Congruences on orthodox semigroups with associate subgroups

Abstract

If S is a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T intersection V(x)| = 1 for every x in S where V(x) denotes the set of inverses of x in S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T intersection A(x)| = 1 for every x in S where A(x) = {y in S: xyx = x} denotes the set of associates (or pre-inverses) of x in S, and showed that such a subsemigroup T is necessarily a maximal subgroup Hα for some idempotent α in S. Throughout what follows, we shall assume that S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y in S). Under these assumptions, we obtained in [1] a structure theorem which generalises that given in [3] for uniquely unit orthodox semigroups. Adopting the notation of [1], we let T intersection A(x) = {x*} and write the subgroup T as Hα = {x*: x xin S}, which we call an associate subgroup of S. For every x x in S we therefore have x*α = x* = αx* and x*x** = α = x**x*. As shown in [1, Theorems 4, 5] we also have (xy)* = y*x* for all x, y in S, and e* = α for every idempotent e.