## 1 Introduction

Arthur Cayley was the first to define in 1854 [2] the notion of a group as well as the table of its operation known as the Cayley table. To describe the structure of a group , Cayley also introduced in 1878 [3] the concept of graph for according to a generating subset , namely the set of labeled oriented edges for every of and of . Such a graph, called Cayley graph, is directed and labeled in (or an encoding of by symbols called letters or colors). A characterization of unlabeled and undirected Cayley graphs was given by Sabidussi in 1958 [5] : an unlabeled and undirected graph is a Cayley graph if and only if we can find a group with a free and transitive action on the graph. Following a question asked by Hamkins in 2010 [4]: ‘Which graphs are Cayley graphs?’, we gave simple graph-theoretic characterizations of Cayley graphs for groups, as well as for left-cancellative and cancellative monoids [1]. In this paper, we generalize this last characterization to Cayley graphs of monoids, then to semigroups. We also strengthen all these characterizations to commutative monoids, semilattices and abelian groups.

To structurally characterize the Cayley graphs (of groups), we selected four basic properties of these graphs. First and by definition, any Cayley graph is deterministic: there are no two edges of the same source and label. The right-cancellative property of groups induces the co-determinism of their graphs: there are no two edges of the same target and label. The left-cancellative property of groups implies that their graphs are simple: there are no two edges of the same source and goal. Finally, any Cayley graph is according to a generating subset hence is connected: there is a chain from the identity element to any vertex. To these four basic conditions is added the well known symmetry property of vertex-transitivity: all the vertices are isomorphic. These five properties satisfied by the Cayley graphs are sufficient to characterize them [1]. Similarly, we obtained a graph-theoretic characterization for the Cayley graphs of cancellative monoids: first, they are now rooted since there is a path from the identity element to any vertex, and then by relaxing the vertex transitivity to the forward vertex-transitivity: all the vertices are accessible-isomorphic i.e. the induced subgraphs by vertex accessibility are isomorphic [1].

To characterize the Cayley graphs of all monoids (not necessarily cancellative), we must weaken the forward vertex-transitivity. We say that a vertex is propagating if there is a homomorphism from its accessible subgraph to the accessible subgraph from any vertex. Thus, the identity of a monoid is a propagating vertex for each of its Cayley graphs. The identity is also an out-simple vertex: it is not source of two edges with the same target. Moreover, any Cayley graph is source-complete: for any label of the graph and from any vertex, there is at least one edge. These properties are sufficient to characterize the Cayley graphs of monoids: they are the deterministic and source-complete graphs with a propagating out-simple root. It follows a graph-theoretic characterization for the Cayley graphs of semigroups (see Theorem 6) and of cancellative semigroups (see Theorem 6).

For the Cayley graphs of commutative monoids, we just have to add the condition that any vertex is locally commutative: for any path from labeled by (two letters) , there is a path from labeled by of the same target. The locally-commutativity can be restricted to a single vertex: the Cayley graphs of commutative monoids are the deterministic and source-complete graphs with a locally commutative propagating out-simple root. It follows a graph-theoretic characterization for the Cayley graphs of semilattices (see Theorem 6). By extending to chains the vertex propagation, we can restrict the vertex-transitivity of a Cayley graph to the existence of a single propagating vertex: the Cayley graphs of (resp. abelian) groups are the deterministic and co-deterministic, simple and connected graphs with a chain-propagating (resp. and locally commutative) source and target-complete vertex.

## 2 Directed labeled graphs

We recall some basic concepts on directed labeled graphs, especially
the vertex-transitivity and the forward vertex-transitivity.

Let be an arbitrary (finite or infinite) set. We denote by
the set of tuples (words) over (the free monoid generated by )
and by the -tuple (the identity element called the
empty word).
A directed -graph is defined by a set of
vertices and a subset of
edges.
Any edge is from the source to the
target with label , and is also written by the
transition or directly if
is clear from the context.
The sources and targets of edges form the set of
non-isolated vertices of and we denote by the set of
edge labels:

and .

We say that is finitely labeled if is finite.
The set is the set of isolated vertices.
From now on, we assume that any graph is without isolated vertex
(i.e. ), hence the graph can be identified with its edge
set . We also exclude the empty graph : every graph is a
non-empty set of labeled edges.
A vertex is an out-simple vertex if there are no two
edges of source with the same target:
.
A graph is simple if all its vertices are out-simple.
We also say that is an in-simple vertex if there are no two
edges with the same source and target :
.
Thus an in-simple vertex for is an out-simple vertex for
the inverse of .
The vertex-restriction of to a set is the
induced subgraph of by :

.

