On distributive sublattices of modular lattices
type: Item
instance of: scholarly article
title: On distributive sublattices of modular lattices
zbMATH Open document ID: 0901.06008
DOI: 10.1007/BF01233920
published in: Algebra Universalis
publication date: 1998-11-24T00:00:00Z
review text: Let \(L\) be a modular lattice and let \(K\) and \(X\) be distributive sublattices of \(L\). \textit{B. Jónsson} [Proc. Am. Math. Soc. 6, 682-688 (1955; Zbl 0066.28301)] gave necessary and sufficient condition for a sublattice of \(L\) generated by \(K\cup X\) to be distributive. The author presents (under some additional conditions on \(K\) and \(X\)) another sufficient condition ensuring distributivity of a sublattice generated by \(K\cup X\). Theorem. Let \(K\) and \(X\) be distributive sublattices of a modular lattice \(L\) such that (i) for all \(x\in X\) there are \(u\), \(v\in K\cap X\) with \(u\leq x\leq v,\) (ii) for all \(x\), \(y\in X\) and \(u\in K\), if \(x\leq u\leq y\) then \(u\in X\). Then the sublattice of \(L\) generated by \(K\cup X\) is distributive.
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 06D05
zbMATH DE Number: 1226564
zbMATH Keywords: distributive sublattice of a modular lattice
author: Zsolt Lengvárszky
reviewed by: Radomír Halaš
MaRDI profile type: Publication
cites work: Q5789040
cites work: Distributive Sublattices of a Modular Lattice
cites work: Distributive sublattices and weakly independent subsets in modular lattices
cites work: Examples of Continuous Geometries
full work available at URL: https://doi.org/10.1007/bf01233920
OpenAlex ID: W87380424
description: scientific article; zbMATH DE number 1226564