Distributivity of strongly representable lattices
type: Item
instance of: scholarly article
title: Distributivity of strongly representable lattices
zbMATH Open document ID: 0358.06024
DOI: 10.1007/BF02485418
author: Pavel Pudlák
published in: Algebra Universalis
publication date: 1977-01-01T00:00:00Z
review text: A lattice is called finitely strongly representable if (1) there is an embedding of it into a finite equivalence lattice and (2) the image of every embedding of it, preserving 0 and 1, in a finite equivalence lattice is a congruence lattice of an algebra. The aim of the paper is to prove the following theorem: If a lattice is finitely strongly representable then it is distributive. It adds, to the previous result of \textit{R. Quackenbush} and \textit{B. Wolk} [Algebra Univers. 1, 165--166 (1971; Zbl 0231.06006)], that the property of finite strong representability is equivalent to distributivity for finite lattices. In the proof, an embedding violating condition (2) is constructed for every non-distributive lattice satisfying (1). This is achieved by joining several copies of the original embedding together.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B15
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 3557846
MaRDI profile type: Publication
cites work: Zur konkreten Charakterisierung von Kongruenzverbänden
cites work: Q3279317
cites work: Strong representation of congruence lattices
cites work: Lattices, equivalence relations, and subgroups
full work available at URL: https://doi.org/10.1007/bf02485418
OpenAlex ID: W1983888041
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description: scientific article; zbMATH DE number 3557846