Semidirect products of pseudovarieties from the universal algebraist's point of view
type: Item
instance of: scholarly article
title: Semidirect products of pseudovarieties from the universal algebraist's point of view
zbMATH Open document ID: 0687.20053
DOI: 10.1016/0022-4049(89)90124-2
author: Jorge Almeida
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: The following question is considered. If membership of two pseudovarieties of finite semigroups can be algorithmically determined, does an effective algorithm exist to determine membership of their semidirect product by a given finite semigroup? Of special interest is the case where one of the pseudovarieties consists of all finite semigroups in which every idempotent is a right zero. In tackling such problems the author illustrates the use of some results from \textit{C. J. Ash} [J. Algebra 92, 104-115 (1985; Zbl 0548.08007)] who explored the relationship between varieties, pseudovarieties and generalized varieties. He begins by proving that a generalized variety is locally finite if and only if it is generated by the class of all its finite members. He also characterizes the class of all finite members of the semidirect product of two locally finite generalized varieties as the semidirect product of the two classes of finite algebras obtained from each of the generalized varieties. Using these and other results he obtains characterizations of various semidirect products of pseudovarieties of semigroups.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 4125704
zbMATH Keywords: pseudovarieties of finite semigroups
zbMATH Keywords: effective algorithm
zbMATH Keywords: membership
zbMATH Keywords: generalized varieties
zbMATH Keywords: semidirect products
reviewed by: Graham T. Clarke
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90124-2
OpenAlex ID: W2063867781
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description: scientific article; zbMATH DE number 4125704