Theorem 3.1] we have $\operatorname{GKdim}L_{K}(H)=\sup\limits_{i}\operatorname{GKdim}L_{K}(H_{i})$.

In general it is not so easy to read off the quasi-cycles from a finite hypergraph. But there is the following algorithm to find all the quasi-cycles: For any vertex $v$ list all the d-paths $x_{1}\dots x_{n}$ starting and ending at $v$ and having the property that $x_{i}\neq x_{j}$ for any $i\neq j$ (there are only finitely many of them). Now delete from that list any $p$ such that $p^{2}$ is not a nod-path. Next delete from the list any $p$ such that $p^{2}$ has a subword $q$ of length $|q|<|p|$ such that $q^{2}$ is a nod-path. The remaining d-paths on the list are precisely the quasi-cycles starting (and ending) at $v$.

Example 30**.**

Consider again the hypergraph

$H:$ $v_{1}$ $v_{2}$ $w_{1}$ $w_{2}$ h

from Example 4. By applying the algorithm described in the paragraph right before this example we find that the only quasi-cycles are $q=h_{22}h_{22}^{*}$ and $q^{\prime}=h_{22}^{*}h_{22}$. Clearly $q\approx q^{\prime}$. It follows from Theorem 28 that $\operatorname{GKdim}L_{K}(H)=1$.

7. Valuations and local valuations

7.1. General results

Definition 31**.**

Let $R$ be a ring. A valuation on $R$ is a map $\nu:R\rightarrow\mathbb{N}_{0}\cup\{-\infty\}$ such that

1. (i)

$\nu(x)=-\infty\Leftrightarrow x=0$ for any $x\in R$, 2. (ii)

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

$\nu(xy)=\nu(x)+\nu(y)$ for any $x,y\in R$.

We use the conventions $-\infty<n$ for any $n\in\mathbb{N}_{0}$ and $(-\infty)+n=n+(-\infty)=-\infty$ for any $n\in\mathbb{N}_{0}\cup\{-\infty\}$.

Remark 32**.**

One checks easily that condition (ii) in Definition 31 is satisfied iff the conditions (iia) and (iib) below are satisfied.

1. (iia)

$\nu(x+y)\leq\max\{\nu(x),\nu(y)\}$ for any $x,y\in R$. 2. (iib)

$\nu(x)=\nu(-x)$ for any $x\in R$.

Recall that a domain is a nonzero ring without zero divisors.

Lemma 33**.**

Let $R$ be a nonzero ring that has a valuation. Then $R$ is a domain.

Proof.

Let $\nu$ be a valuation on $R$. Let $x,y\in R$ such that $xy=0$. Then $-\infty=\nu(0)=\nu(xy)=\nu(x)+\nu(y)$ and hence $\nu(x)=-\infty$ or $\nu(y)=-\infty$. Thus $x=0$ or $y=0$. ∎

Definition 34**.**

A ring with enough idempotents is a pair $(R,E)$ where $R$ is a ring and $E$ is a set of nonzero orthogonal idempotents in $R$ for which the set of finite sums of distinct elements of $E$ is a set of local units for $R$. Note that if $(R,E)$ is a ring with enough idempotents, then $R=\bigoplus\limits_{e\in E}eR=\bigoplus\limits_{f\in E}Rf=\bigoplus\limits_{e,f\in E}eRf$ as additive groups. A ring with enough idempotents $(R,E)$ is called connected if $eRf\neq\{0\}$ for any $e,f\in E$.

Definition 35**.**

Let $(R,E)$ be a ring with enough idempotents. A local valuation on $(R,E)$ is a map $\nu:R\rightarrow\mathbb{N}_{0}\cup\{-\infty\}$ such that

1. (i)

$\nu(x)=-\infty\Leftrightarrow x=0$ for any $x\in R$, 2. (ii)

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

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

A local valuation $\nu$ on $(R,E)$ is called trivial if $\nu(x)=0$ for any $x\in R\setminus\{0\}$ and nontrivial otherwise.

Let $R$ be a ring. Recall that a left ideal $I$ of $R$ is called essential if $I\cap J=\{0\}~{}\Rightarrow~{}J=\{0\}$ for any left ideal $J$ of $R$. If $I$ is an essential left ideal of $R$, then we write $I\subseteq_{e}R$. For any $x\in R$ define the left ideal $\operatorname{ann}(x):=\{y\in R\mid yx=0\}$. The ring $R$ is called left nonsingular, if $\operatorname{ann}(x)\subseteq_{e}R~{}\Leftrightarrow x=0$ for any $x\in R$. A right nonsingular ring is defined similarly. $R$ is called nonsingular if it is left and right nonsingular.

Proposition 36**.**

Let $(R,E)$ be a ring with enough idempotents that has a local valuation. Then $R$ is nonsingular.

Proof.

We show only left singularity of $R$ and leave the right singularity to the reader. Let $\nu$ be a local valuation on $(R,E)$ and $x\in R\setminus\{0\}$. Choose an $e\in E$ such that $ex\neq 0$. Suppose that $ye\in\operatorname{ann}(x)$ for some $y\in R$. Then

[TABLE]

and hence $ye=0$. This shows that $\operatorname{ann}(x)\cap Re=\{0\}$. But $Re\neq\{0\}$ since $e\in Re$. Hence $\operatorname{ann}(x)$ is not essential. ∎

Recall that a nonzero ring $R$ is called a prime ring if $IJ=\{0\}~{}\Rightarrow I=\{0\}\lor J=\{0\}$ for any ideals $I$ and $J$ of $R$. Equivalently, $R$ is a prime ring if $xRy=\{0\}~{}\Rightarrow~{}(x=0\lor y=0)$ for any $x,y\in R$.

Proposition 37**.**

Let $(R,E)$ be a nonzero, connected ring with enough idempotents that has a local valuation. Then $R$ is a prime ring.

Proof.

Let $\nu$ be a local valuation on $(R,E)$ and $x,y\in R\setminus\{0\}$. Clearly there are $e,f\in E$ such that $xe,fy\neq 0$. Since $(R,E)$ is connected, we can choose a $z\in eRf\setminus\{0\}$. Clearly

[TABLE]

Hence $xzy\neq 0$ and thus $xRy\neq\{0\}$. ∎

Recall that a ring $R$ is called von Neumann regular if for any $x\in R$ there is a $y\in R$ such that $xyx=x$.

Proposition 38**.**

Let $(R,E)$ be a ring with enough idempotents that has a nontrivial local valuation. Then $R$ is not von Neumann regular.

Proof.

Let $\nu$ be a nontrivial local valuation on $(R,E)$. Choose an $x\in R$ such that $\nu(x)>0$. By condition (ii) in Definition 35 we may assume that $x\in eRf$ for some $e,f\in E$. Let $y\in R$. Clearly

[TABLE]

It follows that either $\nu(xyx)=-\infty$ (if $\nu(fye)=-\infty$) or $\nu(xyx)\geq 2\nu(x)$ (if $\nu(fye)\geq 0$). Thus $xyx\neq x$. ∎

Recall that a ring is called semiprimitive if its Jacobson radical is the zero ideal.

Proposition 39**.**

Let $(R,E)$ be a connected ring with enough idempotents. Suppose $R$ is a $K$-algebra and there is a local valuation $\nu$ on $(R,E)$ such that $\nu(x)=0$ iff $x\in\operatorname{span}(E)\setminus\{0\}$ where $\operatorname{span}(E)$ denotes the linear subspace of $R$ spanned by $E$. Then $R$ is semiprimitive.

Proof.

