## 1 Introduction

To describe the structure of a group, Cayley introduced in 1878 [9] the concept of graph for any group according to any generating subset . This is simply 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). The study of groups by their Cayley graphs is a main topic of algebraic graph theory [3, 10, 2]. A characterization of unlabeled and undirected Cayley graphs was given by Sabidussi in 1958 [17] : 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. However, this algebraic characterization is not well suited for deciding whether a possibly infinite graph is a Cayley graph. It is pertinent to look for characterizations by graph-theoretic conditions. This approach was clearly stated by Hamkins in 2010: Which graphs are Cayley graphs? [12]. In this paper, we present simple graph-theoretic characterizations of Cayley graphs for firstly left-cancellative and cancellative monoids, and then for groups. These characterizations are then extended to any subset of left-cancellative magmas, left-quasigroups, quasigroups, and groups. Finally, we show that these characterizations are effective for the end-regular graphs of finite degree [15] which are the graphs finitely decomposable by distance from a(ny) vertex or equivalently are the suffix transition graphs of labeled word rewriting systems.

Let us present the main structural characterizations starting with the Cayley graphs of left-cancellative monoids. Among many properties of these graphs, we retain only three basic ones. First and by definition, any Cayley graph is deterministic: there are no two arcs of the same source and label. Furthermore, the left-cancellative condition implies that any Cayley graph is simple: there are no two arcs of the same source and goal. Finally, any Cayley graph is rooted: there is a path from the identity element to any vertex. To these three necessary basic conditions is added a structural property, called arc-symmetric: all the vertices are accessible-isomorphic i.e. the induced subgraphs by vertex accessibility are isomorphic. These four properties characterize the Cayley graphs of left-cancellative monoids. To describe exactly the Cayley graphs of cancellative monoids, we just have to add the co-determinism: there are no two arcs of the same target and label. This characterization is strengthened for the Cayley graphs of groups using the same properties but expressed in both arc directions: these are the graphs that are connected, deterministic, co-deterministic, and symmetric: all the vertices are isomorphic.

We also consider the Cayley graph of a magma according to any subset and that we called generalized. The characterizations obtained require the assumption of the axiom of choice. First, a graph is a generalized Cayley graph of a left-cancellative magma if and only if it is deterministic, simple, source-complete: for any label of the graph and from any vertex, there is at least one edge. This equivalence does not require the axiom of choice for finitely labeled graphs, and in this case, these graphs are also the generalized Cayley graphs of left-quasigroups. Moreover, a finitely labeled graph is a generalized Cayley graph of a quasigroup if and only if it is also co-deterministic and target-complete: for any label of the graph and to any vertex, there is at least one edge. We also characterize all the generalized Cayley graphs of left-quasigroups, and of quasigroups. Finally, a graph is a generalized Cayley graph of a group if anf only if it is simple, symmetric, deterministic and co-deterministic.

## 2 Directed labeled graphs

We consider directed labeled graphs without isolated vertex.
We recall some basic concepts such as determinism, completeness and symmetry.
We introduce the notions of accessible-isomorphic vertices and arc-symmetric
graph.

Let be an arbitrary (finite or infinite) set.
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 .

Thus 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.
For instance is a graph of vertex set and of label set
.
As any graph is a set, there are no two edges with the same source,
target and label. We say that a graph is simple if there are no two
edges with the same source and target:
.
We say that is finitely labeled if is finite.
We denote by the
inverse of .
A graph is deterministic if there are no two edges with the same source
and label: . A graph is co-deterministic if its inverse is deterministic:
there are no two edges with the same target and label:
.
For instance, the graph is simple, not finitely labeled,
deterministic and co-deterministic.
A graph is complete if there is an edge between any couple of
vertices: .
A graph is source-complete if for all vertex
and label , there is an -edge from : .
A graph is target-complete if its inverse is source-complete:
.
For instance, is source-complete, target-complete but not
complete. Another example is given by the graph
represented as follows:

It is simple, deterministic, co-deterministic, complete, source-complete and
target-complete.