The label-restriction of to a set is the
subset of all its edges labeled in :

.

Let be the unlabeled edge relation i.e.
if for some .
We write for the unlabeled adjacency relation
i.e. for
or .
We denote by the set
of successors of .
We write if there is no edge in from
to i.e.
.
The accessibility relation
is the reflexive and
transitive closure under composition of .
A graph is accessible from if for any
, there is such that .
We denote by the induced subgraph of to the
vertices accessible from which is the greatest subgraph of
accessible from .
A root is a vertex from which is accessible i.e.
also denoted by is equal to
. A co-root of is a root of .
A graph is strongly connected if every vertex is a root:
for all .
A vertex of a graph is an -root if
for any vertex of .
A graph is complete if all its vertices are -roots i.e.
there is an edge between any couple of vertices:
.
An -coroot of is an -root of .

A graph is co-accessible from if
is accessible from .
A graph is connected if is strongly
connected.

A path of length
in a graph is a sequence
of consecutive
edges, and we write for indicating the
source , the target and the label word
of the path; such a path is elementary
if it goes through distinct vertices: and we
write .
We write if and
; we also denote by if
for some .

Let and where
and .
We write if and there
exists paths
and forming an
elementary cycle:
which
is illustrated as follows:

In particular for a loop , we have
and for two edges
and of the same source and goal,
we have .

Recall that a morphism from a graph into a graph is a
mapping from into such that
; we write
.
If, in addition, is bijective and is a morphism, is
called an isomorphism from to ; we write
or directly if we do not specify
an isomorphism, and we say that and are isomorphic.
An automorphism of is an isomorphism from to .
For any deterministic graphs and rooted respectively by
and ,

if and with
and then .

###### Proof.

i) Let us check that for any .

The proof is done by induction on
.

: and .

: There is such that
.
As is a morphism, .

By induction hypothesis and as is a morphism,
.

As is deterministic, we get .

ii) By (i), is injective.
By exchanging with and by (i),
for any .

In particular is surjective. Thus is bijective and
. So .
∎

In Lemma 2, even if and are surjective,
the condition and is necessary.
For instance, the two non-isomorphic graphs below are rooted, deterministic,
co-deterministic, and there is a surjective morphism from one into the other.

Two vertices of a graph are isomorphic
and we write if for some automorphism
of . A graph is vertex-transitive if all the
vertices are isomorphic i.e. for every
.
Two vertices of a graph are accessible-isomorphic
and we write if for some isomorphism
from to .
A graph is forward vertex-transitive if all its vertices are
accessible-isomorphic: for every .
Any vertex-transitive graph is forward vertex-transitive.
The semiline is
forward vertex-transitive but is not vertex-transitive.
Any strongly connected forward vertex-transitive graph is vertex-transitive.

We need to circulate in a graph in the direct and inverse direction of
the arrows.

Let be an injective mapping
of image a
disjoint copy of . This allows to define the graph

in order that and with

deterministic and co-deterministic
deterministic and co-deterministic
source and target-complete
source and target-complete
connected
strongly connected
for any

hence is vertex-transitive if and only if is forward
vertex-transitive.

A path of i.e.
with is a chain of
also denoted by where
means that for any
. Thus

for any

such that for with and
,
where
for any ,
and is the mirror of .

Let us give basic properties on (forward) vertex-transitive graphs.
Any forward vertex-transitive graph is source-complete.

Any vertex-transitive graph is source and target-complete.
The forward vertex-transitivity of a rooted graph is reduced to the
accessible-isomorphism of a root with its successors.
The vertex-transitivity of a connected graph is reduced to the isomorphism of
a vertex with its adjacent vertices.
A graph of root is forward vertex-transitive iff
for any .

A connected graph with a vertex is vertex-transitive
if and only if

for any .

###### Proof.

Let be a graph with a root such that
for any .

Let us check that is forward vertex-transitive i.e.
for any .

The proof is done by induction on for .

For , we have . For , let be a vertex such
that .

By induction hypothesis, we have i.e.
for some isomorphism from to
.
As , there exists such that
and . So .

By hypothesis .
By transitivity of , we get .

