Quasiorder lattices of varieties
type: Item
instance of: scholarly article
title: Quasiorder lattices of varieties
zbMATH Open document ID: 1397.08001
author: Gergő Gyenizse
author: M. Maróti
published in: Algebra Universalis
publication date: 2018-07-16T00:00:00Z
review text: The set \(\mathrm{Quo}(A)\) of compatible quasiorders of an algebra \(A\) forms a lattice under inclusion and the congruence lattice \(\mathrm{Con}(A)\) is its sublattice. It is proved that a locally finite variety is congruence distributive (modular) if and only if it is quasiorder distributive (modular). It is shown that the same does not hold for meet semi-distributivity. It is known that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of the congruence lattice isomorphic to \(M_3\). The authors prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for infinite algebras even in the variety of semilattices.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 6904423
zbMATH Keywords: quasiorder
zbMATH Keywords: directed terms
zbMATH Keywords: distributivity
zbMATH Keywords: modularity
zbMATH Keywords: meet semidistributivity
MaRDI profile type: Publication
full work available at URL: s00012-018-0512-1
OpenAlex ID: W2800031695
cites work: Finitely Related Algebras In Congruence Distributive Varieties Have Near Unanimity Terms
cites work: Finitely related algebras in congruence modular varieties have few subpowers
cites work: Q3934450
cites work: Q4002019
cites work: Q3328595
cites work: Q4461652
cites work: On tolerance lattices of algebras in congruence modular varieties
cites work: Congruence modularity is permutability composed with distributivity
cites work: The structure of finite algebras
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Absorption and directed Jónsson terms
cites work: Q5829835
cites work: Q3751631
cites work: Lattices of quasiorders on universal algebras
DOI: 10.1007/S00012-018-0512-1
Recommended article: Varieties having distributive lattices of quasiorders
Recommended article: Q3222893
Recommended article: Quasi-varieties of lattice ordered groups
Recommended article: The structure of the lattices of quasivarieties
Recommended article: Quasiorder lattices in congruence modular varieties
Recommended article: Q4227566
Recommended article: Q4298442
Recommended article: Lattices of quasiorders on universal algebras
Recommended article: Q4695518
description: scientific article; zbMATH DE number 6904423