Lattices and quantum logics with separated intervals, atomicity
type: Item
instance of: scholarly article
title: Lattices and quantum logics with separated intervals, atomicity
zbMATH Open document ID: 0906.06004
DOI: 10.1023/A:1026642028987
author: Z. Riečanová
published in: International Journal of Theoretical Physics
publication date: 1999-02-01T00:00:00Z
review text: A bounded lattice is said to have separated intervals if it admits, for any two disjoint intervals \([a,b]\) and \([c,d]\), a finite covering that consists of closed intervals each of which is disjoint with at least one of the intervals \([a,b]\) and \([c,d]\). The author investigates connections between atomicity of a bounded lattice \(L\), the interval topology on \(L\) and on the MacNeille completion of \(L\), and the condition that \(L\) has separated intervals. For example, a complete lattice \(L\) satisfies the condition iff the interval topology on \(L\) is Hausdorff. Some other results: An atomistic ortholattice has separated intervals iff the interval topology on its MacNeille completion is Hausdorff. A complete orthomodular lattice having separated intervals is atomic. A Boolean lattice has separated intervals iff it is atomic.
reviewed by: Jānis Cīrulis
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06E05
zbMATH DE Number: 1196982
zbMATH Keywords: atomic lattice
zbMATH Keywords: interval topology
zbMATH Keywords: MacNeille completion
zbMATH Keywords: ortholattice
zbMATH Keywords: bounded lattice
zbMATH Keywords: complete lattice
zbMATH Keywords: orthomodular lattice
zbMATH Keywords: Boolean lattice
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1196982