Let $\nu$ be the local valuation on $(R,E)$ such that $\nu(x)=0$ iff $x\in\operatorname{span}(E)\setminus\{0\}$. Assume that the Jacobson radical $J$ of $R$ is not zero. Since $R=\bigoplus\limits_{v,w\in E}eRf$, there are $e,f\in E$ and an $x^{\prime}\in J\cap eRf\setminus\{0\}$. Since $R$ is connected, we can choose an element $z\in fRe\setminus\{0\}$. Then $x:=x^{\prime}z\in J\cap eRe\setminus\{0\}$ since $\nu(x)=\nu(x^{\prime}z)=\nu(x^{\prime})+\nu(z)\geq 0$. Since $J$ does not contain any nonzero idempotents, it follows that $\nu(x)>0$ (if $\nu(x)=0$, then $x=ke$ for some $k\in K$ and hence $J$ contains the nonzero idempotent $e$). Since $x\in J$, we have that $x$ is left quasi-regular, i.e. there is a $y\in R$ such that $x+y=yx$. By multiplying $e$ from the right and from the left one gets $x+eye=eyex$. Hence we may assume that $y\in eRe$. It follows that

[TABLE]

This implies that $\nu(y)=0$ and hence $y=ke$ for some $k\in K$. It follows that $y=yx-x=kex-x=kx-x=(k-1)x$. But this yield a contradiction since $\nu(y)=0$ but either $\nu((k-1)x)=-\infty$, if $k=1$, or $\nu((k-1)x)=\nu(x)>0$, if $k\neq 1$ (note that $\nu((k-1)x)=\nu((k-1)ex)=\nu((k-1)e)+\nu(x)$). Thus the Jacobson radical of $R$ is zero. ∎

7.2. Applications to Leavitt path algebras of hypergraphs

In this subsection $H$ denotes a hypergraph. Note that $(L_{K}(H),H^{0})$ is a ring with enough idempotents. Set $X:=\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$. Let $K\langle X\rangle$ be the free algebra generated by $X$ and $K\langle X\rangle_{\operatorname{nod}}$ the subspace of $K\langle X\rangle$ generated by the nod-paths. By Theorem 17 there is an isomorphism $\operatorname{NF}:L_{K}(H)\rightarrow K\langle X\rangle_{\operatorname{nod}}$ of $K$-vector spaces. Note that for an $a\in L_{K}(H)$, its normal representative $\operatorname{NF}(a)$ can be obtained by taking an arbitrary representative of $a$ in $K\langle X\rangle$ and then applying reductions corresponding to the relations (i’)-(v’) in the proof of Theorem 17 (i.e. replacing an occurrence of a word in the representative of $a$ that equals a LHS of one the relations (i’)-(v’) by the correponding RHS) in an arbitrary order until it is no longer possible. For an $a\in L_{K}(H)$ we define its support $\operatorname{supp}(a)$ as the set of all nod-paths which appear in $\operatorname{NF}(a)$ with nonzero coefficient. Recall that if $p=x_{1}\dots x_{n}$ is a nod-path, then its length $|p|$ is $n$ if $x_{1},\dots,x_{n}\in X\setminus H^{0}$ and [math] if $n=1$ and $x_{1}\in H^{0}$.

Definition 40**.**

$H$ satisfies Condition (LV) if $|s(h)|,|r(h)|\geq 2$ for any $h\in H^{1}$.

Theorem 41**.**

If $H$ satisfies Condition (LV), then the map

[TABLE]

is a local valuation on $(L_{K}(H),H^{0})$. Here we use the convention $\max(\emptyset)=-\infty$.

Proof.

Obviously conditions (i) and (ii) in Definition 35 are satisfied. It remains to show that condition (iii) is satisfied. Let $v\in H^{0}$, $a\in L_{K}(H)v$ and $b\in vL_{K}(H)$. If one of the terms $\nu(a)$ and $\nu(b)$ equals [math] or $-\infty$, then clearly $\nu(ab)=\nu(a)+\nu(b)$. Suppose now $\nu(a),\nu(b)\geq 1$. Clearly $\nu(ab)\leq\nu(a)+\nu(b)$ since a reduction preserves or decreases the length of a monomial. It remains to show that $\nu(ab)\geq\nu(a)+\nu(b)$. Let

[TABLE]

be the elements of $\operatorname{supp}(a)$ with maximal length (namely $\nu(a)$) and

[TABLE]

be the elements of $\operatorname{supp}(b)$ with maximal length (namely $\nu(b)$). We assume that the $p_{k}$’s are pairwise distinct and also that the $q_{l}$’s are pairwise distinct. Note that $x^{k}_{i},y^{l}_{j}\in X\setminus H^{0}$ for any $i,j,k,l$ since $\nu(a),\nu(b)\geq 1$. Since $\operatorname{NF}$ is a linear map, we have

[TABLE]

1. Case 1

Assume that $x^{k}_{\nu(a)}y^{l}_{1}$ is not forbidden for any $k,l$.

Then $p_{k}q_{l}\in\operatorname{supp}(ab)$ for any $k,l$. It follows that $\nu(ab)\geq|p_{k}q_{l}|=\nu(a)+\nu(b)$. 2. Case 2

Assume that there are $k,l$ such that $x^{k}_{\nu(a)}y^{l}_{1}=h_{i1}h_{j1}^{*}$ for some $h\in H^{1}$ and $1\leq i,j\leq|s(h)|$.

1. Case 2.1

*Assume $p_{k^{\prime}}q_{l^{\prime}}\neq x^{k}_{1}\dots x^{k}_{\nu(a)-1}h_{i2}h_{j2}^{*}y^{l}_{2}\dots y^{l}_{\nu(b)}$ for any $k^{\prime},l^{\prime}$.

*Then

[TABLE]

since it does not cancel with another term. It follows that $\nu(ab)\geq\nu(a)+\nu(b)$. 2. Case 2.2

*Assume $p_{k^{\prime}}q_{l^{\prime}}=x^{k}_{1}\dots x^{k}_{\nu(a)-1}h_{i2}h_{j2}^{*}y^{l}_{2}\dots y^{l}_{\nu(b)}$ for some $k^{\prime},l^{\prime}$.

*One checks easily that in this case

[TABLE]

It follows that $\nu(ab)\geq\nu(a)+\nu(b)$. 3. Case 3

Assume that there are $k,l$ such that $x^{k}_{\nu(a)}y^{l}_{1}=h_{1i}^{*}h_{1j}$ for some $h\in h^{1}$ and $1\leq i,j\leq|r(h)|$.

1. Case 3.1

*Assume $p_{k^{\prime}}q_{l^{\prime}}\neq x^{k}_{1}\dots x^{k}_{\nu(a)-1}h_{2i}^{*}h_{2j}y^{l}_{2}\dots y^{l}_{\nu(b)}$ for any $k^{\prime},l^{\prime}$.

*Then

[TABLE]

since it does not cancel with another term. It follows that $\nu(ab)\geq\nu(a)+\nu(b)$. 2. Case 3.2

*Assume $p_{k^{\prime}}q_{l^{\prime}}=x^{k}_{1}\dots x^{k}_{\nu(a)-1}h_{2i}^{*}h_{2j}y^{l}_{2}\dots y^{l}_{\nu(b)}$ for some $k^{\prime},l^{\prime}$.

*One checks easily that in this case

[TABLE]

It follows that $\nu(ab)\geq\nu(a)+\nu(b)$.

Hence condition (iii) also is satisfied and thus $\nu$ is a local valuation on $(L_{K}(H),H^{0})$. ∎

Corollary 42**.**

Suppose that $H$ satisfies Condition (LV). Then $L_{K}(H)$ is nonsingular.

Proof.

Follows from Proposition 36 and Theorem 41. ∎

Recall that a d-path is a path in the double graph $\hat{E}$ of the directed graph $E$ associated to $H$. We call $H$ connected, if for any $v,w\in H^{0}$ there is a d-path $p$ such that $s(p)=v$ and $r(p)=w$.

Corollary 43**.**

Suppose that $H$ satisfies Condition (LV) and is connected. Then $L_{K}(H)$ is a prime ring.

Proof.

