# Connections between Abelian sandpile models and
the $K$-theory of weighted Leavitt path algebras

Gene Abrams  Department of Mathematics
University of Colorado
Colorado Springs, CO 80918, USA<EMAIL_ADDRESS>and  Roozbeh Hazrat
Centre for Research in Mathematics and Data Science
Western Sydney University
Australia<EMAIL_ADDRESS>
###### Abstract.

In our main result, we establish that any conical sandpile monoid
$M=\operatorname{SP}(G)$ of a directed sandpile graph $G$ can be realised as
the $\mathcal{V}$-monoid of a weighted Leavitt path algebra
$L_{\mathsf{k}}(E,w)$, and consequently, the sandpile group as the
Grothendieck group $K_{0}(L_{\mathsf{k}}(E,w))$. We show how to explicitly
construct $(E,w)$ from $G$. Additionally, we describe the conical sandpile
monoids which arise as the $\mathcal{V}$-monoid of a standard (i.e.,
unweighted) Leavitt path algebra.

###### Key words and phrases:

Abelian sandpile model, sandpile monoid, weighted Leavitt path algebra,
Grothendieck group

###### 2010 Mathematics Subject Classification:

16S88, 05C57

##  1\. Introduction

“ _Your descendants will be like the dust of the earth, and you will spread
out to the west and to the east, to the north and to the south._ ” Genesis
28:14.

The notion of sandpile models encapsulates how objects spread and evolve along
a grid. The models were conceived in 1987 in the seminal paper [7] by Bak,
Tang and Wiesenfeld as an example of _self-organized criticality_ , or the
tendency of physical systems to organise themselves without any input from
outside the system, toward critical but barely stable states. The models were
used to describe phenomena such as forest fires, traffic jams, stock market
fluctuations, etc. The book of Bak [8] describes how events in nature
apparently follow this type of behaviour. In a major work [13], D. Dhar
championed the use of an abelian group naturally associated to a sandpile
model as an invariant which proved to capture many properties of the model.
The mathematical formulation of the model is as follows. Consider a sandpile
graph, namely a (finite, directed) graph $E$ with a distinguished sink vertex
$s$ such that there is a (directed) path from any vertex of $E$ to $s$.
Consider a collection of grains of sand placed on each vertex of the sandpile
graph (a configuration). A vertex is unstable if it has the same or more
grains of sand than the number of edges emitting from it. In this case the
vertex topples by sending one grain along each edge emitting from the vertex
to each neighbouring vertex. This toppling may cause neighbouring vertices to
become unstable. The assumption is that grains arriving to the vertex $s$
vanish. A configuration is stable if all vertices are stable. Two foundational
results used in the theory are: the order of toppling does not matter (thus
“abelian” sandpile models); and any configuration can be sequentially toppled
to reach a unique stable configuration. This allows for the construction of a
finite monoid associated to an abelian sandpile model. Let
$\operatorname{SP}(E)$ be the set of all stable configurations of a sandpile
graph $E$. The “sum” of two configurations is interpreted as simply adding the
number of grains of sand at each vertex corresponding to the two
configurations. The operation of addition followed by stabilisation gives the
set $\operatorname{SP}(E)$ the structure of a commutative monoid, called the
_sandpile monoid_ (see Definition 2.10 for a precise definition). The smallest
set of configurations of $\operatorname{SP}(E)$ which is closed under adding
any grains to them (i.e., the recurrent configurations) forms a group, called
the _sandpile group_ $G(E)$ associated to $E$. These algebraic structures
constitute one of the main themes of the subject. A more algebraic study of
these monoids and their groups is carried out in [6, 11]. The books of Klivans
[17] and Corry and Perkinson [12] give self-contained treatments of the
subject of sandpile models.

In a different realm, the notion of Leavitt path algebras $L_{\mathsf{k}}(E)$
associated to directed graphs $E$, with coefficients in a field $\mathsf{k}$,
were introduced in 2005 [1, 5]. These are a generalisation of algebras
(denoted by $L_{\mathsf{k}}(1,1+k)$) introduced by William Leavitt in 1962
[16] as a “universal” ring $A$ of type $(1,1+k)$, so that $A\cong A^{1+k}$ as
right $A$-modules, where $k\in\mathbb{N}^{+}$. The commutative monoid
$\mathcal{V}(B)$ of isomorphism classes of finitely generated projective right
modules of a unital ring $B$ (with operation $\oplus$) goes back to the work
of Grothendieck and Serre [18]. For a Leavitt path algebra
$L_{\mathsf{k}}(E)$, the monoid $\mathcal{V}(L_{\mathsf{k}}(E))$ has received
substantial attention due to two separate impetuses. On the one hand, these
monoids are conical and refinement, and thus related to the realisation
problem for von Neumann regular rings [14]. (The problem asks whether every
countable conical refinement monoid can be realised as the monoid of a von
Neumann regular ring [2, §7.2.3].) On the other hand, the group completion of
$\mathcal{V}(L_{\mathsf{k}}(E))$ is the Grothendieck group
$K_{0}(L_{\mathsf{k}}(E))$. This group has been used effectively as a complete
invariant for a large class of graph $C^{*}$-algebras (these are the analytic
versions of Leavitt path algebras). Thus there is a hope that this group would
play a similar role in the algebraic setting as well. Whether $K_{0}$
classifies these algebras has remained one of the most important and yet
elusive questions in the theory of Leavitt path algebras [2, §7.3.1]. We will
touch on this question in Section 6.

In fact Leavitt established much more in [16] than mentioned in the previous
paragraph; indeed, he showed that for any $n,k\in\mathbb{N}$ that there is a
universal ring $A$ of type $(n,n+k)$ (denoted $L_{\mathsf{k}}(n,n+k)$) for
which $A^{n}\cong A^{n+k}$ as right $A$-modules. When $n\geq 2$, this
universal ring is not realizable as a Leavitt path algebra. With this in mind,
the notion of weighted Leavitt path algebras $L_{\mathsf{k}}(E,w)$ associated
to weighted graphs $(E,w)$ were introduced by the second author in 2011 ([15];
see [21] for a nice overview of this topic). The weighted Leavitt path
algebras $L_{\mathsf{k}}(E,w)$ provide a natural context in which all of
Leavitt’s algebras (corresponding to any pair $n,k\in\mathbb{N}$) can be
realised as a specific example. As well, the monoid
$\mathcal{V}(L_{\mathsf{k}}(E,w))$ and the corresponding Grothendieck group
$K_{0}(L_{\mathsf{k}}(E,w))$ have been completely described in the two works
[15] and [20].

In this note we tie the notions of sandpile models and weighted Leavitt path
algebras together. We start in Section 2 by defining weighted graph monoids,
and show that these include the sandpile monoids. We then show in Section 3
that any weighted graph monoid satisfies the Confluence Property. In
particular, this property will allow us to identify the unit group of the
weighted graph monoid. With these preliminaries established, we move towards
our main result (Theorem 6.1) by casting the theory of sandpile monoids in an
abstract algebraic setting. To do so we utilise two foundational works of
George Bergman: the Diamond Lemma [10], and the Universal Ring Construction
[9]. The first work will allow us to re-establish that sandpile monoids have
unique normal forms (Section 4), while the second will allow us to realise
monoids of certain algebras as graph monoids (Section 5). In Section 6 we
achieve our main result, in which we show that, starting from a sandpile graph
$E$, considering $E$ as a weighted graph $(E,w)$ for an appropriate weight
$w$, we obtain that
$\operatorname{SP}(E)\cong\mathcal{V}(L_{\mathsf{k}}(E,w))$, and consequently
that $G(E)\cong K_{0}(L_{\mathsf{k}}(E,w))$ as well. Furthermore, we identify
those unweighted Leavitt path algebras which arise as sandpile algebras.

The connection established in Theorem 6.1 between sandpile monoids and
weighted Leavitt path algebras allows us to naturally associate an algebra, a
_sandpile algebra_ , to the theory of sandpile models, thereby opening up an
avenue by which to investigate sandpile models via the structure of the
sandpile algebras, and vice versa. We take up such an investigation in [4].

##  2\. Weighted graph monoids and sandpile monoids: basic properties

Throughout we write $\mathbb{N}$ for the set of non-negative integers, and
$\mathbb{N}^{+}$ for the set of positive integers.

We start by setting some terminology and notation about graphs. A _directed
graph_ is a quadruple $E=(E^{0},E^{1},s,r)$, where $E^{0}$ and $E^{1}$ are
sets and $s,r:E^{1}\rightarrow E^{0}$ are maps. Throughout, “graph” will
always mean “directed graph”. The elements of $E^{0}$ are called _vertices_
and the elements of $E^{1}$ _edges_. (We allow the empty set to be viewed as a
graph with $E^{0}=E^{1}=\emptyset.$) If $e$ is an edge, then $s(e)$ is called
its _source_ and $r(e)$ its _range_. If $v$ is a vertex and $e$ an edge, we
say that $v$ emits $e$ if $s(e)=v$, and $v$ receives $e$ if $r(e)=v$. An edge
$e$ is called a loop at $v$ in case $s(e)=v=r(e)$. A vertex is called a sink
if it emits no edges, and is called _irrelevant_ in case it emits exactly one
edge. A graph is called reduced if it contains no irrelevant vertices.

A vertex is called _regular_ if it is not a sink and does not emit infinitely
many edges. The subset of $E^{0}$ consisting of all the regular vertices is
denoted by $E^{0}_{\operatorname{reg}}$. The _out-degree_ of a vertex $v$ is
defined as $|s^{-1}(v)|$. A graph is called _row-finite_ if any vertex emits a
finite number (possibly zero) of edges. The graph $E$ is called _finite_ if
$E^{0}$ and $E^{1}$ are finite sets. In this paper we assume throughout that
all graphs are row-finite, and thus $E^{0}_{\operatorname{reg}}$ consists of
all vertices which are not sinks.

A path $p$ in $E$ is a sequence $p=e_{1}e_{2}\cdots e_{n}$ of edges in $E$
such that $r(e_{i})=s(e_{i+1})$ for $1\leq i\leq n-1$. We define
$s(p)=s(e_{1})$, and $r(p)=r(e_{n})$. By definition, the _length_ $|p|$ of $p$
is $n$. We assign the length zero to vertices. A _closed path_ (based at $v$)
is a path $p$ such that $s(p)=r(p)=v$. A _cycle_ (based at $v$) is a closed
path $p=e_{1}\cdots e_{n}$ based at $v$ such that $s(e_{i})\neq s(e_{j})$ for
any $i\neq j$.

A collection of graphs which will be the primary focus of this article is now
easy to define.

######  Definition 2.1.

A finite directed graph $E$ is called a _sandpile graph_ if $E$ has a unique
sink (denote it by $s$), and for every $v\in E^{0}$ there is a path $p$ with
$s(p)=v$ and $r(p)=s$.

A subset $H\subseteq E^{0}$ is said to be _hereditary_ if for any $e\in
E^{1}$, $s(e)\in H$ implies $r(e)\in H$. A subset $H\subseteq E^{0}$ is called
_saturated_ if whenever $r(s^{-1}(v))\subseteq H$ for a regular vertex $v$,
then $v\in H$. If $H$ is a hereditary and saturated subset, we may consider
the directed graph, also denoted by $H$, consisting of all vertices of $H$ and
all edges emitting from these vertices.

