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# About the power pseudovariety PCS

Autoři 2021 Článek v odborném periodiku Rocky Mountain Journal of Mathematics https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-51/issue-6/About-the-power-pseudovariety-PCS/10.1216/rmj.2021.51.2045.short http://dx.doi.org/10.1216/rmj.2021.51.2045 aggregates of block groups; block groups; completely simple semigroups; Mal’cev products of pseudovarieties of semigroups; power pseudovarieties; power semigroups of finite semigroups; pseudovarieties of finite semigroups; relatively free semigroups 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*CS=J?CS=AgBG has been verified at the same time. Here, J is the pseudovariety of all ??-trivial semigroups, CS is the pseudovariety of all completely simple semigroups, J*CS is the pseudovariety generated by the family of all semidirect products of ??-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*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*CS(H)=J?CS(H) hold.

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