On power varieties of semigroups
type: Item
instance of: scholarly article
title: On power varieties of semigroups
zbMATH Open document ID: 0679.20052
DOI: 10.1016/0021-8693(89)90183-X
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: A semigroup identity \(u=v\) is said to be stable if each u,v is a product of distinct variables. Let V be a variety of semigroups. Then Stab V denotes the smallest variety containing V which is defined by stable identities. The power variety of semigroups is the variety PV generated by the semigroups of all subsets of each semigroup from some variety V. The problem of determination of the power variety PV associated with V is solved in the following way. First, find the variety Stab V. To obtain PV, take the join of Stab V with the variety of all semilattices. Stab V is actually the variety \(P'V\) generated by the semigroups of the nonempty subsets of each member of V. It follows that the operator \(P'\) is idempotent while \(P\neq P^ 2=P^ 3\). Further, \(P^ 2V\) is defined by the permutation identities which hold in V. Via the manipulation of stable identities and using some basic results from the theory of well- quasi-ordering, it is shown that the semilattice of varieties of the form PV is countable and better quasi-ordered under inclusion. Similar questions are also considered in the context of generalized varieties.
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08C15
zbMATH DE Number: 4112898
zbMATH Keywords: stable identities
zbMATH Keywords: power variety of semigroups
zbMATH Keywords: permutation identities
zbMATH Keywords: well-quasi-ordering
zbMATH Keywords: semilattice of varieties
zbMATH Keywords: better quasi-ordered
zbMATH Keywords: generalized varieties
author: Jorge Almeida
reviewed by: Leonid M. Martynov
MaRDI profile type: Publication
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full work available at URL: 0021-8693(89)90183-x
OpenAlex ID: W2037196404
description: scientific article; zbMATH DE number 4112898