Mal'cev algebras with supernilpotent centralizers
type: Item
instance of: scholarly article
title: Mal'cev algebras with supernilpotent centralizers
zbMATH Open document ID: 1219.08001
author: Peter Mayr
published in: Algebra Universalis
publication date: 2011-05-11T00:00:00Z
review text: Let \({\mathcal A}=(A,F)\) be a finite algebra in a congruence permutable variety. A congruence \(\Theta\) on \({\mathcal A}\) is called \(n\)-supernilpotent if the \((n+1)\)-ary commutator \([\Theta,\ldots,\Theta]\) is the equality relation on \(A\). If for every subdirectly irreducible homomorphic image of \({\mathcal A}\) the centralizer of the monolith is \(n\)-supernilpotent then the clone of polynomial functions on \({\mathcal A}\) is determined by relations of arity \(|A|^{n+1}\). By this result, the author obtains finite implicit descriptions of the polynomial functions on finite local rings with \(1\) and on finite groups \({\mathcal G}\) such that in every subdirectly irreducible quotient of \({\mathcal G}\) the centralizer of the monolith is a \(p\)-group.
reviewed by: Helmut Länger
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 13M10
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5886757
zbMATH Keywords: Mal'tsev algebras
zbMATH Keywords: congruence permutable variety
zbMATH Keywords: \(n\)-supernilpotent
zbMATH Keywords: commutator
zbMATH Keywords: subdirectly irreducible
zbMATH Keywords: homomorphic image
zbMATH Keywords: centralizer
zbMATH Keywords: monolith
zbMATH Keywords: clone
zbMATH Keywords: polynomial function
zbMATH Keywords: relation
zbMATH Keywords: local ring
zbMATH Keywords: \(p\)-group
MaRDI profile type: Publication
full work available at URL: s00012-011-0124-5
OpenAlex ID: W2077581599
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DOI: 10.1007/S00012-011-0124-5
description: scientific article; zbMATH DE number 5886757