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Uniform lattices. II: Order continuity and exhaustivity. (English) Zbl 0799.06014
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 §5 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.

MSC:
06B30 Topological lattices
06F30 Ordered topological structures (aspects of ordered structures)
54H12 Topological lattices, etc. (topological aspects)
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