Abstract
This paper may be regarded as a continuation of the investigations begun in [2]; certain intrinsic lattice topologies are studied, especially the order and ideal topologies in Boolean algebras, bicompactly generated lattices, and other more general structures. The results of [1], [2], and [3] are shown to be closely related. It is proved that the ideal topology on any Boolean algebra has a closed subbase consisting of all sublattices, whereas the order topology on an atomic Boolean algebra has a closed subbase consisting of all sub-complete lattices. It is also shown that the order topology on an atomic Boolean algebra is autouniformizable (in the sense defined by Rema [3]) and, if the ground set is infinite, strictly coarser than the ideal topology. The conditions Cl and C3 on a lattice, introduced by Kent [1], are shown to be slightly stronger than the condition “ bicompactly generated ”, and in complete lattices, where these conditions are satisfied, the order topology is shown to be coarser than the ideal topology.
Publisher
Cambridge University Press (CUP)
Cited by
7 articles.
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1. Order Topology and Frink Ideal Topology of Effect Algebras;International Journal of Theoretical Physics;2010-08-05
2. Algebraic Ordered Sets and Their Generalizations;Algebras and Orders;1993
3. Order-Topological lattices;Glasgow Mathematical Journal;1980-01
4. Semilattices having bialgebraic congruence lattices;Pacific Journal of Mathematics;1979-11-01
5. Ideal topology on a distrbutive lattice;Journal of the Australian Mathematical Society;1974-12