On Cayley graphs of algebraic structures

03/22/2018 ∙ by Didier Caucal, et al. ∙ 0

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.

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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