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 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 .
For instance is a complete subgraph of
.
A root is a vertex from which is accessible i.e.
also denoted by is equal to
.
A graph is strongly connected if every vertex is a root:
for all .
A graph is co-accessible from if
is accessible from .
We denote by the
distance between with min.
A graph is connected if is strongly
connected i.e. for any .
Recall that a connected component of a graph is a maximal
connected subset of ; we denote by the set of
connected components of .
A representative set of Comp is a vertex subset
having exactly one vertex in each connected component:
for any ; it induces the
canonical mapping associating with each
vertex the vertex of in the same connected component:
for any .
For instance, is a representative set of
and its canonical mapping is defined by
for any .

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 where is the set of
words over (the free monoid generated by ) and
is the empty word (the identity element).

Recall that a morphism from a graph into a graph is a
mapping from into such that
.
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 .
Two vertices of a graph are isomorphic
and we write if for some automorphism
of .

A graph is symmetric (or vertex-transitive) if all its
vertices are isomorphic: for every .
For instance, the previous graphs and Even are symmetric.

Two vertices of a graph are accessible-isomorphic
and we write if for some isomorphism
from to .
A graph is arc-symmetric if all its vertices are
accessible-isomorphic: for every .
Any symmetric graph is arc-symmetric which is source-complete.
For instance is arc-symmetric but not
symmetric.
On the other hand is not arc-symmetric: two
distinct vertices are not accessible-isomorphic.

## 3 Cayley graphs of left-cancellative and cancellative monoids

We present graph-theoretic characterizations for the Cayley graphs of
left-cancellative monoids (Theorem 3), of cancellative monoids
(Theorem 3), of cancellative semigroups (Theorem 3).

A magma (or groupoid) is a set equipped with a binary
operation that sends any two elements
to the element .

Given a subset and an injective mapping
, we define the graph

which is called a generalized Cayley graph of .
It is of vertex set and of label set
.
We denote by when
[[]] is the identity.
For instance for the magma
.
We also write instead of and
.
Any generalized Cayley graph is deterministic and source-complete.
For instance taking the magma and
, .
By adding ,
.

We say that a magma is left-cancellative if
for any .

Similarly is right-cancellative if
for any .

A magma is cancellative if it is both left-cancellative and
right-cancellative.
Any generalized Cayley graph of a left-cancellative magma is simple.

Any generalized Cayley graph of a right-cancellative magma is
co-deterministic.
Recall also that is a semigroup if is
associative: for any
.
A monoid is a semigroup with an identity element
: for all .
The submonoid generated by is the least submonoid
containing .

When a monoid is left-cancellative, its generalized Cayley graphs are
arc-symmetric.
Any generalized Cayley graph of a left-cancellative monoid is
arc-symmetric.

###### Proof.

Let for some left-cancellative monoid
and some .

Let . We have to check that .

By induction on and for any and
, we have

.

As is associative, we get .
In particular .

We consider the mapping defined by
for any .

As is left-cancellative, is injective.

Furthermore is an isomorphism on its image: for any ,

.

The associativity of gives the necessary condition.

The associativity and the left-cancellative property of gives the
sufficient condition.

Thus restricted to is an isomorphism from
to hence
.
∎

We can not generalize Proposition 3 to the left-cancellative
semigroups. For instance the semigroup with
for any is left-cancellative but the graph
represented below is not arc-symmetric.

A monoid Cayley graph is a generalized Cayley graph
of a monoid generated by which means
that the identity element is a root of .
A monoid is generated by is a root of .
Under additional simple conditions, let us establish the converse of
Proposition 3.

For any graph and any vertex , we introduce the
path-relation as the ternary relation on
defined by

if there exists
such that and .

