Semidirectly closed pseudovarieties of locally trivial semigroups
type: Item
instance of: scholarly article
title: Semidirectly closed pseudovarieties of locally trivial semigroups
zbMATH Open document ID: 0711.20032
DOI: 10.1007/BF02573276
published in: Semigroup Forum
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/135028
review text: Semidirect products of pseudovarieties of semigroups have previously been investigated by the author [J. Pure Appl. Algebra 60, No.2, 113-128 (1989; Zbl 0687.20053)]. In the present paper he considers the question of which pseudovarieties of semigroups are semidirectly closed. In the process, he determines all semidirectly closed varieties of semigroups and all semidirectly closed generalized varieties of nilpotent extensions of rectangular bands. Using the work of \textit{C. J. Ash} [J. Algebra 92, 104-115 (1985; Zbl 0548.08007)] on the relationship between pseudovarieties and generalized varieties, he shows that the generalized variety generated by a semidirectly closed pseudovariety is also semidirectly closed. Finally he describes all semidirectly closed pseudovarieties of locally trivial semigroups, and shows that they form a complete distributive lattice.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08B26
zbMATH DE Number: 4168951
zbMATH Keywords: Semidirect products
zbMATH Keywords: pseudovarieties of semigroups
zbMATH Keywords: semidirectly closed varieties of semigroups
zbMATH Keywords: generalized varieties
zbMATH Keywords: nilpotent extensions of rectangular bands
zbMATH Keywords: locally trivial semigroups
zbMATH Keywords: complete distributive lattice
author: Jorge Almeida
reviewed by: Graham T. Clarke
MaRDI profile type: Publication
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OpenAlex ID: W1965079515
description: scientific article; zbMATH DE number 4168951