A combinatorial representation for a special class of complete distributive lattices
type: Item
instance of: scholarly article
title: A combinatorial representation for a special class of complete distributive lattices
zbMATH Open document ID: 1004.06010
author: L. Ferrari
author: Giorgio Nicoletti
published in: Annals of Combinatorics
publication date: 2002-06-16T00:00:00Z
review text: As is well known, representation theorems by Tarski (for complete atomic Boolean algebras) and Birkhoff (for finite distributive lattices) involve purely lattice-theoretic techniques. For distributive lattices in general, the topological tool of Priestley duality is required. Here the authors produce a representation theorem for a certain class of distributive lattices by using only lattice-theoretic techniques and avoiding topological notions. The (restricted) class concerned consists of completely distributive lattices that are what the authors call Pezzoli lattices, such a lattice being defined as a bounded distributive lattice \(D\) in which there is an order isomorphism \(\gamma\) from the set of completely join-irreducible elements to the set of completely meet-irreducible elements such that if \(b=\gamma(a)\) then \([a,1]\cap [0,b]= \emptyset\) and \([a,1] \cup [0,b]=D\).
Mathematics Subject Classification ID: 06D10
Mathematics Subject Classification ID: 06D50
zbMATH DE Number: 1756475
zbMATH Keywords: infinitely distributive lattices
zbMATH Keywords: duality
zbMATH Keywords: \(P\)-lattices
zbMATH Keywords: completely irreducible elements
zbMATH Keywords: representation theorem
zbMATH Keywords: completely distributive lattices
zbMATH Keywords: Pezzoli lattices
MaRDI profile type: Publication
full work available at URL: s00026-001-8013-9
OpenAlex ID: W2019204971
DOI: 10.1007/S00026-001-8013-9
description: scientific article; zbMATH DE number 1756475