Hagemann's and Herrmann's characterization of strictly affine complete algebras
type: Item
instance of: scholarly article
title: Hagemann's and Herrmann's characterization of strictly affine complete algebras
zbMATH Open document ID: 1013.08002
author: Erhard Aichinger
published in: Algebra Universalis
publication date: 2003-04-23T00:00:00Z
review text: An algebra \(A\) is strictly \(k\)-affine complete if for every finite subset \(T\) of \(A^k\), and for each congruence-preserving function \(f:T\rightarrow A\) there is a \(k\)-ary polynomial function \(p\) over \(A\) such that the restriction \(p|T\) equals \(f\). J. Hagemann and C. Herrmann have characterized these algebras in 1982: \(A\) is strictly \(k\)-affine complete for all \(k\in {\mathbb N}\) if and only if the following two conditions hold: (1) Every subalgebra of a cartesian product of finitely many copies of \(A\) that contains the diagonal is congruence permutable; (2) Each congruence of \(A\), seen as a subalgebra of \(A\times A\), has a distributive congruence lattice. The author proves that (2) can be replaced by the condition that \([x,y]=x\wedge y\) for all congruences \(x,y\in \operatorname {Con} A\) where \([x,y]\) denotes the term condition commutator of \(x,y\).
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 1899953
zbMATH Keywords: strictly affine complete algebra
zbMATH Keywords: congruence permutability
zbMATH Keywords: congruence distributivity
zbMATH Keywords: term condition commutator
MaRDI profile type: Publication
full work available at URL: s000120050173
OpenAlex ID: W1991543193
DOI: 10.1007/S000120050173
description: scientific article; zbMATH DE number 1899953