On Cayley graphs of algebraic structures

TL;DR

This paper provides simple graph-theoretic characterizations of Cayley graphs for various algebraic structures, including monoids, groups, and quasigroups, and demonstrates their effectiveness for finite degree end-regular graphs.

Contribution

It introduces new characterizations of Cayley graphs for multiple algebraic structures and applies them to finite degree end-regular graphs.

Findings

Characterizations are effective for finite degree end-regular graphs.

Unified approach for various algebraic structures.

Simplifies understanding of Cayley graph structures.

Abstract

We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.

Full text

1 Introduction

To describe the structure of a group, Cayley introduced in 1878 [Ca] the concept of graph for any group  $(G,\cdot)$  according to any generating subset  $S$. This is simply the set of labeled oriented edges  g\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{s}}\ g{\cdot}s  for every  $g$  of  $G$  and  $s$  of  $S$. Such a graph, called Cayley graph, is directed and labeled in  $S$  (or an encoding of  $S$  by symbols called letters or colors). The study of groups by their Cayley graphs is a main topic of algebraic graph theory [Bi, GR, BW]. A characterization of unlabeled and undirected Cayley graphs was given by Sabidussi in 1958 [Sa] : 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? [Ha]. 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  $S$  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 [MS] 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  $G$  according to any subset  $S$  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  $A$  be an arbitrary (finite or infinite) set. A directed  $A$-graph  $(V,G)$  is defined by a set  $V$  of vertices  and a subset  G\,\subseteq\,V{\mbox{\scriptsize\times}}A{\mbox{\scriptsize\times}}V  of edges. Any edge  $(s,a,t)\in G$  is from the source  $s$  to the target  $t$  with label  $a$, and is also written by the transition  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ t  or directly  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ t  if  $G$  is clear from the context. The sources and targets of edges form the set  $V_{G}$  of non-isolated vertices  of  $G$  and we denote by  $A_{G}$  the set of edge labels:

      V_{G}\ =\ \{\ s\ |\ \exists\ a,t\ \ (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ t\,\vee\,t\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ s)\ \}    and    A_{G}\ =\ \{\ a\ |\ \exists\ s,t\ \ (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ t)\ \}.

Thus  $V-V_{G}$  is the set of isolated vertices. From now on, we assume that any graph  $(V,G)$  is without isolated vertex  (i.e. $V=V_{G}$), hence the graph can be identified with its edge set  $G$. We also exclude the empty graph  $\emptyset$ : every graph is a non-empty set of labeled edges. For instance  \Upsilon\ =\ \{\ s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{n}}\ s+n\ |\ s\in\mbox{\mathbb{R}}\,\wedge\,n\in\mbox{\mathbb{Z}}\ \}  is a graph of vertex set  $\mathbb{R}$  and of label set  $\mathbb{Z}$. As any graph  $G$  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:  (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ t\,\wedge\,s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{b}}\ t)\ \Longrightarrow\ a=b. We say that  $G$  is finitely labeled  if  $A_{G}$  is finite. We denote by  $G^{-1}\ =\ \{\ (t,a,s)\ |\ (s,a,t)\in G\ \}$  the inverse  of  $G$. A graph is deterministic  if there are no two edges with the same source and label:  (r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ s\,\wedge\,r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ t)\ \Longrightarrow\ s=t. A graph is co-deterministic  if its inverse is deterministic: there are no two edges with the same target and label:  (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ r\,\wedge\,t\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ r)\ \Longrightarrow\ s=t. For instance, the graph  $\Upsilon$  is simple, not finitely labeled, deterministic and co-deterministic. A graph  $G$  is complete  if there is an edge between any couple of vertices: \forall\ s,t\in V_{G}\ \ \exists\ a\in A_{G}\ (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ t). A graph  $G$  is source-complete  if for all vertex  $s$  and label  $a$, there is an  $a$-edge from  $s$ : \forall\ s\in V_{G}\ \ \forall\ a\in A_{G}\ \ \exists\ t\ (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ t). A graph is target-complete  if its inverse is source-complete:  \forall\ t\in V_{G}\ \ \forall\ a\in A_{G}\ \ \exists\ s\ (s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ t). For instance,  $\Upsilon$  is source-complete, target-complete but not complete. Another example is given by the graph  ${\rm Even}\ =\ \{(p,a,q)\ ,\ (p,b,p)\ ,\ (q,a,p)\ ,\ (q,b,q)\}$  represented as follows:

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

The vertex-restriction  $G_{|P}$  of  $G$  to a set  $P$  is the induced subgraph of  $G$  by  $P\cap V_{G}$ :

                                    $G_{|P}\ =\ \{\ (s,a,t)\in G\ |\ s,t\in P\ \}$.

