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December 2021 About the power pseudovariety PCS
Jiří Kaďourek
Rocky Mountain J. Math. 51(6): 2045-2102 (December 2021). DOI: 10.1216/rmj.2021.51.2045


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=JCS=JCS=AgBG has been verified at the same time. Here, J is the pseudovariety of all 𝒥-trivial semigroups, CS is the pseudovariety of all completely simple semigroups, JCS is the pseudovariety generated by the family of all semidirect products of 𝒥-trivial semigroups by completely simple semigroups, and JCS 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=JCS=JCS are given a new proof, while the equality JCS=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))=JCS(H)=JCS(H) hold.


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Jiří Kaďourek. "About the power pseudovariety PCS." Rocky Mountain J. Math. 51 (6) 2045 - 2102, December 2021.


Received: 7 October 2020; Revised: 17 March 2021; Accepted: 23 March 2021; Published: December 2021
First available in Project Euclid: 22 March 2022

Digital Object Identifier: 10.1216/rmj.2021.51.2045

Primary: 20M07
Secondary: 18B40

Keywords: 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 , semidirect products of semigroups

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.51 • No. 6 • December 2021
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