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On semidirectly closed non-aperiodic pseudovarieties of finite monoids

Part of: Representation theory of groups Special aspects of infinite or finite groups Semigroups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  24 August 2020

Jiří Kaďourek
Jiří Kaďourek*
Affiliation:
Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37Brno, Czech Republic (kadourek@math.muni.cz)
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Abstract

It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

Keywords

finite monoidspseudovarieties of finite monoidspseudovarieties of finite groupssemidirect products of monoidssemidirectly closed pseudovarietiesfinite inverse monoidsfinite aperiodic monoidsfiniteℛ-trivial monoidsfinite simple groupsfinite p-groupsfinite solvable groups

MSC classification

Primary: 20M07: Varieties and pseudovarieties of semigroups
Secondary: 20D15: Nilpotent groups, $p$-groups 20E10: Quasivarieties and varieties of groups 20F16: Solvable groups, supersolvable groups 20M18: Inverse semigroups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 913 - 928
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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