We get the second equivalence using the first one for .
∎

## 3 Commutative and propagating graphs

We recall when a graph is deterministic, co-deterministic,
source-complete, target-complete, commutative.
All these notions are equivalent when they are defined globally by paths or
locally by edges.
We introduce the propagation of joined paths which allows to express
differently accessible-isomorphic vertices for deterministic graphs.
The propagation can be restricted to elementary paths for deterministic and
source-complete graphs.
Finally we extend to chains the commutation and the propagation.

A graph is deterministic if there are no two paths with the same source
and label word:

for any and .

This definition coincides with the local property that there are no two edges
with the same source and label:
for any and .

A graph is co-deterministic if its inverse is deterministic:
there are no two paths (resp. edges) with the same target and label word
(resp. label).

A graph is a source-complete graph if for all vertex
and label word , there exists a path from labeled by
: .
Locally a vertex is a source-complete vertex if for any label
there exists such that .
Thus

is source-complete all its vertices are source-complete.

Similarly a vertex of a graph is a
target-complete vertex if is source-complete for
i.e. .
A graph is a target-complete graph if its inverse is
source-complete i.e. all the vertices of are target-complete.

Let us recall the path commutation in a graph.
Let be the binary commutative relation on
defined by for any and
. By reflexivity and transitivity, we extend to
the commutation congruence .
A vertex of a graph is a commutative vertex if
for any
and any such that .
For the following deterministic and source-complete graph:

the vertices and are commutative but is not a
commutative since and .

We say that is a commutative graph if all its vertices are
commutative.
For instance, the following deterministic graph is commutative:

Let us restrict this commutation from paths to edges. We say that a vertex of a graph is a locally commutative vertex if for any and any . The commutation of all vertices may be restricted to the local commutation. A graph is commutative if and only if all its vertices are locally commutative.

###### Proof.

: Any commutative vertex is locally commutative.

: Let a graph whose any vertex is locally
commutative.
Let a path .

We have to check that for any .

By induction on the minimum number of commutations between and ,
we can restrict to i.e. and
for some and .

Thus for some vertices
.

As is locally commutative, we have hence
.
∎

We will now express the accessible-isomorphism of vertices by propagation of
confluent paths. We start with the propagation of loops.
We say that a vertex of a graph is a
loop-propagating vertex when we have the following property:
if has a loop labeled by then any vertex has a loop
labeled by : .
Any locally commutative -root of a deterministic graph is loop-propagating.

###### Proof.

Let be a deterministic graph and be a locally commutative
-root.

Let a loop and a vertex .
As is an -root, there exists such that
.

So .
As is locally commutative, .
As is deterministic, we get .
∎

We extend the propagation of loops to paths.

A vertex is propagating (resp. -propagating) if for
any (resp. ),

.

The restriction of this implication to any with
is the loop-propagating notion.
The restriction of the implication to means that
.

In particular for the graph , the vertex is
not -propagating and is propagating.

For the following deterministic and source-complete graph:

the vertex is propagating but the vertices are not
propagating: we have (resp.
) which is not the case for the other vertices.
All the vertices are -propagating.

For the following two deterministic connected graphs:

the vertices are -propagating (but not propagating) and
is not -propagating.

A propagating vertex of a deterministic graph is a vertex from which there is
a morphism linking it to any vertex.
For any deterministic graph and vertices , we have

for any

if and only if there is a morphism from
to such that .

###### Proof.

: Immediate for any graph.

: As is deterministic, it allows to define the
mapping by

if and
for some .

Thus .
It remains to check that is a morphism.

Let .
There exists such that .

As , we have i.e.
for some vertices .

As is deterministic, and hence
.
∎

The determinism condition in Lemma 3 is necessary:
for the following non deterministic (and non connected) graph :

we have for any
and there is no morphism from into
linking to .
Let us give other basic properties on -propagating vertices.
Any graph with a source-complete -propagating vertex is source-complete.

For any source-complete graph, any out-simple vertex is
-propagating.
Here is a source-complete and deterministic graph without -propagating
vertex.

The existence of a source-complete propagating vertex in a deterministic graph
allows to reduce the commutativity of the graph to the locally commutativity
of the vertex.
Let a deterministic graph with a source-complete and propagating
vertex .

If is locally commutative then is commutative.

###### Proof.

We have to show that is commutative.

Let with .
By Lemma 3, it remains to check that .

