Notes on tolerance relations of lattices: A conjecture of R. N. McKenzie
type: Item
instance of: scholarly article
title: Notes on tolerance relations of lattices: A conjecture of R. N. McKenzie
zbMATH Open document ID: 0723.06009
DOI: 10.1016/0022-4049(90)90138-8
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: If \({\mathcal V}\) and \({\mathcal W}\) are lattice varieties, then \({\mathcal V}\circ {\mathcal W}\) consists of all lattices L for which there is a congruence \(\theta\) on L such that all \(\theta\)-classes of L are in \({\mathcal V}\) and L/\(\theta\in {\mathcal W}\). In general, \(V\circ W\) is not a variety. R. N. McKenzie conjectured that a lattice K belongs to the variety generated by \({\mathcal V}\circ {\mathcal W}\) iff there is a tolerance T on K such that all T- classes of L are in \({\mathcal V}\) and L/T\(\in {\mathcal W}\). The aim of this paper is to disprove this conjecture. It presents a finite (18 element) lattice F of the variety generated by \({\mathcal M}_ 3\circ {\mathcal D}\) which fails the mentioned conjecture.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 06D10
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 4191757
zbMATH Keywords: varieties of lattices
zbMATH Keywords: tolerance relation
zbMATH Keywords: factor lattice by a tolerance
zbMATH Keywords: counterexample
zbMATH Keywords: product of varieties
author: Ervin Fried
author: G. Grätzer
MaRDI profile type: Publication
full work available at URL: 0022-4049(90)90138-8
OpenAlex ID: W2061336362
Wikidata QID: Q123235731
cites work: Q4165427
cites work: Q3348955
description: scientific article; zbMATH DE number 4191757