Cleary $L_{K}(H)$ is not the zero ring since $H$ is not the empty hypergraph. Let $v,w\in H^{0}$. Since $H$ is connected, there is a d-path $p$ such that $s(p)=v$ and $r(p)=w$. Let $\nu$ be the local valuation on $(L_{K}(H),H^{0})$ defined in Theorem 41. Clearly $\nu(p)=|p|\geq 0$ since $\nu(u)=0$ for any $u\in H^{0}$ and $\nu(h_{ij})=\nu(h_{ij}^{*})=1$ for any $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. Hence $vL_{K}(H)w\neq\{0\}$ for any $v,w\in H^{0}$ and therefore $(L_{K}(H),H^{0})$ is a connected ring with enough idempotents. It follows from Proposition 37 that $L_{K}(H)$ is a prime ring. ∎

Corollary 44**.**

Suppose that $H$ satisfies Condition (LV) and that $H^{1}\neq\emptyset$. Then $L_{K}(H)$ is not von Neumann regular.

Proof.

Let $\nu$ be the local valuation on $(L_{K}(H),H^{0})$ defined in Theorem 41. Choose an $h\in H^{1}$. Then $\nu(h_{11})=1$ and therefore $\nu$ is nontrivial. It follows from Proposition 38 that $L_{K}(H)$ is not von Neumann regular. ∎

Corollary 45**.**

Suppose that $H$ satisfies Condition (LV) and is connected. Then $L_{K}(H)$ is semiprimitive.

Proof.

Let $\nu$ be the local valuation on $(L_{K}(H),H^{0})$ defined in Theorem 41. Then clearly $\nu(x)=0$ iff $x\in\operatorname{span}(H^{0})\setminus\{0\}$ where $\operatorname{span}(H^{0})$ denotes the linear subspace of $L_{K}(H)$ spanned by $H^{0}$. Moreover, $(L_{K}(H),H^{0})$ is connected since $H$ is connected (see the proof of Corollary 43). It follows from Proposition 39 that $L_{K}(H)$ semiprimitive. ∎

Example 46**.**

Consider again the hypergraph

$H:$ $v_{1}$ $v_{2}$ $w_{1}$ $w_{2}$ h

from Examples 4 and 30. As shown in Example 30, $\operatorname{GKdim}L_{K}(H)=1$. Clearly $H$ satisfies Condition (LV) and is connected. It follows from the Corollaries 42, 43, 44 and 45 that $L_{K}(H)$ is nonsingular, prime, not von Neumann regular and semiprimitive. Since $L_{K}(H)$ is a finitely generated prime algebra of GK dimension one, $L_{K}(H)$ is fully bounded Noetherian and finitely generated as a module over its center (see [21]). It follows that $L_{K}(H)$ is a PI-ring.

7.3. Classification of the hypergraphs $H$ such that $L_{K}(H)$ is a domain

Definition 47**.**

A hypergraph $H$ satisfies Condition (B) if any $h\in H^{1}$ has the property that either $|s(h)|,|r(h)|\geq 2$ or $|s(h)|=|r(h)|=1$.

Theorem 48**.**

Let $H$ be a hypergraph, then $L_{K}(H)$ is a domain iff $|H^{0}|=1$ and $H$ satisfies Condition (B).

Proof.

If there are distinct $v,w\in H^{0}$. Then $vw=0$ in $L_{K}(H)$ by relation (i) in Definition 2. But $v,w\neq 0$ in $L_{K}(H)$ by Theorem 17 (note that $v$ and $w$ are nod-paths). Hence $L_{K}(H)$ is not a domain.

Suppose now that Condition (B) is not satisfied. Then there is an $h\in H^{1}$ such that $|s(h)|=1~{}\land~{}|r(h)|\geq 2$ or $|s(h)|\geq 2~{}\land~{}|r(h)|=1$. We only consider the first case and leave the second to the reader. Choose $i\neq j\in\{1,\dots,|r(h)|\}$ (possible since $|r(h)|\geq 2$). Then $h_{1i}^{*}h_{1j}=0$ by relation (iv) in Definition 2. But $h_{1i}^{*},h_{1j}\neq 0$ in $L_{K}(H)$ by Theorem 17 (note that $h_{1i}^{*}$ and $h_{1j}$ are nod-paths). Hence $L_{K}(H)$ is not a domain.

Now suppose that $|H^{0}|=1$ and $H$ satisfies Condition (B). For any $h\in H^{1}$ define a hypergraph $H_{h}=(H_{h}^{0},H_{h}^{1},s_{h},r_{h})$ by $H_{h}^{0}=H^{0}$, $H_{h}^{1}=\{h\}$, $s_{h}(h)=s(h)$ and $r_{h}(h)=r(h)$. If $|s(h)|,|r(h)|\geq 2$, then $H_{h}$ satisfies Condition (LV) and hence $L_{K}(H_{h})$ is a domain by Lemma 33 and Theorem 41 (the local valuation $\nu$ is in fact a valuation since $|H^{0}|=1$). If $|s(h)|=|r(h)|=1$, then $L_{K}(H_{h})$ is isomorphic to the Laurent polynomial ring $K[X,X^{-1}]$, which is a domain. Hence all the algebras $L_{K}(H_{h})~{}(h\in H^{1})$ are domains. One checks easily that $L_{K}(H)$ is the coproduct (sometimes also called “free product”) of the $K$-algebras $L_{K}(H_{h})~{}(h\in H^{1})$. It follows from [8, Theorem 3.2] that $L_{K}(H)$ is a domain (note that a “$1$-fir” is the same as a domain). ∎

8. Additional applications of the linear bases

8.1. General results

In this subsection $(A,E)$ denotes a $K$-algebra with enough idempotents, i.e. $A$ is a $K$-algebra and $(A,E)$ is a ring with idempotents. We call an element $a\in A$ homogeneous if $a\in eAf$ for some $e,f\in E$ (recall that $A=\bigoplus\limits_{e,f\in E}eAf$). $B$ denotes a $K$-basis for $A$ which consists of homogeneous elements and contains $E$. Moreover, $\mu$ denotes a map $B\rightarrow\mathbb{N}_{0}$ which has the property that $\mu(b)=0~{}\Leftrightarrow~{}b\in E$.

Definition 49**.**

An element $b\in B\cap eAf$ is called left adhesive if $ab\in B$ for any $a\in B\cap Ae$ and right adhesive if $bc\in B$ for any $c\in B\cap fA$. An element $b\in B$ is called adhesive if it is left and right adhesive. A left valuative basis element is a left adhesive element $b\in B\cap eA$ such that $\mu(ab)=\mu(a)+\mu(b)$ for any $a\in B\cap Ae$. A right valuative basis element is a right adhesive element $b\in B\cap Af$ such that $\mu(bc)=\mu(b)+\mu(c)$ for any $c\in B\cap fA$. A valuative basis element is an adhesive element $b\in B\cap eAf$ such that $\mu(abc)=\mu(a)+\mu(b)+\mu(c)$ for any $a\in B\cap Ae$ and $c\in B\cap fA$.

If $S\in A$ is a subset, then we denote by $I_{l}(S)$ the left ideal of $A$ generated by $S$, by $I_{r}(S)$ the right ideal of $A$ generated by $S$ and by $I(S)$ the ideal of $A$ generated by $S$. The following lemma is straightforward to check.

Lemma 50**.**

The following is true:

1. (i)

If $S_{l}=\{b_{i}\in B\cap e_{i}Af_{i}\mid i\in\Phi\}$ is a set of left adhesive basis elements, then $I_{l}(S_{l})$ is free with basis $\{ab_{i}\mid i\in\Phi,a\in B\cap Ae_{i}\}$. 2. (ii)

If $S_{r}=\{b_{i}\in B\cap e_{i}Af_{i}\mid i\in\Phi\}$ is a set of right adhesive basis elements, then $I_{r}(S_{r})$ is free with basis $\{b_{i}c\mid i\in\Phi,c\in B\cap f_{i}A\}$. 3. (iii)

If $S=\{b_{i}\in B\cap e_{i}Af_{i}\mid i\in\Phi\}$ is a set of adhesive basis elements, then $I(S)$ is free with basis $\{ab_{i}c\mid i\in\Phi,a\in B\cap Ae_{i},c\in B\cap f_{i}A\}$.