By Fact 3, is source-complete.
So for some vertex .

As is locally commutative, .

As and is propagating, we get
for some vertex .

As is deterministic, thus .
∎

When a graph is deterministic, two vertices are accessible-isomorphic means
that each vertex is propagating for the other.
For any deterministic graph and any vertices , we have

.

###### Proof.

: Obvious for any graph .

: Let such that
if and only if for any .

As is deterministic, we define the mappings
and
by

and if
and for some .

As yet done in the proof of Lemma 3, and
are morphisms with and .

By Lemma 2, is an isomorphism from
to i.e. .
∎

A graph is a propagating graph (resp.
-propagating graph) if all its vertices are propagating (resp.
-propagating) i.e. for any (resp. ) and
any , . For the following two connected graphs:

the first graph is propagating but not forward vertex-transitive, and the
second graph is forward vertex-transitive hence propagating.
The -propagating property and the source-complete property coincide for the
simple graphs.
Any -propagating graph is source-complete.

Any -propagating graph with an out-simple vertex is a simple
graph.

Any source-complete and simple graph is -propagating.
Let us restrict the path propagation to elementary paths.

A vertex is an elementary-propagating vertex if for any
,

.

An elementary-propagating graph is a graph whose any vertex is
elementary-propagating.
For instance, the deterministic and simple graph is
elementary-propagating but is not source-complete, hence is not
-propagating.
The elementary-propagating property coincides with the propagating property
for the deterministic and source-complete graphs.
a) A source-complete graph is propagating if and only if

for any and .

b) A deterministic source-complete graph is propagating
iff it is elementary-propagating.

c) A deterministic graph is propagating
if and only if it is forward-vertex transitive.

d) A deterministic graph with a root is a
propagating graph if and only if

for any and .

e) A deterministic target-complete graph with a root
is propagating iff is propagating.

###### Proof.

i) Let us check (a).

: Immediate for any propagating graph.

: Assume that
for any and .

Let us show that for any and .

The proof is done by induction on .

: hence for
any vertex .

: Let with and let
.
We can assume that .

We distinguish the two complemantary cases below.

Case 1 : .
So
for some .

By hypothesis .
By induction hypothesis
hence .

Case 2 : . We distinguish the two subcases below.

Case 2.1 : with
, and .

As is source-complete, for some
vertex .

By induction hypothesis, hence
.

Case 2.2 :
for some with
and .

By hypothesis for some vertex
.

By induction hypothesis hence
.

ii) Let us check (b).

: by (a) for source-complete and
elementary-propagating.

: Let be a deterministic and propagating graph.

Let us check that is elementary-propagating.

Let with and let
.
So and .

As is deterministic and for , the words
and have not the same first letter.

As yet done for (a), there exists and such that
.

So .
For deterministic, hence
.

iii) (c) follows from Lemma 3.

iv) (d) follows from (c) and Lemmas 2 and
3.

v) Let us check (e). Let be a propagating root of a
deterministic and target-complete graph .
We have to show that is propagating.

Let with .
We have to check that .

As is a root, for some .

As is target complete, there exists a vertex such
that .

As and is propagating,
.
As is deterministic, .
∎

Note that the determinism condition in
Proposition 3 (b) and (c) is necessary.
For instance the following propagating and non deterministic graph:

is not elementary-propagating (and not forward vertex-transitive) since
but .

Note also that the rooted condition in
Proposition 3 (e) is necessary.
For instance, the inverse semiline
is deterministic,
target-complete, and is not a propagating graph while is a propagating
co-root.

Let us generalize to the chains the path commutation in a graph.

A vertex is a chain-commutative vertex for a graph if
is commutative for i.e.

for any and any
such that .

We say that is a chain-commutative graph if
is a commutative graph i.e. all the vertices are
chain-commutative for .

As in Lemma 3, we can express locally the chain-commutation.
A vertex is a locally chain-commutative vertex for a graph
if is locally commutative for i.e.
for any and .
Let us apply Lemma 3.
A graph is chain-commutative iff all its vertices are locally
chain-commutative.
Any chain-commutative graph is commutative and the converse is true when the
graph is deterministic, co-deterministic, source and target-complete.
Let be a deterministic, co-deterministic, source and target-complete
graph.

If is commutative then is chain-commutative.

###### Proof.

Let where .
By Corollary 3, it remains to check that
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