Congruences on orthodox semigroups with associate subgroups

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If Sis a regular semigroup then an inverse transversal of S is an inverse subsemigroup T with the property that |T ∩ V(x)| = 1 for every x ∈ S where V(x) denotes the set of inverses of x ∈ S. In a previous publication [1] we considered the similar concept of a subsemigroup T of S such that |T ∩ A(x)| = 1 for every x ∩ S where A(x) = {y∈ S;xyx = x} denotes the set of associates (or pre-inverses) of x ∈ S, and showed that such a subsemigroup T is necessarily a maximal subgroup H_a for some idempotent α ∈ 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 ∈ 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 ∩ A(x) = {x^*} and write the subgroup T as H_α = {x^*;x ∈ S}, which we call an associate subgroup of S. For every x ∈ 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 ∈ S, and e^* = α for every idempotent e.