Proximity frames and regularization. (Q2015817)

By a proximity on a frame \(L\) is usually meant a strong relation \(\vartriangleleft\) on \(L\) [see \textit{B. Banaschewski}, Math. Nachr. 149, 105-115 (1990; Zbl 0722.54018)]. In [Cah. Topologie Géom. Différ. Catég. 31, No. 4, 305-313 (1990; Zbl 0738.18002)] \textit{J. L. Frith} coined the name `\textit{proximal frames}' for the corresponding pairs \((L,\vartriangleleft)\). In the present paper, the authors introduce a different notion of a proximity in frames, as a common generalization of proximities in de Vries algebras and the way below relation on stably compact frames. In their sense, a \textit{proximity} in a frame \(L\) is a binary relation \(\prec\) satisfying (i) \(0\prec 0\) and \(1\prec 1\), (ii) \(a\prec b\) implies \(a\leq b\), (iii) \(a\leq b\prec c\leq d\) implies \(a\prec d\), (iv) \(a,b\prec c\) implies \(a\vee b\prec c\), (v) \(a\prec b,c\) implies \(a\prec b\wedge c\), (vi) \(a\prec b\) implies there exists \(c\in L\) with \(a\prec c\prec b\), (vii) \(a=\bigvee\{b\in L\mid b\prec a\}\). The pair \((L,\prec)\) is called a \textit{proximity frame}. Examples of proximity frames are (1) any de Vries algebra, (2) any frame with its partial ordering, (3) any proximal frame, (4) any stably compact frame with its way below relation, (5) any regular frame with the well inside relation, and (6) any completely regular frame with the really inside relation. The morphisms of the category \(\mathbf{PrFrm}\) of proximity frames are the maps \(h\colon L\to M\) satisfying (i) \(h(0)=0\) and \(h(1)=1\), (ii) \(h(a\wedge b)=h(a)\wedge h(b)\), (iii) \(a\prec b\) and \(c\prec d\) imply \(h(a\vee c)\prec h(b\vee d)\), (iv) \(h(a)=\bigvee\{h(b)\mid b\prec a\}\). The usual category of proximal frames is therefore a (non-full) subcategory of the newly category (Frith's maps are precisely the proximity morphisms that preserve arbitrary joins). As it is well-known, the category \(\mathbf{KHaus}\) of compact Hausdorff spaces is dually equivalent to the category \(\mathbf{KRFrm}\) of compact regular frames. Furthermore, by de Vries duality, \(\mathbf{KHaus}\) is also dually equivalent to the category \(\mathbf{DeV}\) of de Vries algebras. The main goal of this paper is to lift these dual equivalences to the setting of stably compact spaces. In this setting, the role of de Vries algebras is taken by proximity frames (more specifically, its full subcategory \(\mathbf{RPrFrm}\) of regular proximity frames defined by a functorial process of regularization that extends the Booleanization functor). It is shown that the category \(\mathbf{StKSp}\) of stably compact spaces is dually equivalent to the category \(\mathbf{StKFrm}\) of stably compact frames and that \(\mathbf{StKSp}\) is also dually equivalent to \(\mathbf{RPrFrm}\). Restricting back to the compact Hausdorff setting, \(\mathbf{PrFrm}\) provides a new category \(\mathbf{StrInc}\) whose objects are frames with strong inclusions. Both \(\mathbf{KRFrm}\) and \(\mathbf{DeV}\) are subcategories of \(\mathbf{StrInc}\) that are equivalent to \(\mathbf{StrInc}\). The restrictions of these categories are considered also in the setting of spectral spaces, Stone spaces, and extremally disconnected spaces, and links to the categories of distributive lattices and Boolean algebras are discussed.