Varieties generated by modes of submodes
type: Item
instance of: scholarly article
title: Varieties generated by modes of submodes
zbMATH Open document ID: 1270.08004
author: Anna Zamojska-Dzienio
author: Agata Pilitowska
published in: Algebra Universalis
publication date: 2013-01-30T00:00:00Z
review text: Any operation on a set \(A\) may be lifted up to an operation on the set of non-empty subsets of \(A\) in a natural way. In this way, for any algebra \((A,\Omega)\) one obtains its power algebra of subsets of \(A\). G. Grätzer and H. Lakser proved that for a variety \(V\) of algebras, the variety \(V\Sigma\) generated by the power algebras of algebras in \(V\) satisfies precisely the consequences of the linear identities true in \(V\). If \(V\) is a variety of idempotent and entropic algebras, then the set of non-empty subalgebras of any algebra in \(V\) forms a subalgebra of its power algebra. In this way, for any algebra \((A,\Omega)\) in \(V\), one obtains its algebra of subalgebras. The authors solve a long-standing problem concerning identities satisfied in the variety \(VS\) generated by the algebras of subalgebras of algebras in any idempotent and entropic variety. They prove that if \(V\) is idempotent and entropic and \(V\Sigma\) is locally finite, then the variety \(VS\) is defined by the idempotent and linear identities true in \(V\).
Mathematics Subject Classification ID: 08B99
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08A62
zbMATH DE Number: 6132824
zbMATH Keywords: idempotent algebras
zbMATH Keywords: entropic algebras
zbMATH Keywords: modes
zbMATH Keywords: power algebras
zbMATH Keywords: congruence relations
zbMATH Keywords: identities
zbMATH Keywords: varieties of finitary algebras
zbMATH Keywords: locally finite varieties
zbMATH Keywords: algebra of subalgebras
Wikidata QID: Q59289573
reviewed by: Anna Romanowska
MaRDI profile type: Publication
full work available at URL: s00012-012-0201-4
OpenAlex ID: W2003763217
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DOI: 10.1007/S00012-012-0201-4
description: scientific article; zbMATH DE number 6132824