Lemma 51**.**

Distinct left valuative basis elements generate distinct left ideals, distinct right valuative basis elements generate distinct right ideals and distinct valuative basis elements generate distinct ideals.

Proof.

Let $b\in B\cap eAf$ and $b^{\prime}\in B\cap e^{\prime}Af^{\prime}$ be valuative basis elements. W.l.o.g assume that $\mu(b)\geq\mu(b^{\prime})$. Assume that $b^{\prime}\in I(b)$. By the previous lemma, $B_{I(b)}:=\{abc\mid a\in B\cap Ae,c\in B\cap fA\}$ is a basis for $I(b)$. Hence $b^{\prime}=\sum\limits_{x\in B_{I(b)}}k_{x}x$ where almost all $k_{x}$ are zero. Since $b$ is valuative, $\mu(x)>\mu(b)\geq\mu(b^{\prime})$ for any $x\in B_{I(b)}\setminus\{b\}$. It follows that $k_{x}=0$ for any $x\in B_{I(b)}\setminus\{b\}$ and $k_{b}=1$. Thus $b=b^{\prime}$. The other assertions of the lemma can be shown similarly. ∎

Proposition 52**.**

Suppose there exists a valuative basis element $b\in(B\setminus E)\cap eAe$. Then $\dim_{K}(A)=\infty$, $A$ is not simple (in fact it has infinitely many ideals), neither left nor right Artinian and not von Neumann regular.

Proof.

Clear $\mu(b^{n})=n\mu(b)$ since $b$ is valuative. Hence $b^{m}\neq b^{n}$ if $m\neq n$ (note that $\mu(b)\geq 1$ since $b\not\in E$). Therefore $\dim_{K}(A)=\infty$. Lemma 51 implies that we have an infinite descending chain $I(b)\supsetneq I(b^{2})\supsetneq I(b^{3})\dots$ of ideals (note that $b^{n}$ is valuative for any $n\in\mathbb{N}$ since $b$ valuative). Hence $A$ is neither left nor right Artinian. Now assume that $bab=b$ for some $a\in A$. Clearly we may assume that $a\in eAe$. Write $a=\sum\limits_{c\in B\cap eAe}k_{c}c$ where almost all $k_{c}$ are zero. Then $b=bab=\sum\limits_{c\in B\cap eAe}k_{c}bcb$. But $\mu(bcb)=\mu(b)+\mu(c)+\mu(b)>\mu(b)$ for any $c\in B\cap eAe$ and hence we arrived at a contradiction. Thus $A$ is not von Neumann regular. ∎

Definition 53**.**

We say that two basis elements $b\in B\cap eAf$ and $b^{\prime}\in B\cap e^{\prime}Af^{\prime}$ have no common left multiple if there are no $a\in B\cap Ae,a^{\prime}\in B\cap Ae^{\prime}$ such that $ab=a^{\prime}b^{\prime}$. We say that $b$ and $b^{\prime}$ have no common right multiple if there are no $c\in B\cap fA,c^{\prime}\in B\cap f^{\prime}A$ such that $bc=b^{\prime}c^{\prime}$.

Definition 54**.**

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

Proposition 55**.**

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

Proof.

For any $n\in\mathbb{N}$ set $I_{n}:=I_{l}(b^{\prime}b,b^{\prime}b^{2},\dots,b^{\prime}b^{n})$. Then clearly $I_{n}\subseteq I_{n+1}$ for any $n\in\mathbb{N}$. Now assume that $b^{\prime}b^{n+1}\in I_{n}$ for some $n\in\mathbb{N}$. By Lemma 50 the set $\{ab^{\prime}b^{i}\mid 1\leq i\leq n,a\in B\cap Ae\}$ is a basis for $I_{n}$ (note that $b^{\prime}b^{i}$ is adhesive for any $1\leq i\leq n$ since $b$ is adhesive and $b^{\prime}$ is left adhesive). It follows that $b^{\prime}b^{n+1}=ab^{\prime}b^{i}$ for some $1\leq i\leq n$ and $a\in B\cap Ae$. This implies $b^{\prime}b^{n+1-i}=ab^{\prime}$ since $b$ is right cancellative. But this contradicts the assumption that $b$ and $b^{\prime}$ have no common left multiple. Hence $I_{n}\subsetneq I_{n+1}$ for any $n\in\mathbb{N}$ and therefore $A$ is not left Noetherian. Similarly one can prove the second assertion of the proposition. ∎

8.2. Applications to Leavitt path algebras of hypergraphs

Definition 56**.**

A hypergraph $H$ satisfies Condition (A) if there is an $h\in H^{1}$ such that $|s(h)|,|r(h)|\geq 2$.

Remark 57**.**

1. (a)

If $H$ is a hypergraph that does not satisfy Condition (A), then $L_{K}(H)$ is isomorphic to a Leavitt path algebra of a finitely separated graph, cf. Example 5. 2. (b)

If $H$ is a hypergraph, then

[TABLE]

The theorem below shows that a Leavitt path algebra $L_{K}(H)$ that is finite-dimensional as a $K$-vector space or simple or left Artinian or right Artininan or von Neumann regular is isomorphic to a Leavitt path algebra of a finitely separated graph.

Theorem 58**.**

Let $H$ be a hypergraph that satisfies Condition (A). Then $\dim_{K}(L_{K}(H))=\infty$, $L_{K}(H)$ is not simple (in fact it has infinitely many ideals), neither left nor right Artinian and not von Neumann regular.

Proof.

Let $B$ be the basis for $L_{K}(H)$ consisting of all nod-paths. Clearly $B$ consists of elements that are homogeneous with respect to $(L_{K}(H),H^{0})$ and contains $H^{0}$. Define a map $\mu:B\rightarrow\mathbb{N}_{0}$ by $\mu(p)=|p|$ for any nod-path $p$. Then clearly $\mu(b)=0$ iff $b\in H^{0}$. Since $H$ satisfies Condition (A), we can choose an $h\in H^{1}$ such that $|s(h)|,|r(h)|\geq 2$. Set $b:=h_{22}h_{22}^{*}$. One checks easily that $b\in(B\setminus H^{0})\cap s(h)_{2}L_{K}(H)s(h)_{2}$ is a valuative basis elements (note that $h_{22}$ and $h_{22}^{*}$ do not appear in a forbidden word, see Definition 16). The assertion of the theorem follows now from Proposition 52. ∎

Example 46 shows that there is a hypergraph $H$ which satisfies Condition (A) (even Condition (LV)) such that $L_{K}(H)$ is Noetherian. But the next theorem shows that if a hypergraph $H$ satisfies Condition (A’), then $L_{K}(H)$ cannot be left or right Noetherian.

Theorem 59**.**

Let $H$ be a hypergraph that satisfies Condition (A’). Then $L_{K}(H)$ is neither left nor right Noetherian.

Proof.

Define $B$ and $\mu$ as in the proof of the previous theorem. Since $H$ satisfies Condition (A’), we can choose an $h\in H^{1}$ such that $|s(h)|,|r(h)|\geq 2$ and $s(h)$ is a proper multiset or $r(h)$ is a proper multiset or $s(h)$ and $r(h)$ are sets with nontrivial intersection.

First suppose that $s(h)$ is a proper multiset. Choose $1\leq i<j\leq|s(h)|$ such that $s(h)_{i}=s(h)_{j}$. Set $b:=h_{j2}h_{j2}^{*}$, $b^{\prime}:=h_{j2}h_{i2}^{*}$ and $b^{\prime\prime}:=h_{i2}h_{j2}^{*}$. Then clearly $b,b^{\prime},b^{\prime\prime}\in(B\setminus H^{0})\cap vL_{K}(H)v$ where $v=s(h)_{i}=s(h)_{j}$. One checks easily that $b$ is adhesive and both left and right cancellative, $b^{\prime}$ is left adhesive, $b^{\prime\prime}$ is right adhesive, $b$ and $b^{\prime}$ have no common left multiple and $b$ and $b^{\prime\prime}$ have no common right multiple (note that $h_{j2}$ and $h_{j2}^{*}$ do not appear in a forbidden word, see Definition 16). It follows from Proposition 55 that $L_{K}(H)$ is neither left nor right Noetherian. The case that $r(h)$ is a proper multiset can be handled similarly.

