Frankl's conjecture is true for modular lattices
type: Item
instance of: scholarly article
title: Frankl's conjecture is true for modular lattices
zbMATH Open document ID: 0916.06008
author: Tetsuya Abe
author: Bumpei Nakano
published in: Graphs and Combinatorics
publication date: 1999-07-15T00:00:00Z
review text: Frankl's Conjecture states that every finite lattice \(L\) contains a join-irreducible element \(x\) generating a principal filter \(V_x\) of at most half the size of \(L\). The authors observe that \(| V_x|\leq{1\over 2}| L|\) holds for all \(x\) such that for all \(y>x\) there is \(z< y\) such that that \(z\vee x= y\). And they prove that a minimal such \(x\) is join-irreducible if \(L\) is modular. Thus, they establish Frankl's Conjecture for modular lattices. Moreover, for the case of modular lattices as well as the case of sectionally complemented lattices, they show that there is an irreducible \(x\) such that \(| V_x|={1\over 2}| L|\) if and only if \(L\) is Boolean. This paper makes a substantial contribution, but, as the authors remark, there are lattices with no irreducibles \(x\) of the above type.
reviewed by: Christian Herrmann
Mathematics Subject Classification ID: 06C05
zbMATH DE Number: 1234335
zbMATH Keywords: union-closed family
zbMATH Keywords: Boolean lattice
zbMATH Keywords: Frankl's Conjecture
zbMATH Keywords: principal filter
zbMATH Keywords: modular lattices
zbMATH Keywords: sectionally complemented lattices
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1234335