Topological representations of distributive lattices and Brouwerian logics. (English) Zbl 0018.00303

The paper is divided into two parts. The first part generalizes some of Stone’s theory of representations of Boolean algebras by fields of sets [Trans. Am. Math. Soc. 40, 37–111 (1936; Zbl 0014.34002)], to representations of distributive lattices by rings of sets. The theory does not generalize perfectly. Some of the results have been independently published by the reviewer [Duke Math. J. 3, 443–454 (1937; Zbl 0017.19403)], but none of the topological results have. There is a close relation between elements and relatively bicompact subsets of the representing space, and the latter is usually a \(T_0\)-space instead of an \(H\)-space. Also, the author distinguishes “prime ideals” from “divisorless ideals”.
The second part correlates in detail the theory of the first part, with aspects of a Brouwerian logic of Heyting (slightly modified by Stone).


06-XX Order, lattices, ordered algebraic structures
54-XX General topology
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