A conjunction in closure spaces

Studia Logica 43 (4):341 - 351 (1984)
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Abstract

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of -conjunctive closure spaces (X is -conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:1. For every closed and proper subset of an -conjunctive closure space its interior is empty (i.e. it is a boundary set). 2. If X is an -conjunctive closure space which satisfies the -compactness theorem and [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice. 3. A closure space is linear iff it is an -conjunctive and topological space. 4. Every continuous function preserves all conjunctions.

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References found in this work

Lattice Theory.Garrett Birkhoff - 1940 - Journal of Symbolic Logic 5 (4):155-157.
Universal Algebra.George Grätzer - 1982 - Studia Logica 41 (4):430-431.

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