The label-restriction  $G^{|P}$  of  $G$  to a set  $P$  is the subset of all its edges labeled in  $P$ :

                                    $G^{|P}\ =\ \{\ (s,a,t)\in G\ |\ a\in P\ \}$.

Let  \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}_{G}  be the unlabeled edge relation  i.e.  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}_{G}\ t  if  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ t  for some  $a\in A$. We denote by  \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}_{G}(s)\,=\,\{\ t\ |\ s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}_{G}\ t\ \}  the set of successors  of  $s\in V_{G}$ . We write  s\ \mbox{\ {\tiny/}\mathop{\mbox{\small$\longrightarrow$}}\limits_{\mbox{\tiny$$}}}\,\!_{G}\ t  if there is no edge in  $G$  from  $s$  to  $t$  i.e.  G\,\cap\,\{s\}{\mbox{\scriptsize\times}}A{\mbox{\scriptsize\times}}\{t\}\,=\,\emptyset. The accessibility  relation  \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{*}_{G}\,=\,\bigcup_{n\geq 0}\mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{n}_{G}  is the reflexive and transitive closure under composition of  \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}_{G} . A graph  $G$  is accessible  from  $P\subseteq V_{G}$  if for any  $s\in V_{G}$ , there is  $r\in P$  such that  r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{*}_{G}\ s. We denote by  $G_{{\downarrow}P}$  the induced subgraph of  $G$  to the vertices accessible from  $P$  which is the greatest subgraph of  $G$  accessible from  $P$. For instance  \Upsilon_{{\downarrow}\{0\}}\ =\ \{\ m\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{n}}\ m+n\ |\ m,n\in\mbox{\mathbb{Z}}\ \}  is a complete subgraph of  $\Upsilon$. A root  $r$  is a vertex from which  $G$  is accessible  i.e.  $G_{{\downarrow}\{r\}}$  also denoted by  $G_{{\downarrow}r}$  is equal to  $G$. A graph  $G$  is strongly connected  if every vertex is a root:  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{*}_{G}\ t   for all  $s,t\in V_{G}$ . A graph  $G$  is co-accessible  from  $P\subseteq V_{G}$  if  $G^{-1}$  is accessible from  $P$. We denote by  d_{G}(s,t)\,=\,{\rm min}\{\ n\ |\ s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{n}_{G\,\cup\,G^{-1}}\ t\ \}  the distance  between  $s,t\in V_{G}$  with  min$(\emptyset)=\omega$. A graph  $G$  is connected  if  $G\,\cup\,G^{-1}$  is strongly connected  i.e.  d_{G}(s,t)\in\mbox{\mathbb{N}}   for any  $s,t\in V_{G}$ . Recall that a connected component  of a graph  $G$  is a maximal connected subset of  $G$ ;  we denote by  ${\rm Comp}(G)$  the set of connected components of  $G$. A representative set  of  Comp$(G)$  is a vertex subset  $P\subseteq V_{G}$  having exactly one vertex in each connected component:  $|P\,\cap\,V_{C}|=1$  for any  $C\in{\rm Comp}(G)$ ;  it induces the canonical mapping  \pi_{P}:V_{G}\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}\ P  associating with each vertex  $s$  the vertex of  $P$  in the same connected component:  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{*}_{G\,\cup\,G^{-1}}\ \pi_{P}(s)  for any  $s\in V_{G}$ . For instance,  $[0,1[$  is a representative set of  ${\rm Comp}(\Upsilon)$  and its canonical mapping is defined by  $\pi_{[0,1[}(x)\,=\,x-{\lfloor}x{\rfloor}$  for any  x\in\mbox{\mathbb{R}}.

