A characterization of relatively balanced lattices.
type: Item
instance of: scholarly article
title: A characterization of relatively balanced lattices.
zbMATH Open document ID: 1061.06018
author: Tetsuya Abe
published in: Algebra Universalis
publication date: 2005-04-19T00:00:00Z
review text: Let \(L\) be a lattice. \(J(L)\) and \(M(L)\) denote the sets of all join-irreducible \(x \neq 0\) and of all meet-irreducible \(y \neq 1\) in \(L\), respectively. For \(j \in J(L)\), \(j'\) is its unique cover, dually \(m^{*}\) is defined for \(m \in M(L)\). \(A\) lattice \(L\) is balanced if \(j' \leq m\) and \(j \leq m^{*}\) are equivalent for every \(j \in J(L)\) and \(m \in M(L)\). \(L\) is relatively balanced if each of its intervals is balanced. A lattice \(L\) is consistent if for all \(j \in J(L)\) and \(x \in L\), \(j \vee x \in J([x,1]) \cup \{x\}\). Main results: \(L\) is glued by relatively complemented lattices if and only if \(L\) is a relatively balanced lattice. Semimodularity, balancedness and relative balancedness are equivalent in a consistent lattice.
reviewed by: Ivan Chajda
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06C10
zbMATH DE Number: 2158763
zbMATH Keywords: semimodular lattice
zbMATH Keywords: balanced lattice
zbMATH Keywords: consistent lattice
zbMATH Keywords: relatively complemented lattice
MaRDI profile type: Publication
full work available at URL: s00012-002-8172-5
OpenAlex ID: W2025722377
DOI: 10.1007/S00012-002-8172-5
description: scientific article; zbMATH DE number 2158763