On the lattice of completely regular monoid varieties
type: Item
instance of: scholarly article
title: On the lattice of completely regular monoid varieties
zbMATH Open document ID: 0774.20037
DOI: 10.1007/BF02573564
published in: Semigroup Forum
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/135229
review text: For any subvariety \(V\) of the variety \(CR\) of completely regular semigroups, let \(MV\) be the variety of completely regular monoids consisting of the monoids whose completely regular semigroup reducts belong to \(V\), and \(\langle MV\rangle\) the subvariety of \(CR\) generated by the completely regular semigroup reducts of the members of \(MV\). Then \(V\to\langle MV\rangle\) is a complete endomorphism of the lattice \(L(CR)\) of subvarieties of \(CR\), and the image of this endomorphism is in a natural way isomorphic to the lattice \(L(MCR)\) of subvarieties of \(MCR\). In [\textit{L. Polák}, Semigroup Forum 36, 253-284 (1987; Zbl 0638.20032)] a representation of the interval \([SL,CR]\) has been given: each subvariety of \(CR\) containing the variety \(SL\) of semilattices is represented by an order preserving mapping between two fixed posets. The above mentioned observation, together with a translation of Polák's machinery to the new circumstances, allows to establish a corresponding representation theorem for the interval \([MSL,MCR]\) of \(L(MCR)\). The description of the lattice of band monoids given by S. Wismath follows as an easy consequence.
reviewed by: Francis Pastijn
Mathematics Subject Classification ID: 20M07
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 20M17
zbMATH DE Number: 167675
zbMATH Keywords: lattice of subvarieties
zbMATH Keywords: completely regular semigroups
zbMATH Keywords: variety of completely regular monoids
zbMATH Keywords: completely regular semigroup reducts
zbMATH Keywords: complete endomorphism
zbMATH Keywords: semilattices
zbMATH Keywords: representation theorem
zbMATH Keywords: interval
zbMATH Keywords: lattice of band monoids
author: Colleen A. Vachuska
MaRDI profile type: Publication
cites work: Q3934450
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cites work: On varieties of completely regular semigroups. I
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OpenAlex ID: W2084272830
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description: scientific article; zbMATH DE number 167675