For a hereditary and saturated subset $H$ of $E$, we define the quotient graph
$E/H$ as follows:

$(E/H)^{0}=E^{0}\setminus H^{0}\ \ \mbox{and}\ \ (E/H)^{1}=\\{e\in E^{1}\;|\
s(e)\notin H\\}.$

The source and range maps of $E/H$ are the source and range maps restricted
from the graph $E$. We have a natural (inclusion) morphism
$\phi:E/H\rightarrow E$ which is injective on vertices and edges.

######  Definition 2.2.

A _weighted graph_ is a pair $(E,w)$, where $E$ is a graph and
$w:E^{1}\rightarrow\mathbb{N}^{+}$ is a map. If $e\in E^{1}$, then $w(e)$ is
called the _weight_ of $e$.

We set $w(v):=\max\\{w(e)\mid e\in s^{-1}(v)\\}$, for any regular vertex $v$.
This gives a map (called $w$ again)
$w:E_{\operatorname{reg}}^{0}\rightarrow\mathbb{N}^{+}$.

A weighted graph is called a _vertex weighted graph_ if for each regular
vertex $v$ in the graph, the weights of all edges emitting from $v$ are equal
one to the other; i.e., if $w(e)=w(e^{\prime})$ for all $e,e^{\prime}\in
s^{-1}(v)$. In this case $w(v)$ coincides with the weight of any edge emitting
from $v$.

A vertex weighted graph is called a _balanced weighted graph_ if
$w(v)=|s^{-1}(v)|$ for all $v\in E_{\operatorname{reg}}^{0}$.

######  Remark 2.3.

Note that any (row-finite) directed graph can be given the structure of a
balanced weighted graph by assigning $w(v)=|s^{-1}(v)|$, for all $v\in
E_{\operatorname{reg}}^{0}$. (By definition we are not required to assign a
weight to any sinks in $E$.)

In particular, suppose that $H$ is a hereditary and saturated subset of the
weighted graph $(E,w)$. We form the quotient graph $E/H$. Then there are two
natural ways to define a weight function on $E/H$: restrict the weight
function on $E$ to $E/H$, or impose the structure of a balanced weighted graph
on $E/H$. We note that if $w$ is a balanced weight function on $E$, then the
restriction function of $w$ to $E/H$ is not in general the same as the
balanced weight function on $E/H$. In situations where additional clarity is
warranted, we will write $(E/H,w_{r})$ to indicate that the weight function
being considered on $E/H$ is the restriction function of the weight function
$w$ to $E/H$.

We continue by setting some terminology and notation about monoids. Let $M$ be
a commutative monoid. We define the _algebraic_ pre-ordering on $M$ by setting
$a\leq b$ if $b=a+c$, for some $c\in M$. A commutative monoid $M$ is called
_conical_ if $a+b=0$ implies $a=b=0$, and is called _cancellative_ if
$a+b=a+c$ implies $b=c$, where $a,b,c\in M$. The monoid $M$ is called
_refinement_ if, whenever $a,b,c,d\in M$ have $a+b=c+d$, then there are
$e_{1},e_{2},e_{3},e_{4}\in M$ such that $a=e_{1}+e_{2}$, $b=e_{3}+e_{4}$ and
$c=e_{1}+e_{3}$, $d=e_{2}+e_{4}$. An element $0\not=a\in M$ is called an
_atom_ if whenever $a=b+c$ then $b=0$ or $c=0$. We say a commutative monoid is
_atom-cancellative_ if for any atom $a\in M$, the equality $a+m=a+m^{\prime}$
implies $m=m^{\prime}$.

A subset $I$ of $M$ is called an _ideal_ of $M$ if $a+I\subseteq I$, for any
$a\in M$. The ideal $I$ is called an _order-ideal_ if $a+b\in I$ implies
$a,b\in I$. Let $I$ be a submonoid of $M$. For $a,b\in M$, write $a\sim_{I}b$
if there exist $i,j\in I$ such that $a+i=b+j$ in $M$. This is a congruence
relation and thus one can form the quotient monoid $M/\sim$ which we will
denote by $M/I$. The congruence $a\sim_{I}b$ is equivalent to
$(a+I)\cap(b+I)\not=\emptyset$. Observe that $a\sim_{I}0$ in $M$ for any $a\in
I$.

For a commutative monoid $M$, the set

$Z(M)=\\{a\in M\mid a+b=0,\text{ for some }b\in M\\}$

is an abelian group, called the _unit group_ of $M$. Clearly $Z(M)=0$ if and
only if $M$ is conical and $Z(M)=M$ if and only if $M$ is a group. We refer
the reader to [22] for the general theory of commutative monoids and [24] for
refinement commutative monoids.

The following simple lemma will be quite useful later, especially in the proof
of Proposition 6.7.

######  Lemma 2.4.

Any refinement monoid is atom-cancellative. Furthermore, any monoid which is
finite, conical and refinement has no atoms.

###### Proof.

Let $a,m,m^{\prime}\in M$, where $M$ is a refinement monoid and $a$ is an atom
such that $a+m=a+m^{\prime}$. Then there are $e_{1},e_{2},e_{3},e_{4}\in M$
such that $a=e_{1}+e_{2}$, $m=e_{3}+e_{4}$ and $a=e_{1}+e_{3}$,
$m^{\prime}=e_{2}+e_{4}$. Since $a$ is an atom, either $e_{1}=0$ or $e_{2}=0$.
If $e_{1}=0$ then $e_{2}=e_{3}=a$ and thus $m=a+e_{4}=m^{\prime}$. If
$e_{2}=0$ then $e_{1}=a$ and $e_{3}=0$ which gives that $m=e_{4}=m^{\prime}$.

Now let $M$ be finite, conical and refinement. Suppose $a\in M$ is an atom.
Consider the translation map $\phi_{a}:M\backslash\\{0\\}\rightarrow
M\backslash\\{0\\},m\mapsto a+m$. Since $a$ is an atom, the first part of the
lemma implies $\phi_{a}$ is injective. Thus $\phi_{a}$ is surjective as
$M\backslash\\{0\\}$ is finite. But $a$ can’t be in the image of $\phi_{a}$, a
contradiction. ∎

Let $M$ be a commutative monoid. The _group completion_ of $M$, denoted
$M^{+}$, is the free abelian group generated by elements of the set
$\\{[a]\mid a\in M\\}$ subject to the relations $[a+b]-[a]-[b]$, for any
$a,b\in M$. The homomorphism $M\rightarrow M^{+};m\mapsto[m]$ is initial among
all such maps [18]. The Grothendieck group of a unital ring $A$, denoted
$K_{0}(A)$, is by definition the group completion of the monoid
$\mathcal{V}(A)$. When the monoid $M$ is finite, the group completion $M^{+}$
is in fact a subgroup of $M$.

######  Lemma 2.5.

Let $M$ be a finite commutative monoid. Then the group completion of $M$ is
isomorphic to the smallest ideal of $M$.

###### Proof.

Let $I$ be the smallest ideal of $M$. Note that $I=\bigcap_{a\in M}(a+M)$ and,
since $M$ is finite, $\sum_{a\in M}a\in I$ and so $I$ is a nonempty set.
Furthermore $I$ coincides with the set $H(M):=\\{a\in M\mid a\geq b,\text{ for
all }b\in M\\}$. Since every element $x\in H(M)$ has the property that for
every $x^{\prime}\in H(M)$ there exists $y\in H(M)$ with $x+y=x^{\prime}$, it
is standard to show that $H(M)$ is a group, whose identity element we denote
by $z$. For any abelian group $G$, any homomorphism $\psi:M\rightarrow G$
uniquely factors through $\phi:M\rightarrow H(M);m\mapsto z+m$ via the
homomorphism $\psi_{H(M)}.$ This shows that $I\cong M^{+}$. ∎

######  Example 2.6.

We clarify some of the previous observations in the context of a specific
example, one to which we will make reference often in the sequel. For
$n,k\in\mathbb{N}^{+}$, let $M_{n,n+k}$ denote the finite commutative monoid

$M_{n,n+k}:=\\{0,x,2x,\dots,nx,\dots,(n+k-1)x\\},\ \ \ \ \mbox{with relation}\
(n+k)x=nx.$

Then $M_{n,n+k}$ is clearly conical, and not cancellative. The atoms of
$M_{n,n+k}$ consist of the elements $x,2x,\dots,(n-1)x$. (In particular
$M_{1,1+k}$ has no atoms.) The monoid $M_{n,n+k}$ is refinement if and only if
$n=1$. (To show that $M_{n,n+k}$ is not refinement for $n\geq 2$, consider the
equation $x+(n-1)x=x+(n+k-1)x$ in $M_{n,n+k}$.)

Finally, the group completion $M^{+}$ of $M_{n,n+k}$ consists of the subset
$\\{nx,\dots,(n+k-1)x\\}$. So we see that $M^{+}$ is isomorphic to the cyclic
group of order $k$: it is easy to show that the identity element of this group
is $tx$, where $t$ is the unique multiple of $k$ appearing in the integer
interval $[n,n+k-1]$, and that $(t+1)x$ is a generator for $M^{+}$.

We complete this section by tying together the two previously presented
themes, by associating certain monoids to weighted graphs. Once the specific
description and basic properties of these graph monoids have been established,
in the subsequent two sections we’ll look at the Confluence and the Reduction-
Uniqueness properties, respectively.

######  Definition 2.7.

Let $(E,w)$ be a weighted graph. We assign $w(v)=1$ if $v$ is a sink. The
_graph monoid_ $M(E,w)$ associated to $(E,w)$ is defined to be

$M(E,w)=\mathbb{F}_{E}\Big{/}\Big{\langle}w(v)v=\sum_{e\in
s^{-1}(v)}r(e)\,\big{|}\,v\in E^{0}\Big{\rangle}.$ (1)

Here $\mathbb{F}_{E}$ is the free commutative monoid on the set $E^{0}$ of
vertices of $E$. (If $E^{0}$ is the empty set then we interpret $F_{E}$ as the
zero monoid.)

Note that if $s$ is a sink, then since the summation in (1) is over an empty
set, the relation corresponding to $s$ reduces to: $s=0$.

######  Example 2.8.

Let $n,k\in\mathbb{N}^{+}$. Let $E_{n,n+k}$ denote the vertex weighted graph
having one vertex $v$, with $n$ loops at $v$, each having weight $n+k$.
Pictorially, $E_{n,n+k}$ can be viewed as

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{1,1},\dots,y_{n+k,1}}$$\scriptstyle{y_{1,2},\dots,y_{n+k,2}}$$\scriptstyle{y_{1,3},\dots,y_{n+k,3}}$$\scriptstyle{y_{1,n},\dots,y_{n+k,n}}$

So $w(v)=n+k$, and

$M(E_{n,n+k}):=\mathbb{F}_{v}\ \Big{/}\ \langle(n+k)v=nv\rangle\ \cong\
M_{n,n+k},$

where $M_{n,n+k}$ is the monoid defined in Example 2.6.