A path  $(s_{0},a_{1},s_{1},\ldots,a_{n},s_{n})$  of length  $n\geq 0$  in a graph  $G$  is a sequence  s_{0}\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a_{1}}}\ s_{1}\ldots\mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a_{n}}}\ s_{n}  of  $n$  consecutive edges, and we write  s_{0}\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a_{1}{\ldots}a_{n}}}\ s_{n}  for indicating the source  $s_{0}$ ,  the target  $s_{n}$  and the label word  $a_{1}{\ldots}a_{n}\in A_{G}^{*}$  of the path where  $A_{G}^{*}$  is the set of words over  $A_{G}$  (the free monoid generated by  $A_{G}$)  and  $\varepsilon$  is the empty word (the identity element).

Recall that a morphism  from a graph  $G$  into a graph  $H$  is a mapping  $h$  from  $V_{G}$  into  $V_{H}$  such that  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ t\ \Longrightarrow\ h(s)\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{H}\ h(t). If, in addition  $h$  is bijective and  $h^{-1}$  is a morphism, $h$  is called an isomorphism  from  $G$  to  $H$ ;  we write  $G\,\equiv_{h}\,H$  or directly  $G\,\equiv\,H$  if we do not specify an isomorphism, and we say that  $G$  and  $H$  are isomorphic. An automorphism  of  $G$  is an isomorphism from  $G$  to  $G$. Two vertices  $s,t$  of a graph  $G$  are isomorphic  and we write  $s\,\simeq_{G}t$  if  $t=h(s)$  for some automorphism  $h$  of  $G$.

A graph  $G$  is symmetric (or vertex-transitive)  if all its vertices are isomorphic:  $s\,\simeq_{G}t$  for every  $s,t\in V_{G}$ . For instance, the previous graphs  $\Upsilon$  and  Even  are symmetric.

Two vertices  $s,t$  of a graph  $G$  are accessible-isomorphic  and we write  $s\downarrow_{G}t$  if  $t=h(s)$  for some isomorphism  $h$  from  $G_{{\downarrow}s}$  to  $G_{{\downarrow}t}$ . A graph  $G$  is arc-symmetric  if all its vertices are accessible-isomorphic:  $s\downarrow_{G}t$  for every  $s,t\in V_{G}$ .

Fact 1**.**

Any symmetric graph is arc-symmetric which is source-complete.

For instance  \Upsilon^{|\{1\}}_{|\hbox{\scriptsize\rm I\hskip-1.5ptN}}\ =\ \{\ n\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{1}}\ n+1\ |\ n\in\mbox{\mathbb{N}}\ \}  is arc-symmetric but not symmetric. On the other hand  \Upsilon^{|\{-1\}}_{|\hbox{\scriptsize\rm I\hskip-1.5ptN}}\ =\ \{\ n\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{-1}}\ n-1\ |\ n\in\mbox{\mathbb{N}}\ \}  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 LABEL:LeftMonoid), of cancellative monoids (Theorem LABEL:Monoid), of cancellative semigroups (Theorem LABEL:CanSemigroup).

A magma  (or groupoid) is a set  $M$  equipped with a binary operation  \cdot\,:M{\mbox{\scriptsize\times}}M\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}\ M  that sends any two elements  $p,q\in M$  to the element  $p\cdot q$.

Given a subset  $Q\subseteq M$  and an injective mapping  \hbox{{\rm[}\hskip-2.0pt{\rm[}{\ }{\rm]}\hskip-2.0pt{\rm]}}\,:Q\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}\ A, we define the graph

