On Cayley graphs of basic algebraic structures

by   Didier Caucal, et al.

We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.



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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 relationi.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  .


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  .


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.


 : 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.


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  .


 :  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.


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


 :  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.


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.


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