Covers for a class of abundant semigroups

On a semigroup S, the relation R* is defined by aR*b if aRb in some oversemigroup of S. The relation L* is defined dually. We say that a semigroup is abundant if all its R*-classes and L*-classes contain at least one idempotent. We use a+ and a* to denote typical idempotents in the R*-class of a and L*-class of a, respectively. An abundant semigroup S is said to be bountiful if E(S) (its set of idempotents) is a subsemigroup and, for all a in S and e in E(S), the following two conditions hold: 1) for some a+, if e <= a+, then ea=af for some idempotent f; 2) for some a*, if e <= a*, then ae=fa for some idempotent f. For example, full subsemigroups of orthodox semigroups are bountiful. Given a bountiful semigroup S, there is an E-unitary bountiful semigroup P and a surjective homomorphism from P onto S which restricts to an isomorphism from E(P) onto E(S). Moreover, if S is finite, then P can be chosen to be finite. This result is used to generalise a result of McAlister on orthodox semigroups to obtain a criterion for a finite bountiful semigroup to be a member of the pseudovariety A v G, where A is the pseudovariety of finite aperiodic semigroups and G is the pseudovariety of finite groups.