Order-topological complete orthomodular lattices
type: Item
instance of: scholarly article
title: Order-topological complete orthomodular lattices
zbMATH Open document ID: 0817.06009
DOI: 10.1016/0166-8641(94)00040-A
author: Marcel Erné
author: Z. Riečanová
published in: Topology and its Applications
publication date: 1995-07-27T00:00:00Z
review text: The authors prove the equivalence of the following statements for a complete orthomodular lattice \(L\): (i) \(L\) is order-topological; (ii) \(L\) is (Scott-)continuous; (iii) \(L\) is algebraic; (iv) \(L\) is compactly atomistic; (v) \(L\) is in the order topology a totally order-disconnected topological lattice. A lattice \(L\) (element \(a\in L\)) is hypercompact, if its hypercompact elements are join-dense (the complement of \([a)\) is a finite union of principal ideals). For an orthomodular lattice \(L\), the following conditions are equivalent: (i) \(L\) is hypercompactly generated; (ii) \(L\) is atomic and for every atom \(a\in L\), \(N_ a= \{b\in A| a\not\leq b'\}\) is finite; (iii) \(L\) is meet-continuous, atomic with a \(T_ 2\)- interval topology; (iv) the MacNeille completion of \(L\) is a compact \(T_ 1\) (\(T_ 2\), totally disconnected) OML.
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06B35
zbMATH DE Number: 735682
zbMATH Keywords: hypercompact lattice
zbMATH Keywords: complete orthomodular lattice
zbMATH Keywords: order topology
zbMATH Keywords: topological lattice
reviewed by: Gudrun Kalmbach
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 735682