######  Remark 2.9.

We note that even if $E$ is finite, certainly $M(E,w)$ might be infinite. As
an easy example, if $E$ is the graph with one vertex $v$ and one loop $e$
based at $v$ having $w(e)=1$, then $w(v)=1$ and $M(E,w)=\mathbb{F}_{v}/\langle
1v=v\rangle\cong\mathbb{N}$.

There is an explicit description of the congruence on $\mathbb{F}_{E}$ given
by the defining relations of $M(E,w)$ in (1), as follows. For $v\in E^{0}$,
with weight $w(v)$, define the “$\operatorname{\mathbf{r}}$-transform on
$\mathbb{F}_{E}$” by setting

$\operatorname{\mathbf{r}}(w(v)v)\ :=\ \sum_{e\in s^{-1}(v)}r(e).$ (2)

The nonzero elements of $\mathbb{F}_{E}$ can be written uniquely up to
permutation as $\sum_{i=1}^{n}k_{i}v_{i}$, where $v_{i}$ are distinct vertices
and $k_{i}\in\mathbb{N}^{+}$. Define a binary relation $\rightarrow_{1}$ on
$\mathbb{F}_{E}$ by

$\sum_{i=1}^{n}k_{i}v_{i}\ \longrightarrow_{1}\ \Big{(}\sum_{i\neq
j}k_{i}v_{i}\Big{)}+(k_{j}-w(v_{j}))v_{j}+\operatorname{\mathbf{r}}(w(v_{j})v_{j}),$
(3)

whenever $j\in\\{1,\cdots,n\\}$ and $k_{j}\geq w(v_{j})$. (Clearly the
transformation described by $\rightarrow_{1}$ models the toppling process at
$v_{i}$ in the sandpile model.) Let $\rightarrow$ be the transitive and
reflexive closure of $\rightarrow_{1}$ on $\mathbb{F}_{E}$. Namely

$a\rightarrow b\ \ \ \text{ if }a=b,\ \mbox{or}\
a=a_{0}\rightarrow_{1}a_{1}\rightarrow_{1}\dots\rightarrow_{1}a_{k}=b.$ (4)

Finally, let $\sim$ be the congruence on $\mathbb{F}_{E}$ generated by the
relation $\rightarrow$. That is, $a\sim b$ in case there is a string
$a=a_{0},a_{1},\dots,a_{n}=b$ in $\mathbb{F}_{E}$ such that
$a_{i}\rightarrow_{1}a_{i+1}$ or $a_{i+1}\rightarrow_{1}a_{i}$ for each $0\leq
i\leq n-1$. Then

$M(E,w)=\mathbb{F}_{E}/\sim.$

To avoid cumbersome equivalence class notation, is standard (but not
technically correct) to denote the elements of $M(E,w)$ and the elements of
$\mathbb{F}_{E}$ using the same symbols. For instance, we will sometimes write
$a=b$ in $M(E,w)$ for elements $a,b\in\mathbb{F}_{E}$.

######  Definition 2.10.

For a sandpile graph $E$, the sandpile monoid $\operatorname{SP}(E)$ has been
described informally in the Introduction; we revisit that description, and
provide here a formal definition. (See e.g. [6] or [11] for additional
information). For $k_{j}\in\mathbb{N}$ we envision $k_{j}$ grains of sand
sitting on each non-sink vertex $v_{j}$ of $E$. Each vertex $v_{j}$ fires one
grain of sand along each edge emitted by $v_{j}$ to its adjacent vertices
exactly when the number of grains of sand sitting at $v_{j}$ is equal to or
larger than the number of edges emitting from $v_{j}$. Further, any grains
which arrive at the sink $s$ are understood to vanish.

More formally, $\operatorname{SP}(E)$ is the monoid defined by generators and
relations as

$\operatorname{SP}(E)=\mathbb{F}_{E}\Big{/}\Big{\langle}s=0;\
|s^{-1}(v)|v=\sum_{e\in s^{-1}(v)}r(e),\ \mbox{for}\ v\in E^{0}\
\setminus\\{s\\}\Big{\rangle}.$ (5)

######  Example 2.11.

Let $G$ be the graph pictured here.

$\textstyle{\\!\\!\\!\bullet^{s}}$$\textstyle{\bullet^{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{k}}$$\scriptstyle{e_{1}}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{3}}$$\scriptstyle{e_{n}}$

So $G$ has one sink $s$, and one non-sink vertex $x$, with $n$ loops based at
$x$, and $k$ edges from $x$ to $s$. Thus $G$ is a sandpile graph. Moreover,

$\operatorname{SP}(G)=\mathbb{F}_{x,s}\Big{/}\langle s=0;\
(n+k)x=nx+ks\rangle\ \cong M_{n,n+k},$

where $M_{n,n+k}$ is the monoid defined in Example 2.6.

######  Remark 2.12.

(A key connection)  Let $E$ be a sandpile graph. Considering $E$ as a balanced
weighted graph (so $w(v)=|s^{-1}(v)|$ for each non-sink vertex $v$ of $E$),
and defining $w(s)=1$ for the sink $s$, Definitions 1 and 2.10 immediately
give

$\operatorname{SP}(E)=M(E,w).$

So every sandpile monoid is the monoid of a (balanced) weighted graph. The
converse is certainly not true: as one justification, we will show below that
the monoid $\operatorname{SP}(E)$ is finite for any finite graph $E$, while
$M(E,w)$ need not be finite in general (see Remark 2.9).

Of course it is not coincidental that the monoid $M(E,w)$ of Example 2.8 is
the same as the monoid $\operatorname{SP}(G)$ of Example 2.11. We will make
the connection precise in Theorem 6.1.

######  Remark 2.13.

If $E$ is a sandpile graph, let $G$ be the graph gotten from $E$ by
eliminating all irrelevant vertices together with the single edge each emits.
Then clearly $G$ is a sandpile graph, and
$\operatorname{SP}(E)\cong\operatorname{SP}(G)$. In particular, when studying
the structure of $\operatorname{SP}(E)$ we may assume without loss that $E$ is
reduced.

##  3\. The Confluence Property

In this section we show that the graph monoids $M(E,w)$ have a certain
“confluence” property. Informally, this means that whenever two elements $a,b$
are equal when viewed in the quotient monoid $M(E,w)=\mathbb{F}_{E}/\sim$,
then there is a common element in $\mathbb{F}_{E}$ to which both $a$ and $b$
flow. The proof of the Confluence Lemma we present here is similar to the
proof of the corresponding result in the case of the graph monoid $M_{E}$ for
an unweighted graph $E$ [5, §6.3], but with the added complexity of having
weights and sinks in the relations. (We note that the statement of [5, §6.3]
does not generalise verbatim to the weighted case.)

######  Lemma 3.1 (The Confluence Lemma).

Let $E$ be a weighted graph and $M(E,w)$ its associated monoid. For
$a,b\in\mathbb{F}_{E}$, we have $a=b$ in $M(E,w)$ (i.e., $a\sim b$ in
$\mathbb{F}_{E}$) if and only if there exists $c\in\mathbb{F}_{E}$ such that
$a\rightarrow c$ and $b\rightarrow c$.

In particular, any sandpile monoid satisfies this “Confluence Property”.

###### Proof.

The sufficiency follows directly from the definition of $\sim$, as it is the
symmetric and transitive closure of the relation $\rightarrow_{1}$.

To establish the necessity, we start by observing the following fact. Let $a$
be an element of $\mathbb{F}_{E}\backslash\\{0\\}$ such that $kv$ appears in
its presentation, where $v$ is a vertex having weight $k$. Writing
$a=kv+a^{\prime}$ then by definition (3) a transformation $\rightarrow_{1}$
can take either the form
$a\rightarrow_{1}\operatorname{\mathbf{r}}(kv)+a^{\prime}$ or
$a\rightarrow_{1}kv+a^{\prime\prime}$, where
$a^{\prime}\rightarrow_{1}a^{\prime\prime}$.

Now suppose $a=b$ in $M_{(E,w)}$. Then there is a string
$a=a_{0},a_{1},\dots,a_{n}=b$ in $\mathbb{F}_{E}$ such that
$a_{i}\rightarrow_{1}a_{i+1}$ or $a_{i+1}\rightarrow_{1}a_{i}$, where $0\leq
i\leq n-1$. We argue by induction on $n$. If $n=0$, then $a=b$ and there is
nothing to prove. Suppose the statement holds for strings of length $n-1$ and
let $a=a_{0},a_{1},\dots,a_{n}=b$ be a string of length $n$. By induction,
there is a $c$ such that $a_{0}\rightarrow c$ and $a_{n-1}\rightarrow c$. We
consider two cases:

If $b=a_{n}\rightarrow_{1}a_{n-1}$, then clearly $a_{0}\rightarrow c$ and
$b\rightarrow c$ and we are done.

If $a_{n-1}\rightarrow_{1}b$, then by the definition (3), there is a vertex
$v$ with weight $k$ such that $kv$ appears in the presentation of $a_{n-1}$,
i.e., $a_{n-1}=kv+a^{\prime}_{n-1}$ and
$b=\operatorname{\mathbf{r}}(kv)+a^{\prime}_{n-1}$. Since $a_{n-1}\rightarrow
c$ there is a string of transformations

$a_{n-1}=kv+a_{n-1}^{\prime}=c_{0}\longrightarrow_{1}c_{1}\longrightarrow_{1}\dots\longrightarrow_{1}c_{l}=c.$
(6)

We consider two cases.

Case 1. Going along the displayed string (6) if there is no transformation of
this $kv$ in each of the $c_{i}$, $0\leq i\leq l$, then $kv$ appears in the
presentation of each of the $c_{i}$ and therefore we have a chain

$a_{n-1}^{\prime}=c_{0}^{\prime}\longrightarrow_{1}c_{1}^{\prime}\longrightarrow_{1}\dots\longrightarrow_{1}c_{l}^{\prime},$
(7)

where $c_{i}=kv+c_{i}^{\prime}$, $0\leq i\leq l$.

Now we apply one more transformation on
$c_{l}=kv+c_{l}^{\prime}\rightarrow\operatorname{\mathbf{r}}(kv)+c_{l}^{\prime}$.
Since $b=\operatorname{\mathbf{r}}(kv)+a_{n-1}^{\prime}$, applying the same
transformations (7) we have

$b=\operatorname{\mathbf{r}}(kv)+a_{n-1}^{\prime}\longrightarrow_{1}\operatorname{\mathbf{r}}(kv)+c_{1}^{\prime}\longrightarrow_{1}\cdots\longrightarrow_{1}\operatorname{\mathbf{r}}(kv)+c_{l-1}^{\prime}\longrightarrow_{1}\operatorname{\mathbf{r}}(kv)+c_{l}^{\prime}$

Therefore $a\rightarrow
c=c_{l}=kv+c_{l}^{\prime}\rightarrow\operatorname{\mathbf{r}}(kv)+c_{l}^{\prime}$
and $b\rightarrow\operatorname{\mathbf{r}}(kv)+c_{l}^{\prime}$ and we are
done.