Now suppose that $s(h)$ and $r(h)$ are sets with nontrivial intersection. By Remark 3 (a) we may assume that $s(h)_{2}=r(h)_{2}$. Set $b:=h_{22}$ and $b^{\prime}:=h_{22}^{*}$. Then clearly $b,b^{\prime}\in(B\setminus H^{0})\cap vL_{K}(H)v$ where $v=s(h)_{2}=r(h)_{2}$. One checks easily that $b$ and $b^{\prime}$ are adhesive and both left and right cancellative and that $b$ and $b^{\prime}$ have neither a common left multiple nor a common right multiple (note that $h_{22}$ and $h_{22}^{*}$ do not appear in a forbidden word, see Definition 16). It follows from Proposition 55 that $L_{K}(H)$ is neither left nor right Noetherian. ∎

9. The $V$-monoid

Let $R$ be a ring. Recall that a left $R$-module $M$ is called unital if $RM=M$. We denote by $R$-$\operatorname{Mod}$ the category of unital left $R$-modules. Furthermore we denote by $R$-$\operatorname{Mod}_{\operatorname{proj}}$ the full subcategory of $R$-$\operatorname{Mod}$ whose objects are the projective objects of $R$-$\operatorname{Mod}$ that are finitely generated as a left $R$-module. When $R$ has local units, then

[TABLE]

see e.g. [3, Subsection 4A]. $V(R)$ becomes an abelian monoid by defining $[P]+[Q]=[P\oplus Q]$.

Recall from Section 4 that $\operatorname{\mathcal{H}}$ denotes the category of hypergraphs, $\operatorname{\mathcal{A}_{K}^{loc}}$ the category of $K$-algebras with local units and that $L_{K}:\operatorname{\mathcal{H}}\rightarrow\operatorname{\mathcal{A}_{K}^{loc}}$ is a functor that commutes with direct limits. We denote the category of abelian monoids by $\operatorname{\mathcal{M}^{ab}}$. One checks easily that $V$ defines a continuous functor $\operatorname{\mathcal{A}_{K}^{loc}}\rightarrow\operatorname{\mathcal{M}^{ab}}$ in a canonical way. In Subsection 10.1 we define a functor $M:\operatorname{\mathcal{H}}\rightarrow\operatorname{\mathcal{M}^{ab}}$. In Subsection 10.2 we recall some universal ring constructions by G. Bergman which will be used in the proof of the main result of this section, namely Theorem 61. In Subsection 10.3 we prove Theorem 61 which states that $V\circ L_{K}\cong M$ and that $L_{K}(H)$ is left and right hereditary provided $H$ is finite.

9.1. The functor $M:\operatorname{\mathcal{H}}\rightarrow\operatorname{\mathcal{M}^{ab}}$

Definition 60**.**

For any hypergraph $H$ let $M(H)$ be the abelian monoid presented by the generating set $H^{0}$ and the relations $\sum s(h)=\sum r(h)~{}(h\in H^{1})$ where $\sum s(h)=\sum\limits_{i=1}^{|s(h)|}s(h)_{i}$ and $\sum r(h)=\sum\limits_{j=1}^{|r(h)|}r(h)_{j}$. If $\phi:H\rightarrow I$ is a morphism in $\operatorname{\mathcal{H}}$, then there is a unique monoid homomorphism $M(\phi):M(H)\rightarrow M(I)$ such that $M(\phi)(v)=\phi^{0}(v)$ for any $v\in H^{0}$. One checks easily that $M:\operatorname{\mathcal{H}}\rightarrow\operatorname{\mathcal{M}^{ab}}$ is a functor that commutes with direct limits.

9.2. Some universal ring constructions by G. Bergman

In this subsection all rings are assumed to be unital. Let $k$ be a commutative ring and $R$ a $k$-algebra (i.e. $R$ is a ring given with a homomorphism of $k$ into its center). A $R$-ringk is a $k$-algebra $S$ given with a $k$-algebra homomorphism $R\rightarrow S$. By an $R$-module we mean a right $R$-module. In [6], G. Bergman described the following two key constructions:

• •

ADJOINING MAPS Let $M$ be any $R$-module and $P$ a finitely generated projective $R$-module. Then there exists $R$-ringk $S$, having a universal module homomorphism $f:M\otimes S\rightarrow P\otimes S$, see [6, Theorem 3.1]. $S$ can be obtained by adjoining to $R$ a family of generators subject to certain relations, see [6, Proof of Theorem 3.1].

• •

IMPOSING RELATIONS Let $M$ be any $R$-module, $P$ a projective $R$-module and $f:M\rightarrow P$ any module homomorphism. Then there exists an $R$-ringk $S$ such that $f\otimes S=0$ and universal for that property. $S$ can be chosen to be a quotient of $R$, see [6, Proof of Theorem 3.2].

Using the key constructions above Bergman described more complicated constructions. One of them is used in this paper:

• •

ADJOINING ISOMORPHISMS Given two finitely generated projective $R$-modules $P$ and $Q$, one can adjoin a universal isomorphism between $P\otimes$ and $Q\otimes$ by first freely adjoining maps $i:P\otimes\rightarrow Q\otimes$ and $i^{-1}:Q\otimes\rightarrow P\otimes$ (via ADJOINING MAPS) and then setting $1_{Q\otimes}-ii^{-1}$ and $1_{P\otimes}-i^{-1}i$ equal to [math] (via IMPOSING RELATIONS), see [6, p. 38]. Bergman denoted the resulting $R$-ringk by $R\langle i,i^{-1}:\overline{P}\cong\overline{Q}\rangle$.

Set $S:=R\langle i,i^{-1}:\overline{P}\cong\overline{Q}\rangle$. Bergman proved the following (for these results he required that $k$ is a field and that $P$ and $Q$ are nonzero): The abelian monoid $V(S)$ may be obtained from $V(R)$ by imposing one relation $[P]=[Q]$. Further the right global dimension of $S$ equals the right global dimension of $R$, unless the right global dimension of $R$ is [math], in which case the right global dimension of $S$ is $\leq 1$. See [6, Theorem 5.2 and last paragraph of p. 48].

It is easily seen that Bergman’s results mentioned above also apply to left $R$-modules.

9.3. The monoid $V(L_{K}(H))$

Theorem 61**.**

$V\circ L_{K}\cong M$. Moreover, if $H$ is a finite hypergraph, then $L_{K}(H)$ is left and right hereditary.

Proof.

We have divided the proof into two parts, Part I and Part II. In Part I we define a natural transformation $\theta:M\rightarrow V\circ L_{K}$. In Part II we show that $\theta$ is a natural isomorphism and further that $L_{K}(H)$ is left and right hereditary provided that $H$ is finite.

Part I Let $H$ be a hypergraph and $F(H)$ the free abelian monoid generated by $H^{0}$. There is a unique monoid homomorphism $\eta_{H}:F(H)\rightarrow V(L_{K}(H))$ such that $\eta_{H}(v)=[L_{K}(H)v]$ for any $v\in H^{0}$. In order to show that $\eta_{H}$ induces a monoid homomorphism $M(H)\rightarrow V(L_{K}(H))$ we have to check that $\eta_{H}(\sum s(h))=\eta_{H}(\sum r(h))$ for any $h\in H^{1}$, i.e. that

[TABLE]

Given $h\in H^{1}$ let $X_{h}$ be the $|s(h)|\times|r(h)|$-matrix whose entry at position $(i,j)$ is $h_{ij}$ and $Y_{h}$ the $|r(h)|\times|s(h)|$-matrix whose entry at position $(i,j)$ is $h_{ji}^{*}$. It follows from relations (iii) and (iv) in Definition 2 that

[TABLE]

Hence $X_{h}$ defines an isomorphism $\bigoplus\limits_{i=1}^{|s(h)|}L_{K}(H)s(h)_{i}\rightarrow\bigoplus\limits_{j=1}^{|r(h)|}L_{K}(H)r(h)_{j}$ by right multiplication (its inverse is defined by $Y_{h}$). Thus (8) holds and therefore $\eta_{H}$ induces a monoid homomorphism $\theta_{H}:M(H)\rightarrow V(L_{K}(H))$. It is an easy exercise to show that $\theta:M\rightarrow V\circ L_{K}$ is a natural transformation.

