About the power pseudovariety PCS

Abstract

The power pseudovariety $PCS$, that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety $AgBG$ of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties $PCS=AgBG$. In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities $PCS=J\ast CS=J\mathit{ⓜ}CS=AgBG$ has been verified at the same time. Here, $J$ is the pseudovariety of all $\mathcal{𝒥}$-trivial semigroups, $CS$ is the pseudovariety of all completely simple semigroups, $J\ast CS$ is the pseudovariety generated by the family of all semidirect products of $\mathcal{𝒥}$-trivial semigroups by completely simple semigroups, and $JⓜCS$ is the pseudovariety generated by the Mal’cev product of the pseudovarieties $J$ and $CS$. In this paper, another different proof of these equalities is provided first. More precisely, the equalities $PCS=J\ast CS=JⓜCS$ are given a new proof, while the equality $JⓜCS=AgBG$ 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 $H$ of groups, let $CS(H)$ stand for the pseudovariety of all completely simple semigroups whose subgroups belong to $H$. Then it turns out that, for every locally extensible pseudovariety $H$ of groups, the equalities of pseudovarieties $P(CS(H))=J\ast CS(H)=JⓜCS(H)$ hold.