Abstract:

A rectangular group S is a semigroup isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. The Cayley digraph Cay(S,A) of a finite rectangular group S with respect to a nonempty subset A of S is defined as a digraph with vertex set S and arc set consisting of ordered pairs (u,ua) in SxS for some a in A. The set A is called a connection set of Cay(S,A). Let u and v be in S. The distance from u to v in Cay(S,A), denoted by d(u,v), is the number of arcs in the shortest directed u-v path if one exists and is infinity otherwise. In this paper, we characterize the distances of any two vertices in Cayley digraphs of rectangular groups with respect to the Cartesian product connection sets.