Part II We want to show that the natural transformation $\theta:M\rightarrow V\circ L_{K}$ defined in Part I is a natural isomorphism, i.e. that $\theta_{H}:M(H)\rightarrow V(L_{K}(H))$ is an isomorphism for any hypergraph $H$. By Proposition 12 any hypergraph is a direct limit of a direct system of finite hypergraphs. Hence it is sufficient to show that $\theta_{H}$ is an isomorphism for any finite hypergraph $H$ (note that $M$, $V$ and $L_{K}$ commute with direct limits).

Let $H$ be a finite hypergraph. Set $A_{0}:=K^{H^{0}}$. We denote by $\alpha_{v}$ the element of $A_{0}$ whose $v$-component is $1$ and whose other components are [math]. Write $H^{1}=\{h^{1},\dots,h^{n}\}$. We inductively define $K$-algebras $A_{1},\dots,A_{n}$ as follows. Let $1\leq t\leq n$ and assume that $A_{t-1}$ has already been defined. Set $A_{t}:=A_{t-1}\langle i,i^{-1}:\overline{P_{t}}\cong\overline{Q_{t}}\rangle$ (see [6, p. 38]) where

[TABLE]

Investigating the proofs of Theorems 3.1 and 3.2 in [6] we see that $L_{K}(H)\cong A_{n}$. By [6, Theorems 5.1, 5.2], the abelian monoid $V(A_{n})$ is isomorphic to $M(H)$. The monoid isomorphism $M(H)\rightarrow V(L_{K}(H))$ one gets in this way is precisely $\theta_{H}$.

Furthermore, the left global dimension of $A_{n}\cong L_{K}(E,w)$ is $\leq 1$ by [6, Theorems 5.1, 5.2], i.e. $L_{K}(H)$ is left hereditary. Since $L_{K}(H)$ is a ring with involution by Proposition 7 (ii), we have $L_{K}(H)\cong L_{K}(H)^{op}$. Thus $L_{K}(H)$ is also right hereditary. ∎

Recall that a monoid $N$ is called conical if $m+n=0~{}\Rightarrow~{}m=n=0$ for any $m,n\in N$. It is easy to see that $V(L_{K}(H))\cong M(H)$ is conical for any hypergraph $H$. The proposition below shows that conversely one can find for any conical abelian monoid $N$ a hypergraph $H$ such that $N\cong V(L_{K}(H))$ (follows also from Example 5 and [4, Proposition 4.4]).

Proposition 62**.**

For any conical abelian monoid $N$ there is a hypergraph $H$ such that $N\cong V(L_{K}(H))$.

Proof.

Choose a presentation of $N$ with a nonempty set $\{x_{j}\mid j\in J\}$ of generators and a nonempty set $\{r_{i}\mid i\in I\}$ of relations

[TABLE]

where for each $i$ almost all but not all $a_{ij}$ are zero and similarly almost all but not all $b_{ij}$ are zero. The relations can be chosen with these restrictions because $N$ is conical, see [4, Proof of Proposition 4.4]. Define a hypergraph $H=(H^{0},H^{1},s,r)$ by $H^{0}=\{v_{j}\mid j\in J\}$, $H^{1}:=\{h_{i}\mid i\in I\}$, $s(h_{i})(v_{j})=a_{ij}$ and $r(h_{i})(v_{j})=b_{ij}$ (here we consider $s(h)_{i}$ and $r(h_{i})$ as maps $H^{0}\rightarrow\mathbb{N}_{0}$ associating to each $v_{j}\in H^{0}$ its multiplicity in $s(h)$ resp. $r(h)$). It follows from Theorem 61 that $V(L_{K}(H))\cong M(H)\cong N$. ∎

10. The graded $V$-monoid

Throughout this section $\Gamma$ denotes a group with identity $\epsilon$. Let $R$ be a $\Gamma$-graded ring. Recall that a left $R$-module $M$ is called $\Gamma$-graded if there is a decomposition $M=\bigoplus\limits_{\gamma\in\Gamma}M_{\gamma}$ such that $R_{\alpha}M_{\gamma}\subseteq M_{\alpha\gamma}$ for any $\alpha,\gamma\in\Gamma$. We denote by $R$-$\operatorname{Gr}$ the category of $\Gamma$-graded unital left $R$-modules with morphisms the $R$-module homomorphisms that preserve grading. Furthermore we denote by $R$-$\operatorname{Gr}_{\operatorname{proj}}$ the full subcategory of $R$-$\operatorname{Gr}$ whose objects are the projective objects of $R$-$\operatorname{Gr}$ that are finitely generated as a left $R$-module. If $R$ has graded local units, then

[TABLE]

cf. [3, Subsections 2A,4A]. $V^{\operatorname{gr}}(R)$ becomes an abelian monoid by defining $[P]+[Q]=[P\oplus Q]$.

10.1. Smash products

Definition 63**.**

Let $R$ be a $\Gamma$-graded ring. The smash product ring $R\#\Gamma$ is defined as the set of all formal sums $\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}$ where $r^{(\gamma)}\in R$ for any $\gamma\in\Gamma$ and the $p_{\gamma}$’s are symbols. Addition is defined component-wise and multiplication is defined by linear extension of the rule $(rp_{\alpha})(sp_{\beta})=rs_{\alpha\beta^{-1}}p_{\beta}$ where $r,s\in R$ and $\alpha,\beta\in\Gamma$.

Recall that if $R$ is a ring, then $R$-$\operatorname{Mod}$ denotes the category of unital left $R$-modules and $R$-$\operatorname{Mod}_{\operatorname{proj}}$ the full subcategory of $R$-$\operatorname{Mod}$ whose objects are the projective objects of $R$-$\operatorname{Mod}$ that are finitely generated as a left $R$-module. When $R$ has local units, then $V(R)=\{[P]\mid P\in R\text{-}\operatorname{Mod}_{\operatorname{proj}}\}$.

Proposition 64**.**

Let $R$ be a $\Gamma$-graded ring with graded local units. Then the categories $R$-$\operatorname{Gr}_{\operatorname{proj}}$ and $R\#\Gamma$-$\operatorname{Mod}_{\operatorname{proj}}$ are isomorphic. It follows that $V^{\operatorname{gr}}(R)\cong V(R\#\Gamma)$.

Proof.

Fix a set $E$ of graded local units for $R$. By [3, Proposition 2.5], there is an isomorphism of categories $\psi:R\text{-}\operatorname{Gr}\rightarrow R\#\Gamma\text{-}\operatorname{Mod}$ which is defined on objects as follows. If $N$ is an object in $R\text{-}\operatorname{Gr}$, then $\psi(N)=N$ as abelian groups. The left-$R\#\Gamma$ action on $\psi(N)$ is defined by $(rp_{\alpha})n=rn_{\alpha}$ for any $r\in R$, $\alpha\in\Gamma$ and $n\in N$. The inverse functor $\phi:R\#\Gamma\text{-}\operatorname{Mod}\rightarrow R\text{-}\operatorname{Gr}$ of $\psi$ is defined on objects as follows. If $M$ is an object in $R\#\Gamma\text{-}\operatorname{Mod}$, then $\phi(M)=M$ as abelian groups. For each $\gamma\in\Gamma$ set $M_{\gamma}:=\sum\limits_{u\in E}up_{\gamma}M$. Then $M=\bigoplus\limits_{\gamma\in\Gamma}M_{\gamma}$. The left $R$-action on $\psi(M)$ is defined by $rm=(rp_{\alpha})m$ for any $\alpha,\gamma\in\Gamma$, $r\in R_{\gamma}$ and $m\in M_{\alpha}$. One can show that $rm\in M_{\gamma\alpha}$ for any $r\in R_{\gamma}$ and $m\in M_{\alpha}$ and hence $\psi(M)$ is a graded left $R$-module. Below we show that $\psi$ restricts to a functor $\psi_{\operatorname{proj}}:R\text{-}\operatorname{Gr}_{\operatorname{proj}}\rightarrow R\#\Gamma\text{-}\operatorname{Mod}_{\operatorname{proj}}$ and that $\phi$ restricts to a functor $\phi_{\operatorname{proj}}:R\#\Gamma\text{-}\operatorname{Mod}_{\operatorname{proj}}\rightarrow R\text{-}\operatorname{Gr}_{\operatorname{proj}}$.

