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On semidirectly closed pseudovarieties of finite semigroups and monoids

Part of: Semigroups

Published online by Cambridge University Press:  02 August 2021

Jiří Kad’ourek*
Affiliation:
Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

Abstract

For every pseudovariety $\mathbf {V}$ of finite monoids, let $\mathbf {LV}$ denote the pseudovariety of all finite semigroups all of whose local submonoids belong to $\mathbf {V}$ . In this paper, it is shown that, for every nontrivial semidirectly closed pseudovariety $\mathbf {V}$ of finite monoids, the pseudovariety $\mathbf {LV}$ of finite semigroups is also semidirectly closed if, and only if, the given pseudovariety $\mathbf {V}$ is local in the sense of Tilson. This finding resolves a long-standing open problem posed in the second volume of the classic monograph by Eilenberg.

Type
Article
Copyright
© Canadian Mathematical Society 2021

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Footnotes

This research has been supported by the Grant Agency of the Czech Republic under the project GA19-12790S.

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