Case 2. Suppose $0\leq s\leq l-1$ is the first instance in the string (6) that
$\rightarrow_{1}$ transforms this $kv$ to $\operatorname{\mathbf{r}}(kv)$.
Thus we have a chain

$a_{n-1}^{\prime}=c_{0}^{\prime}\longrightarrow_{1}c_{1}^{\prime}\longrightarrow_{1}\dots\longrightarrow_{1}c_{s}^{\prime},$
(8)

where $c_{i}=kv+c_{i}^{\prime}$ and

$a_{n-1}=kv+c_{0}^{\prime}\longrightarrow_{1}kv+c_{1}^{\prime}\longrightarrow_{1}\dots\longrightarrow_{1}kv+c_{s}^{\prime}\longrightarrow_{1}\operatorname{\mathbf{r}}(kv)+c_{s}^{\prime}=c_{s+1}\longrightarrow_{1}\dots\longrightarrow_{1}c_{l}=c$

Since $b=\operatorname{\mathbf{r}}(kv)+a_{n-1}^{\prime}$, applying the same
transformations (8) we have

$b=\operatorname{\mathbf{r}}(kv)+a_{n-1}^{\prime}=\operatorname{\mathbf{r}}(kv)+c_{0}^{\prime}\longrightarrow_{1}\cdots\longrightarrow_{1}\operatorname{\mathbf{r}}(kv)+c_{s}^{\prime}=c_{s+1}\longrightarrow_{1}\dots\longrightarrow_{1}c_{l}=c$

Therefore $a\rightarrow c$ and $b\rightarrow c$ and we are done. ∎

Let $(E,w)$ be a weighted graph and $H$ a hereditary and saturated subset of
$E$. Consider $(H,w)$ as a weighted subgraph of $(E,w)$ consisting of all
vertices of $H$ and all edges emitting from these vertices, with the same
weights as in $E$. Observe that there is a well-defined monoid homomorphism
$M(H,w)\rightarrow M(E,w);a\mapsto a$. For $a,b\in M(H,w)$ if $a=b$ in
$M(E,w)$, an application of Confluence Lemma 3.1 shows there is a
$c\in\mathbb{F}_{E}$ such that $a\rightarrow c$ and $b\rightarrow c$. However
since $H$ is hereditary and saturated, all the transformations occur already
in $H$ and thus $a=b$ in $M(H,w)$. Thus we can consider $M(H,w)$ as a
submonoid of $M(E,w)$. Consequently:

######  Remark 3.2.

Let $E$ be a weighted graph. Let $S$ denote the subset of $E^{0}$ consisting
of those vertices which do not connect to any cycle in $E$. Then clearly $S$
is hereditary and saturated. So by the above observation we can consider the
monoid $M(S,w)$ as a submonoid of $M(E,w)$. Furthermore, we note for later use
that the graph $E/S$ is easily seen to contain no sinks.

As one application, the Confluence Lemma 3.1 allows us to determine the unit
group of graph monoids.

######  Lemma 3.3.

Let $E$ be a finite weighted graph, $M(E,w)$ its associated monoid and $S$ the
set of all vertices in $E$ which do not connect to any cycle in $E$. Then
$Z(M(E,w))=M(S,w)$. Furthermore, if $E$ is a finite vertex weighted graph,
then

$M(E,w)\big{/}Z(M(E,w))=M(E,w)\big{/}M(S,w)\cong M(E/S,w_{r}),$ (9)

where $w_{r}$ is the restriction of the weight function $w$ to $E/S$.

###### Proof.

By Remark 3.2 we have $M(S,w)\subseteq M(E,w)$. If $S$ is not empty, then any
path emitting from an element $v$ in $S$ can be extended to a path that ends
in a sink, as $E$ is finite and $v$ is not connected to a cycle. An easy
induction (on the maximum length of the paths connecting to sinks) shows that
$kv=0$, for some $k\in\mathbb{N}^{+}$. Thus $M(S,w)\subseteq Z(M(E,w))$.

Conversely, let $a\in Z(M(E,w))$. Then $a+b=0$ for some $b\in M(E,w)$. The
Confluence Lemma 3.1 implies that $a+b\rightarrow 0$. Suppose there is a
vertex $w\in E^{0}\backslash S$ appearing in the presentation of $a$. Since
$w$ is connected to a cycle, any possible transformations of $w$ or its
multiple would give a vertex on a cycle and subsequently any further
transformations always contain a vertex on a cycle. Thus $a+b$ can’t be
transformed to $0$, and so $a\in M(S,w)$.

The monoid isomorphism of (9) (which is valid for any hereditary and saturated
subset of $E$, of which $S$ is a particular case) is easy to check, and is
left to the reader. ∎

We can now establish the following corollary, which will be used to connect
sandpile monoids to weighted Leavitt path algebras in Theorem 6.1.

######  Corollary 3.4.

Let $M(E,w)$ be the graph monoid associated to a finite weighted graph $(E,w)$
and $S$ the set of vertices in $E$ which do not connect to any cycle.

  1. (i)

$M(E,w)$ is a group if and only if $E$ is acyclic.

  2. (ii)

$M(E,w)$ is conical if and only if every non-sink vertex in $S$ has weight
one.

###### Proof.

(i) Note that $M(E,w)$ is a group if and only if $M(E,w)=Z(M(E,w))$. The claim
follows now from Lemma 3.3.

(ii) We have that $M(E,w)$ is conical if and only if $Z(M(E,w))=0$, so that by
Lemma 3.3 we get $M(S,w)=0$. Suppose the weights of vertices in $S$ are one
(this in particular implies the weight of each edge emitting from the vertex
is one). Thus the relation (1) for a non-sink vertex $v\in S$ reduces to
$v=\sum_{e\in s^{-1}(v)}r(e)$, whereas for sinks $s$, we have $s=0$. Thus for
$v\in S$, an easy induction (on the maximum length of the paths connecting to
sinks) shows that $v=0$. On the other hand if $M(E,w)$ is conical, then
$M(S,w)=0$. Suppose $v\in S$. Since $v=0$ in $M(S,w)$, by the Confluence Lemma
3.1, $v\rightarrow 0$. However, no transformation can be applied to $v$ if the
weight of $v$ is larger than $1$. This completes the proof. ∎

######  Remark 3.5.

We note that if we consider a finite graph $E$ as a balanced weighted graph,
then Corollary 3.4(ii) says that $M(E,w)$ is conical precisely when every non-
sink vertex in $S$ is irrelevant.

##  4\. Unique normal form: The Diamond Lemma

In addition to the confluence property, one of the main features of sandpile
monoids is that their elements stabilise (i.e., are reduction-finite, see
below) to a unique normal form. In particular, sandpile monoids are
necessarily finite. Although any monoid of the form $M(E,w)$ has been shown to
satisfy the Confluence Property in Lemma 3.1, not every monoid of the form
$M(E,w)$ has the property that each of its elements stabilises (nor do they
need to be finite, as observed previously).

######  Example 4.1.

Let $E$ be the graph

$\textstyle{\bullet^{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet^{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

on which we define the vertex weighting $w(u)=w(v)=2$, then the relations of
weighted graph monoids give $2u=u+2v$ and $2v=u+v$ in $M(E,w)$. One observes
that the element $2u$ never stabilises; specifically, we can continue to apply
the two relations to the element $2u$ and produce any element of the monoid of
the form $2u+nv$ for $n\in\mathbb{N}^{+}$.

We note further that the monoid $M(E,w)$ is finite (equal to
$\\{0,u,v,2u,2v\\}$), and conical, and has atoms $u$ and $v$. But $M(E,w)$ is
not a refinement monoid, as an analysis of the equation $u+u=u+2v$ bears out.

######  Example 4.2.

Let $E$ be the graph

$\textstyle{\bullet^{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet_{z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

We point out here that the size of $M(E,w)$ is highly dependent on the weight
$w$ assigned to the edges of $E$. If we assign weight $w_{1}(e)=1$ to each
edge (i.e., the unweighted case), then $w_{1}(u)=w_{1}(v)=w_{1}(z)=1$, and in
this situation $M(E,w_{1})$ is infinite. (Note: the monoid $M_{E}$ of this
particular unweighted graph has been analyzed as Example $E_{1}^{7}$ in [3,
Section 4].) Similarly, if we assign weight $w_{2}(e)=2$ to each edge, then
$w_{2}(u)=w_{2}(v)=w_{2}(z)=2$, and in this situation $M(E,w_{2})$ is infinite
as well. (We will give the tools in Display 14 below which will allow us to
easily reach these conclusions about $M(E,w_{1})$ and $M(E,w_{2})$.) However,
if we assign weight $w_{3}(e)=3$ to each edge, then
$w_{3}(u)=w_{3}(v)=w_{3}(z)=3$, and in this situation $M(E,w_{3})$ is finite,
indeed $\big{|}M(E,w_{3})\big{|}=27$. This conclusion will follow from Theorem
6.1, as we will see that $M(E,w_{3})$ is isomorphic to the sandpile monoid
$\operatorname{SP}(G)$ of an appropriate graph $G$.

We will employ Bergman’s Diamond Lemma to re-establish that the elements of
sandpile monoids have unique normal forms. Although a proof of this result has
appeared in the literature (see e.g. [6, Section 9]), we include a proof here
for two reasons. First, while previous proofs have made oblique reference to a
“Jordan-Hölder type” “Diamond Lemma” result stemming from Newman [19], we will
show that Bergman’s Diamond Lemma can be applied quite easily and directly
here. Second, with Example 4.2 as context, we will play up exactly where in
this analysis the specific weight function is utilised.

We briefly remind the reader of the setting (see [10]). Let $\mathbb{F}_{X}$
be the free commutative monoid generated by a nonempty set $X$. Let $R$ be a
set of pairs of the form $\sigma=(W_{\sigma},f_{\sigma})$, where
$W_{\sigma},f_{\sigma}\in\mathbb{F}_{X}\backslash\\{0\\}$. The set $R$ is
called a reduction system for $\mathbb{F}_{X}$. For any $\sigma\in R$ and
$A\in\mathbb{F}_{X}$, let
$r_{A+\sigma}:\mathbb{F}_{X}\rightarrow\mathbb{F}_{X}$ denote a map that sends
$A+W_{\sigma}$ to $A+f_{\sigma}$ and fixes all other elements of
$\mathbb{F}_{X}$. The maps
$r_{A+\sigma}:\mathbb{F}_{X}\rightarrow\mathbb{F}_{X}$ are called reductions.
For $a,b\in\mathbb{F}_{X}$, we write $a\rightarrow b$ if there is a sequence
$r_{1},r_{2},\dots,r_{i}$ of reductions, such that $r_{i}\dots r_{1}(a)=b$. An
element $a\in\mathbb{F}_{X}$ is called reduction-finite if for every infinite
sequence $r_{1},r_{2},\dots$ of reductions, $r_{i}$ acts trivially on
$r_{i-1}\dots r_{1}(a)$, for all sufficiently large $i$. We call an element
$a\in\mathbb{F}_{X}$ reduction-unique if it is reduction-finite, and if its
images under all final sequences of reductions are the same. This unique image
is called the _normal form_ of the element $a$.

