Finite distributive lattices and doubly irreducible elements
type: Item
instance of: scholarly article
title: Finite distributive lattices and doubly irreducible elements
zbMATH Open document ID: 0898.06004
DOI: 10.1016/S0012-365X(97)81832-8
author: Gabriela Bordalo
author: Joel Berman
published in: Discrete Mathematics
publication date: 1998-11-01T00:00:00Z
review text: For a finite distributive lattice \(L\), let \(J(L)\) be the poset of its join-irreducible elements. The MacNeille completion of \(J(L)\) is not always distributive: a theorem of Monjardet and Wille says that this is the case if and only if \(L\) is generated by the set \(\text{Irr} (L)\) of its both join and meet irreducible elements. This leads the authors to study the following structure. If \(G\) is a finite poset, let \({\mathcal D}(G)\) be the class of all finite distributive lattices \(L\) such that \(L\) is generated by \(\text{Irr} (L)=G\), quasi-ordered by \(L\leq L'\) if there is an onto homomorphism \(L'\to L\) which is the identity on \(G\). The main result of the paper is that the partially ordered quotient of \({\mathcal D}(G)\) is a Boolean lattice \({\mathcal D}'\) whose greatest element is the free bounded distributive lattice generated by \(G\). A description of the least element of \({\mathcal D}'\) is also obtained as well as a characterization of membership in \({\mathcal D}(G)\).
reviewed by: Georges Hansoul
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 1110083
zbMATH Keywords: doubly irreducible element
zbMATH Keywords: free distributive lattice over a poset
zbMATH Keywords: finite distributive lattice
zbMATH Keywords: Boolean lattice
MaRDI profile type: Publication
cites work: Q4692885
cites work: Bigeneration in complete lattices and principal separation in ordered sets
cites work: Q4165427
cites work: On finite lattices generated by their doubly irreducible elements
cites work: Une construction du réticulé distributif libre sur un ensemble ordonné
description: scientific article; zbMATH DE number 1110083