Fully invariant and verbal congruence relations
type: Item
instance of: scholarly article
title: Fully invariant and verbal congruence relations
zbMATH Open document ID: 1280.08001
author: Clifford Bergman
author: Joel Berman
published in: Algebra Universalis
publication date: 2013-09-09T00:00:00Z
review text: Let \(A\) be an algebra. A congruence \(\theta\) is called verbal if there exists a variety \(V\) such that the factor algebra of \(A\) by \(\theta\) is in \(V\). Every verbal congruence is fully invariant but not conversely. An algebra \(A\) is called verbose if every fully invariant congruence on \(A\) is verbal. A variety is verbose if each algebra in it has this property. The authors prove that for a finite SI-algebra generating congruence-modular (CM in brief) variety its monolith is verbal. Neither finiteness nor CM can be omitted. If \(A\) is a finite algebra in a CM variety and \(\operatorname{Con}A\) is a chain, then \(A\) is verbose. If \(V\) is a locally finite CM minimal variety then \(V\) is verbose. The authors prove several results on verbose varieties which are abelian or semisismple and arithmetical. A number of nice examples is included.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 6206935
zbMATH Keywords: fully invariant congruence
zbMATH Keywords: verbal congruence
zbMATH Keywords: verbose algebra
zbMATH Keywords: verbose variety
MaRDI profile type: Publication
Wikidata QID: Q60143368
full work available at URL: s00012-013-0238-z
OpenAlex ID: W1995223482
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DOI: 10.1007/S00012-013-0238-Z
description: scientific article; zbMATH DE number 6206935