For $\sigma\neq\tau$ in $R$, we call a configuration in which $W_{\sigma}=A+B$
and $W_{\tau}=B+C$ (for some $A,B,C\in\mathbb{F}_{X}$) an overlap ambiguity of
$R$. Such an overlap ambiguity is called resolvable if there exist
compositions of reductions, $r$ and $r^{\prime}$, such that
$r(f_{\sigma}+C)=r^{\prime}(A+f_{\tau})$. Similarly, for $\sigma\neq\tau$ in
$R$, we call a configuration in which $W_{\sigma}=B$ and $W_{\tau}=A+B+C$ (for
some $A,B,C\in\mathbb{F}_{X}$) an inclusion ambiguity. Such an inclusion
ambiguity is called resolvable if there exist compositions of reductions, $r$
and $r^{\prime}$, such that $r(A+f_{\sigma}+C)=r^{\prime}(f_{\tau})$.

In order to use Bergman’s Diamond Lemma result, we require that
$\mathbb{F}_{X}$ be equipped with a semigroup partial ordering $\leq$, i.e.,
if $x<x^{\prime}$ then $x+y<x^{\prime}+y$, for all
$x,x^{\prime},y\in\mathbb{F}$. Furthermore, we need the partial ordering
$\leq$ to be compatible with $R$, namely, that $f_{\sigma}<W_{\sigma}$ for all
$\sigma\in R$.

Bergman’s Diamond Lemma in the setting of commutative monoids [10, Theorem
1.2, §9.1, §10.3] says the following. Suppose $\leq$ is a semigroup partial
ordering on $\mathbb{F}_{X}$ compatible with the reduction system $R$ and
having the descending chain condition. Then all ambiguities of $R$ are
resolvable if and only if all elements of $\mathbb{F}_{X}$ are reduction-
unique under $R$.

The main obstacle in trying to utilise Bergman’s machinery in practice lies in
establishing that the ambiguities are resolvable. We are in position to place
sandpile monoids in the setting of Bergman’s theorem. We will see that neither
type of ambiguity arises in the context of sandpile relations, and thus once
the partial order on the monoid is established, the fact that these monoids
have reduction-unique forms will follow immediately.

######  Proposition 4.3.

Suppose $E$ is a sandpile graph. Then every element of the sandpile monoid
$\operatorname{SP}(E)$ is reduction-unique.

###### Proof.

Let $E$ be a sandpile graph with the unique sink $s$ and consider $E$ as a
balanced weighted graph by assigning $w(v)=|s^{-1}(v)|$, for all $v\in E^{0}$
($v\neq s$).

Let $\mathbb{F}_{E}$ denote the free commutative monoid on the set $E^{0}$.
Let $R$ be the set of pairs of the form $\sigma=(W_{v},f_{v})$, where

$W_{v}=w(v)v,\,\,\text{ and }\,\,f_{v}=\sum_{e\in s^{-1}(v)}r(e),$ (10)

for every non-sink vertex $v$.

We employ here an idea quite similar to one presented in [6]. Let
$D=\max\\{w(v)\ |\ v\in E^{0}\\}$. (Set $D=2$ in case this maximum value is
$1$; i.e., in case each non-sink vertex in $E$ is irrelevant.) Let $n$ denote
$|E^{0}|$, and let $\ell_{v}$ denote the length of the shortest path which
connects $v$ to the sink $s$. Note that $\ell_{v}<n$ for all $v\in E^{0}$, as
no shortest length path can contain a cycle. The assignment $v\mapsto
D^{n-\ell_{v}}$, for any $v\in E^{0}$, induces a homomorphism of monoids
$p:\mathbb{F}_{E}\rightarrow\mathbb{N}$, where for $x=\sum_{v\in
E^{0}}k_{v}v\in\mathbb{F}_{E}$, we define

$p(x):=\sum_{v\in E^{0}}k_{v}D^{n-\ell_{v}}.$

We check that $p(W_{\sigma})<p(f_{\sigma})$ for relations defined in (10). We
have $p(W_{\sigma})=w(v)D^{n-\ell_{v}}$ and $p(f_{\sigma})=\sum_{e\in
s^{-1}(v)}D^{n-\ell_{r(e)}}$.

It is here that we use the assumption that $w(v)=|s^{-1}(v)|$. Since $w(v)\leq
D$ and $\ell_{r(f)}=\ell(v)-1$, for some $f\in s^{-1}(v)$, we obtain

$\displaystyle p(W_{\sigma})=w(v)D^{n-\ell_{v}}$ $\displaystyle=$
$\displaystyle|s^{-1}(v)|D^{n-\ell_{v}}$ $\displaystyle<$ $\displaystyle
D\cdot D^{n-\ell_{v}}+\sum_{e\in s^{-1}(v)\backslash\\{f\\}}D^{n-\ell_{r(e)}}$
$\displaystyle=$ $\displaystyle\sum_{e\in s^{-1}(v)}D^{n-\ell_{r(e)}}$
$\displaystyle=$ $\displaystyle p(f_{\sigma}).$

We now define an order $\geq$ on $\mathbb{F}_{E}$ as follows:

$x\geq y\ \ \ \mbox{ if }\ \ \ x=y\ \ \mbox{or}\ \ x\rightarrow y,$

where the binary relation $\rightarrow$ is defined as in (3) and (4). The
displayed inequality above yields that $x>y$ implies $p(x)<p(y)$, and so
$\geq$ is a partial order on $\mathbb{F}_{E}$. By the definition of the
ordering, it is immediate that $\geq$ is a semigroup partial order which is
compatible with relations (10). The fortunate, obvious observation in this
setting is that for $v\not=v^{\prime}\in E^{0}$, the expressions $W_{v}=w(v)v$
and $W_{v^{\prime}}=w(v^{\prime})v^{\prime}$ in $\mathbb{F}_{E}$ involve
neither overlap ambiguities nor inclusion ambiguities.

Finally, we have $p(x)\leq(\sum_{v\in E^{0}}{k_{v})}D^{n}$, for all
$x=\sum_{v\in E^{0}}k_{v}v\in\mathbb{F}_{E}$. This guarantees that the partial
ordering on $\mathbb{F}_{E}$ defined above has the descending chain condition.

Thus Bergman’s Diamond Lemma guarantees that the monoid generated by
$\mathbb{F}_{E}$ subject to relations (10) is reduction-unique. Since
$\operatorname{SP}(E)$ is this monoid by identifying the sink to be zero,
$\operatorname{SP}(E)$ is also reduction-unique. ∎

######  Remark 4.4.

We note that the sandpile monoid $\operatorname{SP}(E)$ of a sandpile graph
$E$ must be finite. Indeed $|\operatorname{SP}(E)|$ is the product of the
positive integers $|s^{-1}(v_{i})|$ taken over all non-sink vertices $v_{i}$
of $E$. This is clear because each element of $\operatorname{SP}(E)$
corresponds to the (unique) reduced form of each of the elements in the
quotient description of $\operatorname{SP}(E)$ as guaranteed by Proposition
4.3.

##  5\. Weighted Leavitt path algebras

In this section we briefly recall the notion of weighted Leavitt path
algebras. These are algebras associated to weighted graphs (see §2). We refer
the reader to [15, 21] for a detailed analysis of these algebras.

######  Definition 5.1.

Let $(E,w)$ be a weighted graph and $\mathsf{k}$ a field. The free
$\mathsf{k}$-algebra generated by $\\{v,e_{i},e_{i}^{*}\mid v\in E^{0},e\in
E^{1},1\leq i\leq w(e)\\}$ subject to relations

  1. (i)

$uv=\delta_{uv}u,\text{ where }u,v\in E^{0}$,

  2. (ii)

$s(e)e_{i}=e_{i}=e_{i}r(e),~{}r(e)e_{i}^{*}=e_{i}^{*}=e_{i}^{*}s(e),\text{
where }e\in E^{1},1\leq i\leq w(e)$,

  3. (iii)

$\sum_{e\in s^{-1}(v)}e_{i}e_{j}^{*}=\delta_{ij}v,\text{ where }v\in
E_{\operatorname{reg}}^{0}\text{ and }1\leq i,j\leq w(v)$,

  4. (iv)

$\sum_{1\leq i\leq w(v)}e_{i}^{*}f_{i}=\delta_{ef}r(e),\text{ where }v\in
E_{\operatorname{reg}}^{0}\text{ and }e,f\in s^{-1}(v)$,

is called the weighted Leavitt path algebra of $(E,w)$, and denoted
$L_{\mathsf{k}}(E,w)$. In relations (iii) and (iv) we set $e_{i}$ and
$e_{i}^{*}$ zero whenever $i>w(e)$.

Note that if the weight of each edge is $1$, then $L_{\mathsf{k}}(E,w)$
reduces to the usual Leavitt path algebra $L_{\mathsf{k}}(E)$ of the graph
$E$. Also, if $E$ is the empty graph then $L_{K}(E,w)$ is defined to be the
zero ring.

######  Example 5.2.

Weighted Leavitt path algebras were originally conceived in [15] in order to
provide a context in which to generalise the algebras $L_{\mathsf{k}}(n,n+k)$
constructed by W. Leavitt mentioned in the Introduction. Specifically, let
$E_{n,n+k}$ denote the weighted graph consisting of one vertex $v$ and $n$
loops of weight $n+k$ at $v$. Then it is shown in [15, Example 5.5] that

$L_{\mathsf{k}}(E_{n,n+k})\cong L_{\mathsf{k}}(n,n+k).$

In particular, let $(E,w)$ be the weighted graph consisting of one vertex and
one loop of weight $n$. Then the weighted Leavitt path algebra of $E$ is
isomorphic to the Leavitt path algebra of a graph with one vertex and $n$
loops, which in turn is the Leavitt algebra $L_{\mathsf{k}}(1,n)$. ([2,
Definition 1.1.3]).

######  Remark 5.3.

(1) Let $n,k\in\mathbb{N}^{+}$, and let $(E,w)$ denote the weighted graph
$E_{n,n+k}$ described in Example 5.2. So $(E,w)$ has one vertex $v$, and $n$
loops at $v$ each having weight $n+k$. Let $(E^{\prime},w^{\prime})$ denote
the weighted graph $E_{n+k,n}$. So $(E^{\prime},w^{\prime})$ has one vertex
$v^{\prime}$, and $n+k$ loops at $v^{\prime}$ each having weight $n$. Because
of the symmetry involved in the definition of weighted Leavitt path algebras,
it is easy to show that $L_{\mathsf{k}}(E,w)\cong
L_{\mathsf{k}}(E^{\prime},w^{\prime})$ as $\mathsf{k}$-algebras. Our choice
here to focus on the $n$ loops of weight $n+k$ point of view is to make the
connection we will establish in Theorem 6.1 below between sandpile monoids and
weighted Leavitt path algebras more transparent. We note, however, that the
algebras $L_{\mathsf{k}}(E,w)$ and $L_{\mathsf{k}}(E^{\prime},w^{\prime})$ are
different if these are viewed as graded $\mathsf{k}$-algebras.

