On undirected Cayley graphs

We determine all periodic (and, therefore, all finite) semigroups G for which there exists a non-empty subset S of G such that the Cayley graph of G relative to S is an undirected Cayley graph. Let G be a semigroup, and let S be a nonempty subset of G. The Cayley graph Cay(G, S) of G relative to S is defined as the graph with vertex set G and edge set E(S) consisting of those ordered pairs (x, y) such that sx = y for some s ∈ S. Cayley graphs of groups are significant both in group theory and in constructions of interesting graphs with nice properties. They have received serious attention in the literature (see, in particular, [1], [2], [5]). The Cayley graph of a semigroup has been introduced by Bohdan Zelinka [9]. In the investigation of the Cayley graphs of semigroups it is first of all interesting to find the analogues of natural conditions which have been used in the group case. For example, it is well known that the Cayley graph Cay(G, S) of a group G is symmetric or undirected if and only if S = S−1. A graph D = (V, E) is said to be undirected if and only if, for every (u, v) ∈ E, the edge (v, u) belongs to E, too. It is a common practice even to include the condition S = S−1 in the definition of a Cayley graph if the undirected case is being considered. In [8] the authors characterise all vertex-transitive directed Cayley graphs produced by periodic semigroups. (A semigroup G is periodic if, for each g ∈ G, there exist positive integers m, n such that g = g.) The aim of this paper is to determine all periodic (and, therefore, all finite) semigroups G for which there exists a non-empty subset S of G such that Cay(G, S) is an undirected Cayley graph (see Theorem 1). In order to investigate when there is some kind of a substitute for the group-theoretic inversion map, it is convenient to Australasian Journal of Combinatorics 25(2002), pp.73–78 think of every element s of S as ‘colour’ of all edges (g, sg). This is in fact a multicolouring of the graph, i.e., each edge may have several colours. We find conditions necessary and sufficient for each edge of colour s always to have a reverse edge of a fixed colour s′ depending only on s (see Theorem 2). We use standard concepts and notation of semigroup theory following [3] and [6]. If S ⊆ G, then the subsemigroup generated by S in G is denoted by 〈S〉. A semigroup is said to be completely simple if it has no proper ideals and has an idempotent minimal with respect to the natural partial order defined on the set of all idempotents by e ≤ f ⇔ ef = fe = e. Theorem 1 For a periodic semigroup G, the following conditions are equivalent: (i) there exists a subset S of G such that the Cayley graph Cay(G, S) is undirected; (ii) G has a completely simple subsemigroup C such that CG = G. A right zero band is a semigroup satisfying the identity xy = y. Theorem 2 Let G be a finite semigroup, and let S be a subset of G. Then the following conditions are equivalent: (i) there exists a one-to-one mapping s → s′ from S to S such that, for every edge (g, sg) of the Cayley graph, there is a reversed edge (sg, s′sg); (ii) SG = G, 〈S〉 is isomorphic to a direct product H ×R of a group H and a right zero band R, and for each g ∈ H |S ∩ ({g} × R)| = |S ∩ ({g−1} × R)|. Suppose that H is a group, I and Λ are nonempty sets, and P = [pλi] is a (Λ × I)-matrix with entries pλi ∈ H for all λ ∈ Λ, i ∈ I. The Rees matrix semigroup M(H; I, Λ; P ) over H with sandwich-matrix P consists of all triples (h; i, λ), where i ∈ I, λ ∈ Λ, and h ∈ H, with multiplication defined by the rule (h1; i1, λ1)(h2; i2, λ2) = (h1pλ1i2h2; i1, λ2), for all h1, h2 ∈ H, i1, i2 ∈ I, λ1, λ2 ∈ Λ. A semigroup is simple if it has no proper ideals. All information on Rees matrix semigroups and completely simple semigroups required for the proofs is collected in the following Rees theorem (see [6], Theorems 3.3.1, 3.2.3 and 3.2.11). Theorem 3 (Rees Theorem) Every completely simple semigroup is isomorphic to a Rees matrix semigroup M(H; I, Λ; P ) over a group H. Conversely, every semigroup M(H; I, Λ; P ) is completely simple if and only if each row and column of P contains at least one nonzero entry. All periodic simple semigroups are completely simple.