Affine complete varieties
type: Item
instance of: scholarly article
title: Affine complete varieties
zbMATH Open document ID: 0641.08007
DOI: 10.1007/BF01188385
author: Kalle Kaarli
author: Alden F. Pixley
published in: Algebra Universalis
publication date: 1987-01-01T00:00:00Z
review text: For a congruence relation \(\theta\) on an algebra \({\mathcal A}=(A;F)\), a k- ary function f on A is said to be compatible with \(\theta\) if \[ (\forall (x\quad 1,...,x\quad k),(y\quad 1,...,y\quad k)\in A\quad k)(((\forall i)(x\quad i\theta y\quad i))\to f(x\quad 1,...\quad,x\quad k)\theta f(y\quad 1,...,y\quad k)). \] f is compatible if it is compatible with all \(\theta\in con(A)\), \({\mathcal A}\) is affine complete if each compatible function is a polynomial function of \({\mathcal A}\) and a variety is affine complete if all its members are. This paper investigates the properties of affine complete varieties; some of the results obtained are: (1) the subdirectly irreducible algebras in an affine complete variety are finite; (2) the subdirectly irreducible algebras in a congruence distributive affine complete variety have no proper subalgebras; (3) an arithmetical variety of finite type is affine complete if and only it is generated by a single finite algebra having no proper subalgebras. The paper mentions a number of open problems, and ends with some results on variants of the notion of affine completeness.
Mathematics Subject Classification ID: 08B99
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 4043915
zbMATH Keywords: affine complete algebra
zbMATH Keywords: congruence relation
zbMATH Keywords: compatible function
zbMATH Keywords: affine complete varieties
zbMATH Keywords: subdirectly irreducible algebras
zbMATH Keywords: arithmetical variety
zbMATH Keywords: affine completeness
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
cites work: Q4177625
cites work: Characterization of algebraic functions in equational classes generated by independent primal algebras
cites work: Compatible function extension property
cites work: Completeness in arithmetical algebras
cites work: Characterizations of arithmetical varieties
cites work: Q3689232
cites work: Produkte von Kongruenzklassengeometrien universeller Algebren
description: scientific article; zbMATH DE number 4043915