If for any there exists a unique such that
, we denote by
the binary path-operation
on defined by for any
; we also write when we need to specify .
Let us give conditions so that this path-operation exists and is associative
and left-cancellative.
Let be a root of a deterministic and arc-symmetric graph .

Then is a left-cancellative monoid of
identity and generated by .

If is co-deterministic then is
cancellative.

If is simple then with for any .

###### Proof.

i) Let . Let us check that there is a unique
such that .

As is a root, there exists such that .

As is source-complete, there exists such that
. Hence .

Let . There exists such that
and .

As is arc-symmetric, we have and as is
deterministic, we get .

As is deterministic, it follows that .

Thus exists and is denoted by in the rest of this
proof.

Let us show that is a left-cancellative monoid.

ii) Let us show that is associative.

Let .
We have to check that .

As is a root, there exists such that
and .

By (i), and

So hence .

As is deterministic, we get
.

iii) Let us check that is an identity element.

Let . As , we get
i.e. .

For , we have .
As is deterministic, we get .

iv) Let us check that is left-cancellative.
Let such that .

There exists such that and
.

So and .
As and , we get
.

As is deterministic, we have .

v) Let us check that is a generating
subset of . Let .

There exists , and
such that
.

By Fact 2, there exists such
that .

For every , hence
.

vi) Assume that is co-deterministic.
Let us check that is right-cancellative.

Let such that .

There exists such that .
So and .

As is co-deterministic, we get .

vii) Assume that is simple.
Let .

As is simple and deterministic, we define the following injection
[[]] from into by

for .

Let . Let us show that .

: Let .
As , there exists such that
.

So .
As is deterministic, .

Furthermore and .
So .

: Let .
So for some .

Thus and .
So .
∎

For instance let us consider a graph of the following representation:

It is a skeleton of the graph of where is the
successor and goes to the next limit ordinal:
is isomorphic to .
By Proposition 3, it is a Cayley graph of a left-cancellative
monoid. Precisely to each word , we associate the unique vertex
accessible from the root by the path labeled by
. Thus

.

By Proposition 3, is a
left-cancellative monoid where for any ,

and we have with
and .

Propositions 3 and 3 give a graph-theoretic
characterization of the Cayley graphs of left-cancellative monoids.
A graph is a Cayley graph of a left-cancellative monoid if and only if

it is rooted, simple, deterministic and arc-symmetric.

###### Proof.

We obtain the necessary condition by Proposition 3 with
Facts 3,3,3.

The sufficient condition is given by Proposition 3.
∎

We can restrict Theorem 3 to cancellative monoids.
A graph is a Cayley graph of a cancellative monoid if and only if

it is rooted, simple, deterministic, co-deterministic,
arc-symmetric.

###### Proof.

: By Theorem 3 and Fact 3.

: By Proposition 3.
∎

The previous graph is not co-deterministic hence, by Theorem 3
or Fact 3, is not a Cayley graph of a cancellative monoid.
On the other hand and according to Proposition 3, a
quater-grid of the following representation:

is a Cayley graph of a cancellative monoid. Precisely and as for the previous
graph, we associate to each word the unique vertex
accessible from the root by the path labeled by .
By Proposition 3, is a
cancellative monoid where

for any .

and we have with
and .

Recall that a Cayley graph of a semigroup is a generalized Cayley
graph such that whose
is the subsemigroup generated by .
Theorem 3 can be easily extended into a characterization of the
Cayley graphs of cancellative semigroups. Indeed, a semigroup without an
identity is turned into a monoid by just adding an identity.
Precisely a monoid-completion of a semigroup
is defined by if has an identity element,
otherwise whose is an
identity element of :
for any . This natural completion does not preserve the
left-cancellative property but it preserves the cancellative property.
Any monoid-completion of a cancellative semigroup is a cancellative monoid.

###### Proof.

Let be a monoid-completion of a
cancellative semigroup without an identity element.

i) Suppose there are such that .

In this case, let us check that is an identity element.