Let $P\in R\text{-}\operatorname{Gr}_{\operatorname{proj}}$. Then $\psi(P)$ is a projective object in $R\#\Gamma\text{-}\operatorname{Mod}$ since $\psi$ is an isomorphism of categories. Since $P$ is finitely generated as a left $R$-module, there are $x_{1},\dots,x_{n}\in P$ such that $P=\sum\limits_{i=1}^{n}Rx_{i}$. Clearly we can assume that the $x_{i}$’s are homogeneous (replace each $x_{i}$ by its homogeneous components). Let $y=\sum\limits_{i=1}^{n}r_{i}x_{i}\in P$ where $r_{1},\dots,r_{n}\in R$. Then $y=\sum\limits_{i=1}^{n}(r_{i}p_{\deg(x_{i})})x_{i}$ in $\psi(P)$. This shows that $\psi(P)$ is finitely generated by the $x_{i}$’s as a left $R\#\Gamma$-module. Thus $\psi(P)\in R\#\Gamma$-$\operatorname{Mod}_{\operatorname{proj}}$.

Let now $Q\in R\#\Gamma\text{-}\operatorname{Mod}_{\operatorname{proj}}$. Then $\phi(Q)$ is a projective object in $R\text{-}\operatorname{Gr}$ since $\phi$ is an isomorphism of categories. Since $Q$ is finitely generated as a left $R\#\Gamma$-module, there are $x_{1},\dots,x_{n}\in Q$ such that $Q=\sum\limits_{i=1}^{n}(R\#\Gamma)x_{i}$. Clearly we can assume that the $x_{i}$’s are homogeneous with respect to the grading on $\phi(Q)$. Let $y=\sum\limits_{i=1}^{n}r_{i}x_{i}\in Q$ where $r_{1},\dots,r_{n}\in R\#\Gamma$. For $1\leq i\leq n$ write $r_{i}=\sum\limits_{\gamma\in\Gamma}r_{i,\gamma}p_{\gamma}$ where $r_{i,\gamma}\in R$ for any $\gamma\in\Gamma$. Then $y=\sum\limits_{i=1}^{n}r_{i,\deg(x_{i})}x_{i}$ in $\phi(Q)$ (note that $(r_{i,\gamma}p_{\gamma})x_{i}=0$ if $\gamma\neq\deg(x_{i})$ since $x_{i}\in Q_{\deg(x_{i})}$). This shows that $\phi(Q)$ is finitely generated by the $x_{i}$’s as a left $R$-module. Thus $\phi(Q)\in R\text{-}\operatorname{Gr}_{\operatorname{proj}}$.

By the previous two paragraphs, $\psi$ restricts to a functor $\psi_{\operatorname{proj}}:R\text{-}\operatorname{Gr}_{\operatorname{proj}}\rightarrow R\#\Gamma\text{-}\operatorname{Mod}_{\operatorname{proj}}$ and $\phi$ restricts to a functor $\phi_{\operatorname{proj}}:R\#\Gamma\text{-}\operatorname{Mod}_{\operatorname{proj}}\rightarrow R\text{-}\operatorname{Gr}_{\operatorname{proj}}$. Clearly $\psi_{\operatorname{proj}}\circ\phi_{\operatorname{proj}}=\operatorname{Id}_{R\#\Gamma\text{-}\operatorname{Mod}_{\operatorname{proj}}}$ and $\phi_{\operatorname{proj}}\circ\psi_{\operatorname{proj}}=\operatorname{Id}_{R\text{-}\operatorname{Gr}_{\operatorname{proj}}}$ since $\psi\circ\phi=\operatorname{Id}_{R\#\Gamma\text{-}\operatorname{Mod}}$ and $\phi\circ\psi=\operatorname{Id}_{R\text{-}\operatorname{Gr}}$. Thus $\psi_{\operatorname{proj}}$ is an isomorphism of categories. We leave it to the reader to deduce that the map

[TABLE]

is a monoid isomorphism. ∎

10.2. Admissible weight maps

Definition 65**.**

Let $H$ be a hypergraph and $E$ the directed graph associated to $H$ (see Remark 3 (b)). Let $w:E^{1}\rightarrow\Gamma$ be a map such that $w(h_{ij})=w(h_{i1})w(h_{11})^{-1}w(h_{1j})$ for any $h\in H^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. Then $w$ is called an admissible weight map for $H$.

Remark 66**.**

Let $H$ be a hypergraph. Set $X:=\{h_{1j},h_{i1}\mid h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}\subseteq E^{1}$. Clearly there is a $1$-$1$ correspondence between the set of all maps $X\rightarrow\Gamma$ and the set of all admissible weight maps $E^{1}\rightarrow\Gamma$ for $H$.

Lemma 67**.**

Let $H$ be a hypergraph and $w:E^{1}\rightarrow\Gamma$ an admissible weight map for $H$. Then

1. (i)

$w(h_{ik})w(h_{jk})^{-1}=w(h_{il})w(h_{jl})^{-1}$* for any $h\in H^{1}$, $1\leq i,j\leq|s(h)|$ and $1\leq k,l\leq|r(h)|$ and* 2. (ii)

$w(h_{ki})^{-1}w(h_{kj})=w(h_{li})^{-1}w(h_{lj})$* for any $h\in H^{1}$, $1\leq i,j\leq|r(h)|$ and $1\leq k,l\leq|s(h)|$.*

Proof.

Straightforward computation. ∎

Lemma 68**.**

Let $H$ be a hypergraph and $w:E^{1}\rightarrow\Gamma$ an admissible weight map for $H$. Then $w$ induces a $\Gamma$-grading on $L_{K}(H)$ such that $\deg(v)=\epsilon$, $\deg(h_{ij})=w(h_{ij})$ and $\deg(h_{ij}^{*})=w(h_{ij})^{-1}$ for any $v\in H^{0}$ and $h_{ij}\in E^{1}$.

Proof.

Set $X:=\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$ and let $K\langle X\rangle$ denote the free $K$-algebra generated by $X$. Clearly there is a $\Gamma$-grading on $K\langle X\rangle$ defined by $\deg(v):=\epsilon$, $\deg(h_{ij}):=w(h_{ij})$ and $\deg(h_{ij}^{*}):=w(h_{ij})^{-1}$ for any $v\in E^{0}$, $h\in E^{1}$, $1\leq i\leq|s(h)|$ and $1\leq j\leq|r(h)|$. It follows from Lemma 67 that the relations (i)-(iv) in Definition 2 are homogeneous. Hence the $\Gamma$-grading on $K\langle X\rangle$ induces a $\Gamma$-grading on $L_{K}(H)$. ∎

Example 69**.**

Let $H$ be a hypergraph. Set $n:=\sup\{|s(h)|\mid h\in H^{1}\}$. Define a map $w:E^{1}\rightarrow\mathbb{Z}^{n}$ by $w(h_{ij})=\alpha_{i}$ where $\alpha_{i}$ denotes the element of $\mathbb{Z}^{n}$ whose $i$-th component is $1$ and whose other components are [math]. The map $w$ induces the standard grading of $L_{K}(H)$.