                               {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,Q}{\rm]}\hskip-2.0pt{\rm]}}\ =\ \{\ p\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{\footnotesize\hbox{{\rm[}\hskip-1.5pt{\rm[}{$q$}{\rm]}\hskip-1.5pt{\rm]}}}}\ p\cdot q\ |\ p\in M\,\wedge\,q\in Q\ \}

which is called a generalized Cayley graph  of  $M$. It is of vertex set  $M$  and of label set  \hbox{{\rm[}\hskip-2.0pt{\rm[}{Q}{\rm]}\hskip-2.0pt{\rm]}}\,=\,\{\ \hbox{{\rm[}\hskip-2.0pt{\rm[}{q}{\rm]}\hskip-2.0pt{\rm]}}\ |\ q\in Q\ \}. We denote  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,Q}{\rm]}\hskip-2.0pt{\rm]}}  by  ${\cal C}(M,Q)$  when  [[  ]]  is the identity. For instance  \Upsilon\,=\,{\cal C}(\mbox{\mathbb{R}},\mbox{\mathbb{Z}})  for the magma  (\mbox{\mathbb{R}},+). We also write  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M}{\rm]}\hskip-2.0pt{\rm]}}  instead of  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,M}{\rm]}\hskip-2.0pt{\rm]}}  and  {\cal C}(M)\,=\,{\cal C}(M,M)\,=\,\{\ p\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{q}}\ p\cdot q\ |\ p,q\in M\ \}.

Fact 2**.**

Any generalized Cayley graph is deterministic and source-complete.

For instance taking the magma  (\mbox{\mathbb{Z}},-)  and  \hbox{{\rm[}\hskip-2.0pt{\rm[}{-1}{\rm]}\hskip-2.0pt{\rm]}}=a, {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{\mbox{$\mathbb{Z}$},{-1}}{\rm]}\hskip-2.0pt{\rm]}}\ =\ \{\ n\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ n+1\ |\ n\in\mbox{\mathbb{Z}}\ \}. By adding  \hbox{{\rm[}\hskip-2.0pt{\rm[}{1}{\rm]}\hskip-2.0pt{\rm]}}=b,  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{\mbox{$\mathbb{Z}$},{1,-1}}{\rm]}\hskip-2.0pt{\rm]}}\,=\,\{\ n\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}\ n+1\ |\ n\in\mbox{\mathbb{Z}}\ \}\,\cup\,\{\ n\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{b}}\ n-1\ |\ n\in\mbox{\mathbb{Z}}\ \}.

We say that a magma  $(M,\cdot)$  is left-cancellative  if  $r\cdot p=r\cdot q\ \Longrightarrow\ p=q$  for any  $p,q,r\in M$.

Similarly  $(M,\cdot)$  is right-cancellative  if  $p\cdot r=q\cdot r\ \Longrightarrow\ p=q$  for any  $p,q,r\in M$.

A magma is cancellative  if it is both left-cancellative and right-cancellative.

Fact 3**.**

*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  $(M,\cdot)$  is a semigroup  if  $\cdot$  is associative:  $(p\cdot q)\cdot r\,=\,p\cdot(q\cdot r)$  for any  $p,q,r\in M$. A monoid  $(M,\cdot)$  is a semigroup with an identity element  $1$ :  $1{\cdot}p\,=\,p{\cdot}1\,=\,p$  for all  $p\in M$. The submonoid generated  by  $Q\subseteq M$  is the least submonoid  $Q^{*}\,=\,\{\ q_{1}{\cdot}\ldots{\cdot}q_{n}\mid n\geq 0\,\wedge\,q_{1},\ldots,q_{n}\in Q\ \}$  containing  $Q$.

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

Proposition 3.1**.**

Any generalized Cayley graph of a left-cancellative monoid is arc-symmetric.

Proof 3.2**.**

*Let  G\,=\,{\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,Q}{\rm]}\hskip-2.0pt{\rm]}}  for some left-cancellative monoid  $(M,\cdot)$  and some  $Q\subseteq M$.

Let  $r\in M$. We have to check that  $1\downarrow_{G}r$.

By induction on  $n\geq 0$  and for any  $q_{1},\ldots,q_{n}\in Q$  and  $s\in M$, we have

                              r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{\tiny\hbox{{\rm[}\hskip-3.15pt{\rm[}{$q_{1}$}{\rm]} \hskip-3.5pt{\rm]}}\ldots\tiny\hbox{{\rm[}\hskip-3.15pt{\rm[}{$q_{n}$}{\rm]} \hskip-3.5pt{\rm]}}}}_{G}\ s\ \ \Longleftrightarrow\ \ s\,=\,({\ldots}(r{\cdot}q_{1}){\ldots}){\cdot}q_{n} .

