Normal cryptogroups with an associate subgroup

Abstract

Let S be a semigroup. For a, x ∈ S such that a = axa, we say that x is an associate of a. A subgroup G of S which contains exactly one associate of each element of S is called an associate subgroup of S. It induces a unary operation in an obvious way, and we speak of a unary semigroup satisfying three simple axioms.

A normal cryptogroup S is a completely regular semigroup whose H -relation is a congruence and S/H is a normal band. Using the representation of S as a strong semilattice of Rees matrix semigroups, in a previous communication we characterized those that have an associate subgroup.

In this paper, we use that result to find three more representations of this semigroup. The main one has a form akin to the one of semigroups in which the identity element of the associate subgroup is medial.