Uniform lattices. II: Order continuity and exhaustivity
type: Item
instance of: scholarly article
title: Uniform lattices. II: Order continuity and exhaustivity
zbMATH Open document ID: 0799.06014
DOI: 10.1007/BF01765846
author: Hans Weber
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1994-11-17T00:00:00Z
review text: This paper is a continuation of Part I [ibid. 160, 347-370 (1991; Zbl 0790.06006)]. For the definition of lattice topologies and lattice uniformities we refer to Section 1.1 of Part I. The main aim of this paper is to give conditions under which a lattice topology is weaker than another one. For that order continuity plays a decisive role. Our general setting leads to common generalizations of results for \(FN\)-topologies on Boolean rings and locally solid topologies on Riesz spaces or on \(l\)-groups. In \S5 the problem of comparing lattice topologies is examined under an order-completeness assumption on the lattice. That assumption can be dropped if one of the lattice topologies is induced by an exhaustive lattice uniformity. In Section 8.1 the problem above is examined under additional countability assumptions. In Section 8.2 we briefly point out that also Lebesgue's dominated convergence theorem can be formulated and proved in our abstract setting as consequence of previous considerations. Some results of this paper can also be used as key in the study of modular functions, which will be explained in another paper.
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 06F30
Mathematics Subject Classification ID: 54H12
zbMATH DE Number: 590103
zbMATH Keywords: lattice topologies
zbMATH Keywords: lattice uniformities
zbMATH Keywords: order continuity
zbMATH Keywords: \(FN\)- topologies on Boolean rings
zbMATH Keywords: locally solid topologies on Riesz spaces
zbMATH Keywords: \(l\)-groups
zbMATH Keywords: Lebesgue's dominated convergence theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590103