The decidability of the affine completeness generation problem.
type: Item
instance of: scholarly article
title: The decidability of the affine completeness generation problem.
zbMATH Open document ID: 1062.08006
author: Kalle Kaarli
author: Alden F. Pixley
published in: Algebra Universalis
publication date: 2005-04-19T00:00:00Z
review text: An algebra \(A\) is affine complete if each congruence-compatible operation on \(A\) is a polynomial of \(A\). A variety is affine complete if each of its members is affine complete. The authors give an explicit proof of the theorem: There is an effective procedure for deciding whether or not a given finite algebra of finite type generates an affine complete variety. The proof is based on the fact that if such an algebra \(A\) generates an affine complete variety then \(A\) must have an \((n+1)\)-ary near unanimity term. Then the variety is congruence distributive. On the other hand, there exist congruence non-distributive affine complete algebras, thus the problem of characterizing affine complete algebras remains still open.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 2158800
zbMATH Keywords: affine complete variety
zbMATH Keywords: near unanimity term
zbMATH Keywords: decidability
zbMATH Keywords: variety generation
MaRDI profile type: Publication
full work available at URL: s00012-002-8208-x
OpenAlex ID: W2069329186
DOI: 10.1007/S00012-002-8208-X
description: scientific article; zbMATH DE number 2158800