A bitopological point-free approach to compactifications
A d-frame consists of two frames \(L_{-}\) and \(L_{+}\) together with two relations \(\mathsf{tot}\subseteq L_-\times L_+\) and \(\mathsf{con}\subseteq L_+\times L_-\) satisfying certain axioms (imposing, in particular, that \(\mathsf{tot}\) is an upper set in \(L_-\times L_+\) and a sublattice of \(L_{-}^{\text{op}}\times L_+\) and that \(\mathsf{con}\) is a lower set in \(L_+\times L_-\), a sublattice of \(L_{+}\times L_{-}^{\text{op}}\) and closed under directed joins in \(L_+\times L_-\)). D-frames were introduced by the last two authors of the present paper in [On the bitopological nature of Stone duality. Technical Report CSR-06-13, School of Computer Science, The University of Birmingham (2006)] and are a kind of point-free analogue of bitopological spaces (different from the well-known biframes of \textit{B. Banaschewski, G. C. L. Brümmer} and \textit{K. A. Hardie} [``Biframes and bispaces'', Quaest. Math. 6, 13--25 (1983; Zbl 0513.06005)]. Indeed, the corresponding category is dually adjoint to the category of bitopological spaces and bicontinuous maps. This adjunction yields a general bitopological version of Stone duality, encompassing a number of classical Stone-type dualities (namely, those of Stone for Boolean algebras and bounded distributive lattices, those of the two authors for strong proximity lattices with negation and the well-known Ehresmann-Bénabou duality for ordinary frames). In the paper under review, the authors investigate d-frames and point-free compactifications of d-frames. They study a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames by their associated proximities. Further, they show that the category of d-frames has a normal coreflection (a somewhat surprising fact) taking completely regular d-frames to regular normal d-frames, and that the Stone-Čech compactification of d-frames factors through it. They end the paper with an adjunction between the category of d-frames and the category of ordinary frames that allows to subsume \textit{B. Banaschewski}'s compactifications of frames [``Compactification of frames'', Math. Nachr. 149, 105--115 (1990; Zbl 0722.54018)] as well as \textit{M. B. Smyth}'s stable compactifications [``Stable compactification. I.'', J. Lond. Math. Soc., II. Ser. 45, No. 2, 321--340 (1992; Zbl 0760.54018)] within their compactifications of d-frames as special cases.
scientific article; zbMATH DE number 5935899
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