Directed Graphs and Combinatorial Properties of Semigroups

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Combinatorial properties of words in groups and semigroups have been investigated by many authors. For recent results on this topic we refer to the book [4] and survey [11] (see also [10]). In particular, Justin [9] described all repetitive commutative semigroups (a section of [4] and a chapter of [12] are devoted to this combinatorial property, see also [3, 4]). In this paper we obtain complete descriptions of all commutative semigroups satisfying three other combinatorial properties defined in terms of directed graphs. Throughout, by a graph we mean a directed graph without loops or multiple edges. The power graph Pow S of a semigroup S has all elements of S as vertices and has edges u v for all u v ∈ G such that u = v and v is a power of u. The divisibility graph Div S has vertex set S and edges u v , where u = v and u divides v; i.e., u belongs to the ideal generated by v. The annihilator graph Ann S of a semigroup S with 0 has vertex set S and the set of edges u v ∈ S × S uv = 0 u = v .