On the existence of congruence-uniform structures on universal algebras
type: Item
instance of: scholarly article
title: On the existence of congruence-uniform structures on universal algebras
zbMATH Open document ID: 0870.08003
DOI: 10.1007/BF01192714
published in: Algebra Universalis
publication date: 1997-08-07T00:00:00Z
review text: Let \(\mathcal A=(A,F)\) be an algebra and \(\mathcal N\) a uniform structure on \(A\). \(\mathcal N\) is called a proper congruence-uniform structure (pcus) if there exists a fundamental system of entourages consisting of congruence relations and this uniform structure is neither discrete nor indiscrete. A question which arises is whether or not for a given \(\mathcal A\) there exists a pcus. Sample result: There exists a pcus on \(\mathcal A\) iff Con\(\mathcal A\) is not atomic or Con\(\mathcal A\) is atomic but the greatest element of \(\mathcal L\) is not compact. In particular, if Con\(\mathcal A\) is distributive, then there exists a pcus iff Con\(\mathcal A\) is not atomic or it has infinitely many of atoms. The author gives a complete solution for multioperator groups and for lattices.
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 1046547
zbMATH Keywords: uniform structure
zbMATH Keywords: congruence lattice
zbMATH Keywords: congruence-uniform structure
author: Harald Woracek
reviewed by: Ivan Chajda
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1046547