On Semigroups Whose Idempotent-Generated Subsemigroup is Aperiodic

In this paper, all semigroups are assumed to be finite unless otherwise stated. If H is a pseudovariety of groups (that is, a class of groups closed under formation of finite direct products, subgroups and quotients groups), then it is natural to define a group to be Hsolvable if it has a subnormal series each of whose quotients belongs to H. For instance a group is solvable in the classical sense if and only if it is Ab-solvable where Ab is the pseudovariety of Abelian groups. A group is a p-group if and only if it is Zp-solvable where Zp is the pseudovariety generated by a cyclic group of order p. One should not confuse Zp-solvablity with the classical notion of p-solvability [16]. Recall [9] that a group G is psolvable if it has a normal series each of whose quotients is a p-group or a p′-group (the latter meaning the order is relatively prime to p). If H is the pseudovariety generated by Zp and the simple p′-groups, then the p-solvable groups are precisely the H-solvable groups in our sense. It is easy to see that a group isH-solvable if and only if its simple group divisors belong to H and moreover the collection of H-solvable groups is the smallest pseudovariety of groups containing H and closed under extension (or equivalently wreath product); see [6]. This pseudovariety is denoted WH. Alternatively a group is H-solvable if iteration of the operation of taking the verbal subgroup corresponding to H eventually yields the trivial subgroup. For instance, a group is solvable if and only if iteration of the derived subgroup eventually arrives at the trivial subgroup. One way to generalize these notions to semigroups is to consider all semigroups whose simple group divisors belong to H. Equivalently, one is considering all semigroups whose subgroups belong toWH. This set forms a pseudovariety of semigroups (defined analogously