(2) The previous paragraph notwithstanding, we caution the reader that in
general one cannot cavalierly modify the edges and weights and expect that the
corresponding Leavitt path algebras will be isomorphic. For instance, let
$(F,w)$ be the graph with one vertex $v$, and two loops at $v$ each having
weight $2$; and let $(F^{\prime},w^{\prime})$ be the graph with one vertex
$v^{\prime}$ and four loops at $v$ each having weight $1$. Then
$L_{\mathsf{k}}(F,w)\not\cong L_{\mathsf{k}}(F^{\prime},w^{\prime})$. (Indeed,
$M(F,w)\not\cong M(F^{\prime},w^{\prime})$ as monoids, since the latter is
finite (as noted in Example 2.6), while the former monoid is infinite. Since
these two monoids are not isomorphic, neither can the two weighted Leavitt
path algebras be isomorphic; see the $\mathcal{V}$-monoid Realization Theorem
below.)

For a ring $A$ with identity, the commutative monoid
$\operatorname{\mathcal{V}}(A)$ is defined as the set of isomorphism classes
of finitely generated projective right $A$-modules equipped with direct sum as
the binary operation. The group completion of this monoid is the Grothendieck
group $K_{0}(A)$ of $A$. We refer the reader to the book of Magurn for a
comprehensive treatment of these ideas [18].

The Leavitt algebras $L_{\mathsf{k}}(n,n+k)$ can be produced from Bergman’s
machinery of the Universal Construction of Rings [9], which subsequently
thereby also describes the structure of the $\mathcal{V}$-monoids of these
algebras. Let $\mathsf{k}$ be a field, $A$ a $\mathsf{k}$-algebra and let $P$
and $Q$ be nonzero finitely generated projective right $A$-modules. Then there
is a $\mathsf{k}$-algebra $B$, with an algebra homomorphism $A\rightarrow B$
such that there is a universal isomorphism
$i:\overline{P}\rightarrow\overline{Q}$, where $\overline{M}=M\otimes_{A}B$
for any right $A$-module $M$ ([9, p. 38 and Theorem 3.3]). Bergman’s Theorem
[9, Theorem 5.2] states that $\operatorname{\mathcal{V}}(B)$ is the quotient
of $\operatorname{\mathcal{V}}(A)$ modulo the relation $[P]=[Q]$.

Using this, starting with a field $A=\mathsf{k}$, and the finitely generated
projective modules $A$-modules $P=\mathsf{k}^{n}$ and $Q=\mathsf{k}^{n+k}$,
Bergman’s machinery applied to this data gives that $B=L_{\mathsf{k}}(n,n+k)$,
and subsequently that

$\mathcal{V}(L_{\mathsf{k}}(n,n+k))\cong M_{n,n+k}=\\{0,x,2x,\dots,(n+k-1)x\
|\ (n+k)x=nx\\}.$

We note that the process of explicitly identifying the $\mathcal{V}$-monoid
$\mathcal{V}(B)$ is extremely delicate, requiring the full use of the powerful
machinery of Bergman. Indeed, while Leavitt had established in [16] that the
monoid of free modules over $L_{\mathsf{k}}(n,n+k)$ was isomorphic to
$M_{n,n+k}$, it was not established until Bergman’s fundamental work fifteen
years later in [9] that $M_{n,n+k}$ actually represented the full
$\mathcal{V}$-monoid of $L_{\mathsf{k}}(n,n+k)$.

######  Remark 5.4.

It is germane to point out that the original observation which led us to
connect sandpile monoids to the $\mathcal{V}$-monoids of weighted Leavitt path
algebras was made in the context of the monoid $M_{n,n+k}$. Specifically,
$M_{n,n+k}$ arises on the one hand (as noted in Example 5.2 together with the
previous paragraph) as the $\mathcal{V}$-monoid of the most elementary type of
vertex weighted graph, namely the vertex weighted graph having exactly one
vertex. On the other hand (as noted in Example 2.11), $M_{n,n+k}$ also arises
as the sandpile monoid of the most elementary type of sandpile graph, namely
the sandpile graph having exactly one non-sink vertex. Once this connection
was realised, the roadmap to connect general sandpile monoids and weighted
Leavitt path algebras became evident.

Although the computation of the $\mathcal{V}$-monoid can be carried out for
the Leavitt path algebra of arbitrary (finite) weighted graphs (see e.g.
[20]), for the current work we need only describe the $\mathcal{V}$-monoid of
weighted Leavitt path algebras associated to vertex weighted graphs, which we
now do.

######  Definition 5.5.

Let $(E,w)$ be a row-finite vertex weighted graph. We define the monoid

$M_{(E,w)}=\mathbb{F}_{E}\Big{/}\Big{\langle}w(v)v=\sum_{e\in
s^{-1}(v)}r(e)\,\big{|}\,v\in E_{\operatorname{reg}}^{0}\Big{\rangle}.$ (11)

Note: while clearly similar one to the other, the definitions of the monoids
$M(E,w)$ and $M_{(E,w)}$ are not identical. Specifically, there is no relation
in $M_{(E,w)}$ whose left hand side involves any sinks which may exist in $E$;
while in $M(E,w)$, there is a relation of the form $1\cdot s=0$ for each sink
in $E$. However, if $E$ contains no sinks, then $M(E,w)=M_{(E,w)}$.

$\mathcal{V}$-monoid Realization Theorem. (cf. [20],[21]) Let $(E,w)$ be a
row-finite vertex weighted graph. Then there is an isomorphism of monoids

$\displaystyle M_{(E,w)}\longrightarrow\mathcal{V}(L_{\mathsf{k}}(E,w)),\ \ \
\ \mbox{via}\ \ \ \ v\longmapsto[vL_{\mathsf{k}}(E,w)],$ (12)

where $vL_{\mathsf{k}}(E,w)$ is the finitely generated projective right
$L_{\mathsf{k}}(E,w)$-module generated by $v$. Furthermore, due to its
construction, by [9, Theorem 6.2], $L_{\mathsf{k}}(E,w)$ is the universal
$\mathsf{k}$-algebra which realises the monoid $M_{(E,w)}$ of (11) as its
$\mathcal{V}$-monoid.

In particular, for a directed (unweighted) row-finite graph $E$ we have

$\mathcal{V}(L_{\mathsf{k}}(E))\cong
M_{E}\cong\mathbb{F}_{E}\Big{/}\Big{\langle}v=\sum_{e\in
s^{-1}(v)}r(e)\,\big{|}\,v\in E_{\operatorname{reg}}^{0}\Big{\rangle}.$ (13)

######  Remark 5.6.

Let $E$ be a finite graph, and let $N^{\prime}$ denote the adjacency matrix of
$E$, i.e., $(n_{ij})\in M_{|E^{0}|}({\mathbb{Z}})$, where $n_{ij}$ is the
number of edges from $v_{i}$ to $v_{j}$. Furthermore let $I^{\prime}_{w}$ be
the _weighted identity matrix of $(E,w)$_, defined to be the
$|E^{0}|\times|E^{0}|$ matrix $(a_{ij})$, where $a_{ij}=0$ for $i\not=j$ and
$a_{ii}=w(v_{i})$. Let $N^{t}$ and $I_{w}$ be the matrices obtained from
$N^{\prime}$ and $I^{\prime}_{w}$ by first taking the transpose and then
removing the columns corresponding to sinks, respectively. Since the
Grothendieck group of $L_{\mathsf{k}}(E,w)$ is the group completion of the
commutative monoid $M_{(E,w)}$, we obtain that

$K_{0}(L_{\mathsf{k}}(E,w))\cong\operatorname{coker}\big{(}N^{t}-I_{w}:\mathbb{Z}^{E_{\operatorname{reg}}^{0}}\longrightarrow\mathbb{Z}^{E^{0}}\big{)}.$
(14)

Let $(E,w)$ be any weighted graph. If $E$ has no sinks, then
$M_{(E,w)}=M(E,w)$. On the other hand, if $E$ does contain sinks, then one may
construct the weighted graph $E^{\prime}$ by attaching a loop to each sink,
and defining the weight $w^{\prime}$ on $E^{\prime}$ to be $w$ for any edge
$e\in E^{1}$, and $w^{\prime}(e)=1$ for any added loop. Then clearly
$M_{(E,w)}=M(E^{\prime},w^{\prime})$. Consequently, by the Confluence Lemma
3.1 together with the $\mathcal{V}$-monoid Realization Theorem, we get

######  Corollary 5.7.

Let $(E,w)$ be a weighted graph. Then $\mathcal{V}(L_{\mathsf{k}}(E,w))$ has
the Confluence Property.

##  6\. Sandpile monoids and weighted Leavitt path algebras: the connection

Let $E$ be a graph, and let $S$ denote the hereditary and saturated subset of
vertices of $E$ which do not connect to any cycle. Consider $E$ as a balanced
weighted graph $(E,w)$, so that $w(v)=|s^{-1}(v)|$ for all $v\in
E_{\operatorname{reg}}^{0}$, and consider $(E/S,w_{r})$ as a weighted subgraph
of $(E,w)$ as described in Remark 2.3.

We are now in position to realise the conical sandpile monoids as the
$\mathcal{V}$-monoids of weighted Leavitt path algebras.

######  Theorem 6.1.

Let $E$ be a sandpile graph, and let $S$ denote the set of vertices of $E$
which do not connect to any cycle. Let $w$ denote a balanced weighting on $E$.
Then

$\operatorname{SP}(E)\big{/}Z(\operatorname{SP}(E))\cong\mathcal{V}(L_{\mathsf{k}}(E/S,w_{r})).$
(15)

In particular $\operatorname{SP}(E)$ is conical if and only if any vertex not
connected to a cycle is irrelevant. In this case

$\operatorname{SP}(E)\cong\mathcal{V}(L_{\mathsf{k}}(E/S,w_{r}))\ \ \ \ \
\mbox{and}\ \ \ \ \ G(E)\cong K_{0}(L_{\mathsf{k}}(E/S,w_{r})).$ (16)

###### Proof.

As noted in Remark 2.12, $\operatorname{SP}(E)\cong M(E,w)$. By Lemma 3.3,
$\operatorname{SP}(E)\big{/}Z(\operatorname{SP}(E))\cong M(E/S,w_{r}).$ By the
definition of $S$ we see that $E/S$ has no sinks. So
$M(E/S,w_{r})=M_{(E/S,w_{r})}$. But by (12),
$M_{(E/S,w_{r})}\cong\mathcal{V}(L_{\mathsf{k}}(E/S,w_{r}))$. This gives the
desired isomorphism (15).

The second part of the theorem follows from Corollary 3.4(ii) and Lemma 2.5
(using the fact that $G(E)$ is the smallest ideal of $\operatorname{SP}(E)$).
∎

The statement that conicality of a sandpile monoid is equivalent to the
property that vertices not connected to cycles are irrelevant has been
established previously in a number of articles, see e.g. [23, Proposition
4.2.5] and [6, Proposition 5.7].

Here are some examples which clarify how Theorem 6.1 plays out in some
specific important situations.

