The power pseudovariety , that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties . In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities has been verified at the same time. Here, is the pseudovariety of all -trivial semigroups, is the pseudovariety of all completely simple semigroups, is the pseudovariety generated by the family of all semidirect products of -trivial semigroups by completely simple semigroups, and is the pseudovariety generated by the Mal’cev product of the pseudovarieties and . In this paper, another different proof of these equalities is provided first. More precisely, the equalities are given a new proof, while the equality is quoted from a foregoing paper. Subsequently in this paper, this new proof of the mentioned equalities is further refined to yield a proof of the following more general result: For any pseudovariety of groups, let stand for the pseudovariety of all completely simple semigroups whose subgroups belong to . Then it turns out that, for every locally extensible pseudovariety of groups, the equalities of pseudovarieties hold.
"About the power pseudovariety ." Rocky Mountain J. Math. 51 (6) 2045 - 2102, December 2021. https://doi.org/10.1216/rmj.2021.51.2045