Example 70**.**

Let $H$ be a hypergraph. Set $m:=\sup\{|s(h)|\mid h\in H^{1}\}$ and $n:=\sup\{|r(h)|\mid h\in H^{1}\}$. Define a map $w:E^{1}\rightarrow\mathbb{Z}^{m}\oplus\mathbb{Z}^{n}$ by $w(h_{ij})=(\alpha_{i},\alpha_{j})$ where $\alpha_{i}$ is defined as in the previous example. One checks easily that $w$ is an admissible weight map for $H$ and therefore it induces a $\mathbb{Z}^{m}\oplus\mathbb{Z}^{n}$-grading on $L_{K}(H)$.

10.3. Covering hypergraphs

Definition 71**.**

Let $H$ be a hypergraph and $w:E^{1}\rightarrow\Gamma$ an admissible weight map for $H$. Define a hypergraph $\overline{H}$ by $\overline{H}^{0}=\{v_{\gamma}\mid v\in H^{0},\gamma\in\Gamma\}$, $\overline{H}^{1}=\{h_{\gamma}\mid h\in H^{1},\gamma\in\Gamma\}$,

[TABLE]

and

[TABLE]

The hypergraph $\overline{H}$ is called the covering hypergraph of $H$ defined by $w$.

Proposition 72**.**

Let $H$ be a hypergraph and $w:E^{1}\rightarrow\Gamma$ an admissible weight map for $H$. Let $\overline{H}$ be the covering hypergraph of $H$ defined by $w$. Then there is an isomorphism $\phi:L_{K}(\overline{H})\rightarrow L_{K}(H)\#\Gamma$ such that

[TABLE]

for any $\gamma\in\Gamma$, $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$, $1\leq j\leq|r(h)|$.

Proof.

Set $a_{v_{\gamma}}:=vp_{\gamma}$, $b_{h_{\gamma},i,j}:=h_{ij}p_{w(h_{1j})^{-1}\gamma}$ and $c_{h_{\gamma},i,j}:=h_{ij}^{*}p_{w(h_{i1})w(h_{11})^{-1}\gamma}$ for any $v_{\gamma}\in\overline{H}^{0}$, $h_{\gamma}\in\overline{H}^{1}$, $1\leq i\leq|\overline{s}(h_{\gamma})|=|s(h)|$, $1\leq j\leq|\overline{r}(h_{\gamma})|=|r(h)|$. In order to show that

[TABLE]

is an $\overline{H}$-family in $L_{K}(H)\#\Gamma$ one has to show that the relations (i)-(iv) in Remark 3 (c) are satisfied. We leave (i) and (ii) to the reader and show only (iii) and (iv).

Let $h_{\gamma}\in\overline{H}^{1}$ and $1\leq i,j\leq|\overline{s}(h_{\gamma})|$. Clearly

[TABLE]

and hence (iii) holds. Let now $h_{\gamma}\in\overline{H}^{1}$ and $1\leq i,j\leq|\overline{r}(h_{\gamma})|$. Clearly

[TABLE]

and hence (iv) holds. Thus $X$ is an $\overline{H}$-family in $L_{K}(H)\#\Gamma$ and therefore there is a unique $K$-algebra homomorphism $\phi:L_{K}(\overline{H})\rightarrow L_{K}(H)\#\Gamma$ such that

[TABLE]

for any $\gamma\in\Gamma$, $v\in H^{0}$, $h\in H^{1}$, $1\leq i\leq|s(h)|$, $1\leq j\leq|r(h)|$.

Clearly the image of $\phi$ contains the set $S:=\{vp_{\gamma},h_{ij}p_{\gamma},h^{*}_{ij}p_{\gamma}\mid v\in H^{0},h\in H^{1},1\leq i\leq|s(h)|,1\leq j\leq|r(h)|\}$. But $S$ generates $L_{K}(H)\#\Gamma$ as a $K$-algebra and therefore $\phi$ is surjective. It remains to show that $\phi$ is injective. Let $A$ denote the set of all nod-paths $\theta$ for $L_{K}(\overline{H})$ and $B$ the set of all nod-paths $\eta$ for $L_{K}(H)$. It is an easy exercise to show that there is an injective map $f:A\rightarrow B\times\Gamma$, $\theta\mapsto(\eta_{\theta},\gamma_{\theta})$ that has the property that $\phi(\theta)=\eta_{\theta}p_{\gamma_{\theta}}$. Now let $x=\sum\limits_{\theta\in A}k_{\theta}\theta\in L_{K}(\overline{H})$ where almost all coefficients $k_{\theta}\in K$ are zero. Then $\phi(x)=\sum\limits_{\theta\in A}k_{\theta}\eta_{\theta}p_{\gamma_{\theta}}=\sum\limits_{\gamma\in\Gamma}(\sum\limits_{\begin{subarray}{c}\theta\in A,\\ \gamma_{\theta}=\gamma\end{subarray}}k_{\theta}\eta_{\theta})p_{\gamma}$. Assume now that $\phi(x)=0$. Then $\sum\limits_{\begin{subarray}{c}\theta\in A,\\ \gamma_{\theta}=\gamma\end{subarray}}k_{\theta}\eta_{\theta}=0$ in $L_{K}(H)$ for any $\gamma\in\Gamma$. But since $f$ is injective, we have $\eta_{\theta_{1}}\neq\eta_{\theta_{2}}$ for any $\theta_{1},\theta_{2}\in A$ such that $\theta_{1}\neq\theta_{2}$ and $\gamma_{\theta_{1}}=\gamma_{\theta_{2}}$. It follows from Theorem 17 that $k_{\theta}=0$ for any $\theta\in A$ and hence $x=0$. Thus $\phi$ is injective. ∎

10.4. The monoid $V^{\operatorname{gr}}(L_{K}(H))$

In this subsection $H$ denotes a hypergraph. We fix an admissible weight map $w:E\rightarrow\Gamma$ for $H$. Recall that $w$ induces a $\Gamma$-grading on $L_{K}(H)$.

Definition 73**.**

We define $M^{\operatorname{gr}}(H)$ as the abelian monoid presented by the generating set $\{v_{\gamma}\mid v\in H^{0},\gamma\in\Gamma\}$ and the relations

[TABLE]

Theorem 74**.**

$V^{\operatorname{gr}}(L_{K}(H))\cong M^{\operatorname{gr}}(H)$.

Proof.

Set $L:=L_{K}(H)$ and $\overline{L}:=L_{K}(\overline{H})$ where $\overline{H}$ is the covering hypergraph of $H$ defined by $w$. By Propositions 64 and 72 we have $V^{\operatorname{gr}}(L)\cong V(L\#\Gamma)\cong V(\overline{L})$. By Theorem 61 we have $V(\overline{L})\cong M^{\operatorname{gr}}(H)$. Thus $V^{\operatorname{gr}}(L_{K}(H))\cong M^{\operatorname{gr}}(H)$. ∎

Corollary 75**.**

Let $(E,C)$ be a finitely separated graph. Then the abelian monoid $V^{\operatorname{gr}}(L_{K}(E,C))$ defined with respect to the standard $\mathbb{Z}$-grading of $L_{K}(E,C)$ is presented by the generating set $\{v_{\gamma}\mid v\in E^{0},n\in\mathbb{Z}\}$ and the relations

[TABLE]

where $s(X)$ denotes the common source of the edges in $X$.

Corollary 76**.**

Let $(E,w)$ be a row-finite vertex-weighted graph. Then the abelian monoid $V^{\operatorname{gr}}(L_{K}(E,w))$ defined with respect to the standard $\mathbb{Z}^{n}$-grading of $L_{K}(E,w)$ (where $n$ is the supremum of the set of all weights) is presented by the generating set $\{v_{\gamma}\mid v\in E^{0},\gamma\in\mathbb{Z}^{n}\}$ and the relations

[TABLE]

where $\alpha_{i}$ denotes the element of $\mathbb{Z}^{n}$ whose $i$-th component is $1$ and whose other components are zero and $\operatorname{Reg}(E)$ denotes the set of all vertices in $E$ which emit at least one edge.