Compactifications and A-compactifications of frames. Proximal frames

Abstract

The aim of this paper is to give two new descriptions of the ordered set of all (up to equivalence) regular compactifications of a completely regular frame. F and to introduce and study the notion of A-frame as a generalization of the notion of Alexandroff space (known also as zero-set space) (Alexandroff[l], Gordon[15]). A description of the ordered set of all (up to equivalence) A-compactifications of an A-frame by means of an ordered by inclusion set of some distributive lattices (called AP-sublattices) is obtained. It implies that any A-frame has a greatest A-compactification and leads to the descriptions of A new category isomorphic to the category of proximal frames is introduced. A question for compactifications of frames analogous to the R. Chandler's question [8, p. 71] for compactifications of spaces is formulated and solved. Many results of [1], [3], [15], [23], [9], [10] and [11] are generalized.

Hausdorff compactifications and zero-one measures

“…one cannot reconstruct cX only on the base of Coz(P cX ) (see [8]). In the present paper we answer the question posed above in the affirmative (see Theorem 3•17(a) and Theorem 3•8), using the technique developed in our papers [5,6]. Then we apply the obtained results for giving a construction of all Hausdorff compactifications of a space X by means of zero-one measures, which is the main purpose of the paper.…”

“…It is well known that the Wallman-type compactifications of a Tychonoff space X can be obtained as spaces of all regular zero-one measures on suitable lattices of subsets of X (see [1,2,4,12]). Using the technique developed in [5,6], we find for any Tychonoff space X a Boolean algebra B X and a set L X of sublattices of B X having the following property: for any Hausdorff compactification cX of X there exists a (unique) L cX ∈ L X such that the maximal spectrum of L cX and the space of all u-regular zero-one measures on the Boolean subalgebra b(L cX ) of B X , generated by L cX , are Hausdorff compactifications of X equivalent to cX. Let us give more details now.…”

Topobooleans and Boolean contact algebras with interpolation property

“…Just as Boolean algebras can be seen as models for the classical propositional logic, frames can be seen as models for the geometric propositional logic, which is a logic with finite conjunctions and infinite disjunctions. Let us also recall that Banaschewski [2] and Frith [29] introduced the notion of proximity frame and in [15] a category isomorphic to that of proximity frames was constructed. Proximity frames were studied as well in [4] and [7].…”