About the power pseudovariety \(\mathbf{PCS}\). (English) Zbl 1508.20077
Summary: The power pseudovariety \(\mathbf{PCS}\), that is, the pseudovariety of finite semigroups generated by all power semigroups of finite completely simple semigroups has recently been characterized as the pseudovariety \(\mathbf{AgBG}\) of all so-called aggregates of block groups. This characterization can be expressed as the equality of pseudovarieties \(\mathbf{PCS} = \mathbf{AgBG}\). In fact, a longer sequence of equalities of pseudovarieties, namely the sequence of equalities \(\mathbf{PCS} = \mathbf{J} \ast \mathbf{CS} = \mathbf{J} \text{\textcircled{m}} \mathbf{CS} = \mathbf{AgBG}\) has been verified at the same time. Here, \(\mathbf{J}\) is the pseudovariety of all \(\mathcal{J}\)-trivial semigroups, \(\mathbf{CS}\) is the pseudovariety of all completely simple semigroups, \(\mathbf{J} \ast \mathbf{CS}\) is the pseudovariety generated by the family of all semidirect products of \(\mathcal{J}\)-trivial semigroups by completely simple semigroups, and \(\mathbf{J} \text{\textcircled{m}} \mathbf{CS}\) is the pseudovariety generated by the Mal’cev product of the pseudovarieties \(\mathbf{J}\) and \(\mathbf{CS}\). In this paper, another different proof of these equalities is provided first. More precisely, the equalities \(\mathbf{PCS} = \mathbf{J} \ast \mathbf{CS} = \mathbf{J} \text{\textcircled{m}} \mathbf{CS}\) are given a new proof, while the equality \(\mathbf{J} \text{\textcircled{m}} \mathbf{CS} = \mathbf{AgBG}\) 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 \(\mathbf{H}\) of groups, let \(\mathbf{CS}(\mathbf{H})\) stand for the pseudovariety of all completely simple semigroups whose subgroups belong to \(\mathbf{H}\). Then it turns out that, for every locally extensible pseudovariety \(\mathbf{H}\) of groups, the equalities of pseudovarieties \(\mathbf{P}(\mathbf{CS}(\mathbf{H})) = \mathbf{J} \ast \mathbf{CS}(\mathbf{H}) = \mathbf{J} \text{\textcircled{m}} \mathbf{CS} (\mathbf{H})\) hold.
MSC:
| 20M07 | Varieties and pseudovarieties of semigroups |
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
Keywords:
pseudovarieties of finite semigroups; power semigroups of finite semigroups; power pseudovarieties; completely simple semigroups; block groups; aggregates of block groups; relatively free semigroups; semidirect products of semigroups; Mal’cev products of pseudovarieties of semigroupsReferences:
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