As  $\cdot$  is associative, we get V_{G_{{\downarrow}r}}\,=\,\{\ s\mid r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}^{*}_{G}\ s\ \}\,=\,r{\cdot}Q^{*}. In particular  $V_{G_{{\downarrow}1}}\,=\,Q^{*}$.

We consider the mapping  f_{r}:\,M\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}\ M  defined by  $f_{r}(p)=r{\cdot}p$  for any  $p\in M$.

As  $\cdot$  is left-cancellative,  $f_{r}$  is injective.

Furthermore  $f_{r}$  is an isomorphism on its image: for any  $p,q,p^{\prime}\in M$,

                              p\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{\tiny\hbox{{\rm[}\hskip-3.15pt{\rm[}{$q$}{\rm]} \hskip-3.5pt{\rm]}}}}_{G}\ p^{\prime}\ \ \Longleftrightarrow\ \ f_{r}(p)\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{\tiny\hbox{{\rm[}\hskip-3.15pt{\rm[}{$q$}{\rm]} \hskip-3.5pt{\rm]}}}}_{G}\ f_{r}(p^{\prime}).

The associativity of  $\cdot$  gives the necessary condition.

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

Thus  $f_{r}$  restricted to  $Q^{*}$  is an isomorphism from  $G_{{\downarrow}1}$  to  $G_{{\downarrow}r}$  hence  $1\downarrow_{G}\,r$. ∎

We can not generalize Proposition 3.1 to the left-cancellative semigroups. For instance the semigroup  $M=\{a,b\}$  with  $x{\cdot}y=y$  for any  $x,y\in M$  is left-cancellative but the graph  ${\cal C}(M)$  represented below is not arc-symmetric.*

A monoid Cayley graph  is a generalized Cayley graph  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,Q}{\rm]}\hskip-2.0pt{\rm]}}  of a monoid  $M$  generated by  $Q$  which means that the identity element  $1$  is a root of  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,Q}{\rm]}\hskip-2.0pt{\rm]}}.

Fact 4**.**

A monoid  $M$  is generated by  $Q$   $\Longleftrightarrow$   $1$  is a root of  {\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{M,Q}{\rm]}\hskip-2.0pt{\rm]}}.

*Under additional simple conditions, let us establish the converse of Proposition 3.1.

For any graph  $G$  and any vertex  $r$, we introduce the path-relation  ${\rm Path}_{G}(r)$  as the ternary relation on  $V_{G}$  defined by

         $(s,t,x)\in{\rm Path}_{G}(r)$  if there exists  $u\in A_{G}^{*}$  such that  r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{u}}_{G}\ t  and  s\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{u}}_{G}\ x.

If for any  $s,t\in V_{G}$  there exists a unique  $x$  such that  $(s,t,x)\in{\rm Path}_{G}(r)$, we denote by  \ast_{r}:V_{G}{\mbox{\scriptsize\times}}V_{G}\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}\ V_{G}  the binary path-operation  on  $V_{G}$  defined by  $(s,t,s\ast_{r}t)\in{\rm Path}_{G}(r)$  for any  $s,t\in V_{G}$ ; we also write  ${}_{G}\ast_{r}$  when we need to specify  $G$. Let us give conditions so that this path-operation exists and is associative and left-cancellative.*

Proposition 3.3**.**

*Let  $r$  be a root of a deterministic and arc-symmetric graph  $G$.

  Then  $(V_{G},\ast_{r})$  is a left-cancellative monoid of identity  $r$  and generated by  \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits}_{G}(r).

  If  $G$  is co-deterministic  then  $\ast_{r}$  is cancellative.

  If  $G$  is simple  then  G\,=\,{\cal C}\hbox{{\rm[}\hskip-2.0pt{\rm[}{V_{G},\mbox{\mathop{\mbox{\small\longrightarrow}}\limits}_{G}(r)}{\rm]}\hskip-2.0pt{\rm]}}  with  \hbox{{\rm[}\hskip-2.0pt{\rm[}{s}{\rm]}\hskip-2.0pt{\rm]}}\,=\,a  for any  r\ \mbox{\mathop{\mbox{\small$\longrightarrow$}}\limits^{a}}_{G}\ s.*