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McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames. [ABSTRACT FROM AUTHOR]

Stone duality generalizes to an equivalence between the categories Stone R of Stone spaces and closed relations and BA S of boolean algebras and subordination relations. Splitting equivalences in Stone R yields a category that is equivalent to the category KHaus R of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BA S yields a category that is equivalent to the category De V S of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then gives that KHaus R is equivalent to De V S , thus resolving a problem recently raised in the literature. The equivalence between KHaus R and De V S further restricts to an equivalence between the category KHaus of compact Hausdorff spaces and continuous functions and the wide subcategory De V F of De V S whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition. [ABSTRACT FROM AUTHOR]

Abstract For a topological space X , let L (X) be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [3] that the modal logics S4 , S4.1 , S4.2 , S4.1.2 , S4.Grz , S4. Grz n (n ≥ 1), and their intersections arise as L (X) for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or S4. Grz n for some n ≥ 1. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space. [ABSTRACT FROM AUTHOR]

We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space X and a totally ordered algebra A , we introduce the concept of a finitely valued normal function f : X → A . We show that the operations of A lift to the set FN ( X , A ) of all finitely valued normal functions, and that there is a canonical proximity relation ≺ on FN ( X , A ) . This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain A are axiomatized as proximity Baer Specker A -algebras; that is, the pairs ( S , ≺ ) , where S is a torsion-free A -algebra generated by its idempotents which is a Baer ring, and ≺ is a proximity relation on S . We introduce the category of proximity Baer Specker A -algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker A -algebras. [ABSTRACT FROM AUTHOR]

Abstract: We introduce zero-dimensional de Vries algebras and show that the category of zero-dimensional de Vries algebras is dually equivalent to the category of Stone spaces. This shows that Stone duality can be obtained as a particular case of de Vries duality. We also introduce extremally disconnected de Vries algebras and show that the category of extremally disconnected de Vries algebras is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. As a result, we give a simple construction of the Gleason cover of a compact Hausdorff space by means of de Vries duality. We also discuss the insight that Stone duality provides in better understanding of de Vries duality. [Copyright &y& Elsevier]

Abstract: We introduce the bimodal logic , which is the extension of Bennett’s bimodal logic by Grzegorczyk’s axiom and show that the lattice of normal extensions of the intuitionistic modal logic WS5 is isomorphic to the lattice of normal extensions of , thus generalizing the Blok–Esakia theorem. We also introduce the intuitionistic modal logic WS5.C, which is the extension of WS5 by the axiom , and the bimodal logic , which is the extension of Shehtman’s bimodal logic by Grzegorczyk’s axiom, and show that the lattice of normal extensions of WS5.C is isomorphic to the lattice of normal extensions of . [Copyright &y& Elsevier]

Abstract: We introduce zero-dimensional proximities and show that the poset of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that is isomorphic to the poset of Boolean bases of X, and derive Dwinger''s theorem that is isomorphic to as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone–Čech compactification of X is a unique up to equivalence extremally disconnected compactification of X. [Copyright &y& Elsevier]

Abstract: We develop duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces. We show that these binary relations are in 1–1 correspondence with subframes of Heyting spaces. We introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear iff it is a subframe variety, and that it is dense nuclear iff it is a cofinal subframe variety. We give an alternative proof that every (cofinal) subframe variety of Heyting algebras is generated by its finite members. [Copyright &y& Elsevier]

We prove that S4 is complete with respect to Boolean combinations of countable unions of convex subsets of the real line, thus strengthening a 1944 result of McKinsey and Tarski (Ann. of Math. (2) 45 (1944) 141). We also prove that the same result holds for the bimodal system S4+S5+C, which is a strengthening of a 1999 result of Shehtman (J. Appl. Non-Classical Logics 9 (1999) 369). [Copyright &y& Elsevier]

We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible T1-space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally compact Hausdorff, and spectral spaces. [Copyright &y& Elsevier]