The Kurosh-Ore exchange property
type: Item
instance of: scholarly article
title: The Kurosh-Ore exchange property
zbMATH Open document ID: 0675.06003
DOI: 10.1007/BF02170062
author: S. H. Smith
published in: Acta Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: For a lattice L we denote by J(L) the set of all join irreducibles of L; next let \(J_ 0(L)=J(L)\cup \{O_ L\}\). If \(a\in L\), \(a=x_ 1\vee x_ 2\vee...\vee x_ n\) and \(x_ i\in J_ 0(L)\) for \(i=1,2,...,n\), then \(x_ 1\vee...\vee x_ n\) is said to be a \(\vee\)-representation of a. The Kurosh-Ore exchange property for \(\vee\)-representations will be denoted by KOP. \textit{J. P. S. Kung} [Order 2, 105-112 (1985; Zbl 0582.06008)] investigated the notion of consistent lattices. The present author proves the following theorem: In a lattice L, in which each element has a \(\vee\)-representation, the KOP holds if and only if L is consistent. Next, the author studies the following form of KOP for a closure structure (X,Cl): If A,B\(\subseteq X\) and \(Cl(A)=Cl(B)\), then for each \(a\in A\) there exists \(b\in B\) such that \(Cl(A)=Cl(A\setminus \{a\})\cup \{b\}\). In the last section of the paper, several conditions for a finite lattice L are found which are equivalent with the condition saying that L is semimodular and has the KOP.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 06C10
zbMATH DE Number: 4105027
zbMATH Keywords: semimodular lattice
zbMATH Keywords: closure structure
zbMATH Keywords: join irreducibles
zbMATH Keywords: \(\vee \)- representation
zbMATH Keywords: Kurosh-Ore exchange property
zbMATH Keywords: consistent lattices
reviewed by: Ján Jakubík
MaRDI profile type: Publication
cites work: Decomposition Theory for Nonsemimodular Lattices
cites work: Q3956998
cites work: Meet-distributive lattices and the anti-exchange closure
cites work: Q3875978
cites work: S-verklebte Summen von Verbänden
cites work: Matchings and Radon transforms in lattices. I: Consistent lattices
cites work: Q3872514
cites work: Matchings for linearly indecomposable modular lattices
cites work: Subdirect decomposition of concept lattices
description: scientific article; zbMATH DE number 4105027