######  Example 6.2.

Let $G$ denote the graph of Example 2.11. So $S=\\{s\\}$. Then $(G/S,w_{r})$
is the graph $E_{n,n+k}$ (defined in Example 2.8) having one vertex $v$, and
$n$ loops at $v$ each of weight $n+k$. So by Example 5.2 we have that
$L_{\mathsf{k}}(G/S,w_{r})\cong L_{\mathsf{k}}(n,n+k)$. Then Theorem 6.1
(re)establishes that

$\mathcal{V}(L_{\mathsf{k}}(n,n+k))\cong\mathcal{V}(L_{\mathsf{k}}(G/\\{s\\},w_{r}))\cong\operatorname{SP}(G)\cong
M_{n,n+k}\ ,$

as expected.

######  Example 6.3.

Let $(E,w_{3})$ be the weighted graph described in Example 4.2. Now consider
the sandpile graph $T$ pictured here. Then clearly as graphs $E=T/S$. But
$w_{3}$ on $E=T/S$ is precisely $w_{r}$ inherited from $T$, where $w$ is the
balanced weight function. So $M(E,w_{3})=\operatorname{SP}(T)$. In particular,
by Theorem 6.1, $\operatorname{SP}(T)$ arises as the $\mathcal{V}$-monoid of
the Leavitt path algebra of the weighted graph $(E,w_{3}).$

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Remark 4.4 gives that the size of this monoid is $3\times 3\times 3=27$.

######  Example 6.4.

Let $E$ be a reduced sandpile graph with $|SP(E)|=p$, a prime number. We show
that either $\operatorname{SP}(E)\cong\mathbb{Z}_{p}$, or
$\operatorname{SP}(E)\cong
M_{p-l,p}\cong\mathcal{V}(L_{\mathsf{k}}(E_{p-l,p}))$, for some $1\leq l<p$.

Since the number of elements of the sandpile monoid is prime, by Remark 4.4
the graph $E$ must have only two vertices, namely a sink $s$ and a non-sink
vertex $v$ which connects to the sink.

If all the edges from $v$ are connected to the sink, then by the relations of
sandpile monoid, $\operatorname{SP}(E)\cong M(E,w)\cong\langle v\mid
pv=0\rangle$, where $E$ is the balanced weighted graph. Clearly then
$\operatorname{SP}(E)\cong\mathbb{Z}_{p}$. We note that in this case
$\operatorname{SP}(E)$ is not conical (as e.g. $1+(p-1)=0$), so cannot be
realised as the $\mathcal{V}$-monoid of any associative ring.

On the other hand, if $E$ consists of $p-l$ loops at $v$ and $l$ edges from
$v$ to $s$, the relations of the monoid of the balanced weighted graph gives
$\operatorname{SP}(E)\cong M(E\setminus\\{s\\},w_{r})\cong\langle v\mid
pv=(p-l)v\rangle=M_{p-l,p}$. But this is exactly
$\mathcal{V}(L_{\mathsf{k}}(E_{p-l,p}))$ by Example 6.2.

Consequently, if $|SP(E)|$ is prime, then either $\operatorname{SP}(E)$ is not
conical, or, in case $\operatorname{SP}(E)$ is conical, then by the
observation made in Example 2.6, $G(E)$ is a cyclic group of order $p-l$ for
some $1\leq l<p$. Now let $F$ be the graph

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The graph $F$ appears in [3, Example 3.8], where it is shown that
$|\mathcal{V}(L_{\mathsf{k}}(F))|=5$, and that $K_{0}(L_{\mathsf{k}}(F))$ is
isomorphic to the non-cyclic group $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. In
particular, $F$ is a graph for which $\mathcal{V}(L_{K}(F))$ is finite, but
for which $\mathcal{V}(L_{K}(F))$ is not isomorphic to $\operatorname{SP}(E)$
for any sandpile graph $E$.

######  Example 6.5.

Let $E$ be a _weighted cycle_. That is, $E$ is a vertex weighted graph $(E,w)$
for which $E^{0}=\\{v_{1},v_{2},\dots,v_{m}\\}$,
$E^{1}=\\{e_{1},e_{2},\dots,e_{m}\\}$ with $s(e_{i})=v_{i}$ and
$r(e_{i})=v_{i+1}$ for $1\leq i\leq m-1$, $s(e_{m})=v_{m}$, $r(v_{m})=e_{1}$,
and $w(e_{i})=w(s(e_{i}))$ is some positive integer for each $1\leq i\leq m$.
We assume (for reasons described below) that $w(v_{i})\geq 2$ for at least one
$v_{i}$.

Consider the unweighted directed graph $F:=\hat{E}^{\operatorname{op}}$. Here
$\hat{E}$ denotes the (unweighted) directed graph obtained from $E$ by
replacing any edge in $E$ of weight $w(e)>1$ with $w(e)$ edges; and
$\operatorname{op}$ stands for the opposite of the graph, i.e. the graph with
all edge orientations reversed. Comparing the generators and relations of the
algebra $L_{\mathsf{k}}(E,w)$ with those of $L_{\mathsf{k}}(F)$ shows that
$L_{\mathsf{k}}(E,w)\cong L_{\mathsf{k}}(F)$ as $\mathsf{k}$-algebras.

Therefore by (12)

$M_{(E,w)}\cong\mathcal{V}(L_{\mathsf{k}}(E,w))\cong\mathcal{V}(L_{\mathsf{k}}(F))\cong
M_{F}.$

Now consider the graph $G$ defined as follows. The vertex set of $G$ is
$G^{0}:=\\{s,v_{1},v_{2},\dots,v_{m}\\}$. The vertex $s$ is a sink in $G$. The
edge set $G^{1}$ consists of the following. As in $E$, for each $v_{i}\in
G^{0}$, $s^{-1}(v_{i})$ includes the edge $e_{i}$. In addition, in $G$, each
$s^{-1}(v_{i})$ also includes $w(v_{i})-1$ edges in $G^{1}$ from $v_{i}$ to
$s$.

Then by considering the generators and relations of the indicated monoids, we
get

$\operatorname{SP}(G)\cong
V(L_{\mathsf{k}}(E,w))\cong\mathcal{V}(L_{\mathsf{k}}(F)).$ (17)

In particular, $\operatorname{SP}(G)$ arises as the $\mathcal{V}$-monoid of an
unweighted Leavitt path algebra.

For any positive integer $n$ let $C_{n}$ denote the monoid

$C_{n}=\\{0,x,2x,\dots,(n-1)x\\}\ \mbox{ with relation }nx=x.$

So $C_{n}=M_{1,n}$ when $n\geq 2$. Then we have that each of the three
displayed monoids in (17) is isomorphic to the monoid $C_{N}$, where
$N=w(v_{1})\cdot w(v_{2})\cdots w(v_{m})$. We note that if $n=1$ then the
monoid $C_{1}$ is infinite; thus the condition $w(v_{i})\geq 2$ for some $i$
is necessary to ensure that this monoid is finite.

As a specific instance of the ideas presented in Example 6.5, suppose $(E,w)$
is the weighted cycle indicated below, where $E^{0}=\\{a,b,c\\}$,
$w(a)=w(b)=2$ and $w(c)=1$. The associated (unweighted) graphs $F$ and $G$ are
pictured as well. Since $N=2\cdot 2\cdot 1=4$ we have each of the monoids in
(17) is isomorphic to $C_{4}$. Of course this is not hard to see directly:
note that in $M(E,w)$ we have $2a=b,2b=c,$ and $c=a$, and thus
$M(E,w)=\\{0,a,2a,3a\\}$, where $4a=a$.

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In Theorem 6.1 we established the ubiquity of weighted Leavitt path algebras
in the context of sandpile monoids, namely, that every conical sandpile monoid
$\operatorname{SP}(G)$ arises as the $\mathcal{V}$-monoid of a suitably
defined weighted Leavitt path algebra. Perhaps surprisingly, the unweighted
Leavitt path algebras (i.e., the classical Leavitt path algebras) turn out to
be quite sparse in this context. We will establish the precise statement in
Proposition 6.9.

We now assume that $G$ denotes a conical sandpile graph; i.e. $G$ contains a
unique sink $s$, and every vertex in $G$ connects to $s$, and (by Theorem 6.1)
every relevant vertex connects to a cycle.

For context, we note that it is easy to find examples of reduced conical
sandpile graphs $G$ for which there exists $v\in G^{0}\setminus\\{s\\}$ and
$x\in\mathbb{F}_{G}$ having $x\neq v$ in $\mathbb{F}_{G}$, but $x=v$ in
$\operatorname{SP}(G)$. For instance, let $v,w\in G^{0}$ for which
$s^{-1}(w)=\\{e,f_{1},\dots,f_{k}\\}$ where $k\geq 1$ and $r(e)=v$ and
$r(f_{i})=s$ for all $1\leq i\leq k$. Then $(k+1)w\neq v$ in $\mathbb{F}_{G}$
(even if $v=w$), because $v$ is an atom in $\mathbb{F}_{G}$. But $(k+1)w=v$ in
$\operatorname{SP}(G)$.

We will now show that the example described in the previous paragraph
essentially describes the only possible such configurations.

######  Lemma 6.6.

Let $G$ be a reduced conical sandpile graph. Let $\mathbb{F}_{G}$ denote the
free abelian monoid on the set $G^{0}$. Suppose $v\in G^{0}$ and suppose there
exists a nonzero element $x\in\mathbb{F}_{G}$ for which $x\neq v$ in
$\mathbb{F}_{G}$, but $x=v$ in $\operatorname{SP}(G)$. Then there exists a
vertex $u_{v}\in G^{0}\setminus\\{s\\}$ for which

$s^{-1}(u_{v})=\\{e_{v},e_{1},\dots,e_{n_{u_{v}}-1}\\},$

where $n_{u_{v}}$ denotes $|s^{-1}_{G}(u_{v})|$, $r(e_{v})=v$ and $r(e_{i})=s$
for all $1\leq i\leq n_{u_{v}}-1$. (Note that we allow $u_{v}=v$, and $e_{v}$
to be a loop at $v$.)

###### Proof.

If $x=v$ in $\operatorname{SP}(G)$ but $x\neq v$ in
$\mathbb{F}_{G^{0}\setminus\\{s\\}}$, then by the definition of $\sim$ (see
(4)) there exists a sequence of positive length $x=a_{0}\rightsquigarrow
a_{1}\rightsquigarrow\dots\rightsquigarrow a_{i-1}\rightsquigarrow
a_{i}\rightsquigarrow\dots\rightsquigarrow a_{n}=v$ where
$a_{i-1}\rightsquigarrow a_{i}$ denotes either $\rightarrow$ or $\leftarrow$.
We focus on the last move in the sequence, $a_{n-1}\rightsquigarrow a_{n}=v$.
Because $G$ is reduced each non-sink vertex of $G$ emits at least two edges,
so the $\rightsquigarrow$ connecting $a_{n-1}$ to $v$ cannot be $\leftarrow$.
So we must have $a_{n-1}\rightarrow v$. That is, for some $u_{v}\in
G^{0}\setminus\\{s\\}$ we have that $n_{u_{v}}u_{v}$ is in ${\rm
supp}(a_{n-1})$, and

$(a_{n-1}-n_{u_{v}}u_{v})+\sum_{\\{e\in E^{1}|s(e)=u_{v},r(e)\neq
s\\}}r(e)=v.$

Since $v$ is an atom in $\mathbb{F}_{G}$, there are two possibilities:

Case 1: $a_{n-1}-n_{u_{v}}u_{v}=v$, and $\sum_{\\{e\in
E^{1}|s(e)=u_{v},r(e)\neq s\\}}r(e)=0$ ; or

Case 2: $a_{n-1}-n_{u_{v}}u_{v}=0$, and $\sum_{\\{e\in
E^{1}|s(e)=u_{v},r(e)\neq s\\}}r(e)=v$.

