An inequality for the sizes of prime filters of finite distributive lattices
type: Item
instance of: scholarly article
title: An inequality for the sizes of prime filters of finite distributive lattices
zbMATH Open document ID: 0937.06004
DOI: 10.1016/S0012-365X(98)00313-6
author: Dwight Duffus
author: Bill Sands
published in: Discrete Mathematics
publication date: 2000-06-01T00:00:00Z
review text: The results in this paper were inspired by an old problem which became known as Frankl's conjecture, i.e., for every (nonempty) finite union-closed family of finite sets, there is an element contained in at least half of them. There is an equivalent lattice-theoretic statement: every finite lattice \(L\) contains a join-irreducible element \(a\) such that the number \(u(a)\) of elements in the prime filter \(\{b \in L: a\leq b\}\) is \(\leq |L|/2\). The main result (Theorem 1) in the paper shows that for any finite distributive lattice \(L\), \[ \sum_{a\in J(L)}4^{u(a)}\geq j(L)4^{|L|/2}, \] where \(J(L)\) denote the set of all join-irreducible elements of \(L\) and \(j(L)=|J(L)|\). It should be noted that the result implies Frankl's conjecture for finite distributive lattices. The authors prove that every finite non-Boolean lattice \(L\) contains either a prime filter of size at most \(|L|/3\) or a prime filter of size at least \(2|L|/3\).
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06B05
zbMATH DE Number: 1334540
zbMATH Keywords: finite distributive lattice
zbMATH Keywords: prime filter
zbMATH Keywords: join-irreducible element
zbMATH Keywords: Frankl's conjecture
zbMATH Keywords: finite non-Boolean lattice
reviewed by: Leo Esakia
MaRDI profile type: Publication
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full work available at URL: s0012-365x(98)00313-6
OpenAlex ID: W2063023801
description: scientific article; zbMATH DE number 1334540