We have .
As is left-cancellative, we get .

Let . So .
As is right-cancellative, we get .

Finally .
As is left-cancellative, we get .

ii) By hypothesis has no identity element. By (i), there are no
such that .
Let us show that is left-cancellative.

Let for some .
Let us check that .

As is left-cancellative, we only have to consider the case where
.

If then .

Otherwise and . By (i), we get .

iii) Similarly there are no such that
hence remains also
right-cancellative.
∎

Let us translate the monoid-completion of cancellative semigroups into their
Cayley graphs.
A root-completion of a graph is a graph
defined by if is rooted, otherwise
and is the root of
; we say that is rootable
into . For instance the following non connected graph:

is arc-symmetric but is not rootable into an arc-symmetric graph. On the other hand, a graph consisting of two (isomorphic) deterministic and source-complete trees over is rootable into a deterministic source-complete tree over . Finally the following graph:

is also rootable into a simple, deterministic, co-deterministic,
arc-symmetric graph.
We can apply Theorem 3.
A graph is a Cayley graph of a cancellative semigroup if and only if

it is rootable into a simple, deterministic, co-deterministic,
arc-symmetric graph.

###### Proof.

: Let for some
cancellative semigroup and some generating subset of
i.e. . We have the following two complementary cases.

Case 1 : has an identity element.
By Theorem 3, is rooted, simple, arc-symmetric,
deterministic and co-deterministic.
As has a root, it is rootable into itself.

Case 2 : is not a monoid.
Let be a monoid-completion
of .

By Lemma 3, , remains cancellative.
Furthermore . Let

.

By Theorem 3,
is rooted, simple, arc-symmetric, deterministic and
co-deterministic.
Moreover is rootable into .

: Let a graph rootable into a simple,
deterministic, co-deterministic, arc-symmetric graph .
We have the following two complementary cases.

Case 1 : is rooted. By Theorem 3
(or Proposition 3), is a Cayley graph of a cancellative
monoid.

Case 2 : has no root.
Let be the root of and
.

So and .

By Proposition 3,
for the associative
and cancellative path-operation on of
identity element with generated by .

As is not the target of an edge of and by
definition, remains an internal operation on i.e.
for any .

Finally and
is a cancellative semigroup.
∎

For instance by Theorem 3, the previous graph is a Cayley
graph of a cancellative semigroup. It is isomorphic to

.

We have with and
for the following associative and cancellative
path-operation defined by

for any
.

We can now restrict Theorem 3 to the Cayley graphs of groups.

## 4 Cayley graphs of groups

We present a graph-theoretic characterization for the Cayley graphs of groups:
they are the deterministic, co-deterministic, symmetric, simple and
connected graphs (Theorem 4).
By removing the connectivity condition and under the assumption of the axiom
of choice, we get a characterization for the generalized Cayley graphs of
groups (Theorem 4).

Recall that a group is a monoid whose each element has an inverse : . So is strongly connected hence by Proposition 3 is symmetric. Any generalized Cayley graph of a group is symmetric.

###### Proof.

Let be a group and be an
injective mapping.

By Proposition 3, is
arc-symmetric.

As is strongly connected, is
symmetric.

For any ,
remains
symmetric.
∎

We start by considering the monoid Cayley graphs of a group which are
the generalized Cayley graph with .
Any monoid Cayley graph of a group is strongly connected.

###### Proof.

Let for some group and some
with .

Let .
We have to check that .

There exists and such that
.
So .

For any , we have
for some
and .

Thus for .
∎

Let us complete Proposition 3 in the case where the graph is
symmetric. In this case, the path-operation is also invertible.
For any root of a deterministic and symmetric graph ,

is a group.

###### Proof.

It suffices to complete the proof of Proposition 3 when
is in addition symmetric.

Let . Let us show that has an inverse.

There exists such that .

As , is also a root hence there exists such
that .

Let be the vertex such that .
So

and .

As

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