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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 Structure and classification of infinite or finite groups Semigroups

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 ([email protected])
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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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References

Almeida, J., Finite semigroups and universal algebra (Singapore, World Scientific, 1994).Google Scholar
Almeida, J. and Weil, P., Profinite categories and semidirect products, J. Pure Appl. Algebra 123 (1998), 1–50.CrossRefGoogle Scholar
Alperin, J. L. and Bell, R. B., Groups and representations (New York, Springer-Verlag, 1995).CrossRefGoogle Scholar
Eilenberg, S., Automata, languages and machines, Vol. B (New York, Academic Press, 1976).Google Scholar
Higgins, P. M. and Margolis, S. W., Finite aperiodic semigroups with commuting idempotents and generalizations, Israel J. Math. 116 (2000), 367–380.CrossRefGoogle Scholar
Lallement, G., Semigroups and combinatorial applications (New York, John Wiley & Sons, 1979).Google Scholar
Selberg, A., An elementary proof of Dirichlet's theorem about primes in an arithmetic progression, Ann. of Math. (2) 50 (1949), 297–304.CrossRefGoogle Scholar
Stiffler, Jr. P., Extension of the fundamental theorem of finite semigroups, Advances in Math. 11 (1973), 159–209.CrossRefGoogle Scholar
Teixeira, M. L., On semidirectly closed pseudovarieties of aperiodic semigroups, J. Pure Appl. Algebra 160 (2001), 229–248.CrossRefGoogle Scholar
Tilson, B., Categories as algebra: An essential ingredient in the theory of monoids, J. Pure Appl. Algebra 48 (1987), 83–198.CrossRefGoogle Scholar