The equation $\sum_{\\{e\in E^{1}|s(e)=u_{v},r(e)\neq s\\}}r(e)=0$ of Case 1
would give that all edges $e$ for which $s(e)=u_{v}$ have $r(e)=s$; since
$n_{u_{v}}\geq 2$ this would give $n_{u_{v}}u_{v}=0$ in
$\operatorname{SP}(G)$, which would violate that $G$ is conical. So Case 1
cannot occur.

But the equations given in Case 2 describe exactly the desired configuration
of vertices and edges in $G$. ∎

######  Proposition 6.7.

Let $G$ be a reduced conical sandpile graph. Suppose in addition that
$\operatorname{SP}(G)$ is a refinement monoid. Then there is a partition
$\mathcal{P}$ of $G^{0}\setminus\\{s\\}$ so that the graph $G$ has the
following form.

Let $P=\\{v_{1},v_{2},\dots,v_{t}\\}$ be one of the equivalence classes in
$\mathcal{P}$. Then each $v_{i}$ emits all but one of its edges to the sink
$s$, and this one not-to-the-sink edge $e_{i}$ has $s(e_{i})=v_{i}$ for each
$1\leq i\leq t$, $r(e_{i})=v_{i-1}$ for each $2\leq i\leq t$, and
$r(e_{1})=v_{t}$. Call this subgraph $G_{P}$.

Then $G$ is the disjoint union of $\\{s\\}$ with the subgraphs $\\{G_{P}\ |\
P\in\mathcal{P}\\}$.

###### Proof.

By Lemma 2.4, the monoid $\operatorname{SP}(G)$ contains no atoms. In
particular each vertex $v\in G^{0}\setminus\\{s\\}$ can be written as the sum
of two nonzero elements in $\operatorname{SP}(G)$. Since $v$ is an atom in
$\mathbb{F}_{G}$ we have that the hypotheses of Lemma 6.6 are satisfied. So
for each $v\in G^{0}\setminus\\{s\\}$ there exists a vertex $u_{v}$ with the
indicated configuration. By considering the indicated configuration, if a
vertex $u\in G^{0}\setminus\\{s\\}$ arises as $u_{v}$ for some vertex $v$,
then $v$ is unique. (In other words, if $u\in G^{0}\setminus\\{s\\}$ has
$u=u_{v}=u_{v^{\prime}}$ for vertices $v,v^{\prime}$ in
$G^{0}\setminus\\{s\\}$, then $v=v^{\prime}$.) Since such a configuration must
occur for every $v\in G^{0}\setminus\\{s\\}$, by the Pigeon Hole Principle
applied to $G^{0}\setminus\\{s\\}$ (recall that by definition any sandpile
graph is finite) we have that each $u\in G^{0}\setminus\\{s\\}$ appears
exactly once as $u_{v}$ for some (unique) $v\in G^{0}\setminus\\{s\\}$. Now
partition $G^{0}\setminus\\{s\\}$ using the transitive closure of this
relationship. ∎

######  Corollary 6.8.

Suppose $G$ is a conical sandpile graph for which $\operatorname{SP}(G)$ is a
refinement monoid. Then

$\operatorname{SP}(G)\cong\bigoplus_{i=1}^{t}C_{n_{i}},$

for some integers $n_{1},\dots,n_{t}\geq 2$. Conversely, any monoid of the
form $\bigoplus_{i=1}^{t}C_{n_{i}}$ with $n_{1},\dots,n_{t}\geq 2$ is a
conical sandpile monoid which is also a refinement monoid.

###### Proof.

This follows directly from the observations made in Example 6.5, together with
Remark 2.13 and Proposition 6.7. ∎

######  Proposition 6.9.

Let $E$ be a finite graph. Let $L_{\mathsf{k}}(E)$ denote the (unweighted)
Leavitt path algebra. Then the monoid $\mathcal{V}(L_{\mathsf{k}}(E))$ is
isomorphic to the sandpile monoid $\operatorname{SP}(G)$ of a sandpile graph
$G$ if and only if
$\mathcal{V}(L_{\mathsf{k}}(E))\cong\oplus_{i=1}^{t}C_{n_{i}}$ for some
positive integers $n_{1},\dots,n_{t}$.

###### Proof.

Sufficiency of the statement follows from Corollary 6.8.

For necessity, we note that the $\mathcal{V}$-monoid of any ring is conical,
while the $\mathcal{V}$-monoid of the (unweighted) Leavitt path algebra
$L_{\mathsf{k}}(E)$ of any graph $E$ is refinement (see e.g. [2, Theorem
3.6.8]). So if $\mathcal{V}(L(E))\cong\operatorname{SP}(G)$ for some sandpile
graph $G$ then Corollary 6.8 gives the result. ∎

######  Remark 6.10.

By Proposition 6.9 we know the structure of the $\mathcal{V}$-monoid of any
unweighted Leavitt path algebra whose $\mathcal{V}$-monoid is a finite
sandpile monoid. We discuss the specific case where
$\mathcal{V}(L_{\mathsf{k}}(E))\cong C_{n}$ (i.e., that there is only one term
in the direct sum given in Proposition 6.9). By [2, Theorem 3.1.10 or Lemma
6.3.14], $L_{\mathsf{k}}(E)$ is a purely infinite simple algebra. Moreover,
$K_{0}(L_{\mathsf{k}}(E))\cong\mathbb{Z}_{n-1}$. Fix any isomorphism
$\varphi:K_{0}(L_{\mathsf{k}}(E))\to\mathbb{Z}_{n-1}$, and let $t$ denote
$\varphi([L_{\mathsf{k}}(E)])$.

Let $R_{n}$ denote the graph having one vertex and $n$ loops at that vertex.
It is well known that $K_{0}({\rm
M}_{\ell}(L_{\mathsf{k}}(R_{n})))\cong\mathbb{Z}_{n-1}$, for each
$\ell\in\mathbb{N}^{+}$, and that under this isomorphism $[{\rm
M}_{\ell}(L_{\mathsf{k}}(R_{n}))]\mapsto\ell\in\mathbb{Z}_{n-1}$. Therefore
$K_{0}(L_{\mathsf{k}}(E))\cong K_{0}({\rm M}_{t}(L_{\mathsf{k}}(R_{n})))$, via
a map that takes $[L_{\mathsf{k}}(E)]$ to $[{\rm
M}_{t}(L_{\mathsf{k}}(R_{n}))]$. Let $R_{n,t}$ denote the standard graph
having $L_{\mathsf{k}}(R_{n,t})\cong{\rm M}_{t}(L_{K}(R_{n}))$. It is well-
known that ${\rm det}(I-A_{R_{n,t}})$ is negative.

If $E$ has the property that ${\rm det}(I-A_{E})<0$, then by the
Classification Theorem of purely infinite simple Leavitt path algebras [2,
Theorem 6.3.32], we obtain that $L_{\mathsf{k}}(E)$ is isomorphic to
$M_{t}(L_{K}(R_{n}))$.

On the other hand, if $E$ has the property that ${\rm det}(I-A_{E})>0$, then
as of the writing of this article we do not know whether $L_{\mathsf{k}}(E)$
is necessarily isomorphic to ${\rm M}_{t}(L_{K}(R_{n}))$. We note that this
barrier lies at the heart of the longstanding Algebraic Kirchberg Phillips
Question for Leavitt Path Algebras of Finite Graphs, see e.g. [2, Question
6.3.3].

Summary Observation.

Now that our main results have been established, we present a brief overview
of a monoid-theoretic aspect of this article. Assume here that “monoid” means
“finitely generated commutative monoid written additively.”

A “distinguished element” in a monoid $M$ is an element $I\in M$ for which for
each $a\in M$ there exists $b\in M$ and $n\in\mathbb{N}^{+}$ with $a+b=nI$. If
$R$ is a unital ring then $[R]$ is a distinguished element in
$\mathcal{V}(R)$. For a sandpile monoid $\operatorname{SP}(G)$, both
$I_{1}=\sum_{v\in G^{0}\setminus\\{s\\}}v$ and $I_{2}=\sum_{v\in
G^{0}\setminus\\{s\\}}(|s^{-1}(v)-1|)v$ are easily seen to be distinguished
elements. (The element $I_{2}$ is sometimes denoted MAX in the literature.)

Let $\mathcal{U}$ denote the class of monoids that are isomorphic to

the $\mathcal{V}$-monoid of some associative unital ring.

Let $\mathcal{S}_{1}$ denote the class of monoids that are isomorphic to

the $\mathcal{V}$-monoid of a weighted Leavitt path algebra of a finite graph.

Let $\mathcal{S}_{2}$ denote the class of monoids that are isomorphic to

the $\mathcal{V}$-monoid of a standard (i.e., unweighted) Leavitt path algebra
of a finite graph.

Let $\mathcal{S}_{3}$ denote the class of monoids that are isomorphic to

a sandpile monoid, and are also conical.

Then:

  1. (1)

By Bergman’s Theorem [9, Theorem 6.2], $\mathcal{U}$ is precisely the class of
monoids that are conical and have a distinguished element.

  2. (2)

In our main result (Theorem 6.1) we established that
$\mathcal{S}_{3}\subseteq\mathcal{S}_{1}$, and we have given examples (see
e.g. Remark 2.9) to show that this inclusion is proper.

  3. (3)

Clearly $\mathcal{S}_{2}\subseteq\mathcal{S}_{1}$, and this inclusion is
proper (e.g. $M_{n,n+k}$ for any $n\geq 2$).

  4. (4)

Subsequently we identified in Proposition 6.9 the monoids in the class
$\mathcal{S}_{2}\cap\mathcal{S}_{3}$; these turn out to be direct sums of
monoids of the form $C_{n}$.

We conclude the article by presenting the following natural definition. An
investigation into various properties of these structures will be taken up in
[4].

######  Definition 6.11.

Let $(E,w)$ be a vertex weighted graph. We say the weighted Leavitt path
algebra $L_{\mathsf{k}}(E,w)$ is a _sandpile algebra_ if
$\mathcal{V}(L_{K}(E,w))$ is a sandpile monoid.

Theorem 6.1 gives that if $E$ is a sandpile graph, and $S$ is the set of
vertices of $E$ which do not connect to any cycles, then
$L_{\mathsf{k}}(E/S,w_{r})$ is a sandpile algebra. In particular if $E$ is a
sandpile graph such that each non-sink vertex connects to a cycle, then
considering $E$ as a balanced weighted graph,
$L_{\mathsf{k}}(E\setminus\\{s\\},w_{r})$ is a sandpile algebra.

##  7\. Acknowledgements

A part of this work was done when the second author was an Alexander von
Humboldt Fellow at the University of Münster. He would like to thank both
institutions for an excellent hospitality.

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