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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]

By Isbell duality, each compact regular frame L is isomorphic to the frame of opens of a compact Hausdorff space X. In this note we study the spectrum Spec(L) of prime filters of a compact regular frame L. We prove that X is realized as the minimum of Spec(L) and the Gleason cover of X as the maximum of Spec(L). We also characterize zero-dimensional, extremally disconnected, and scattered compact regular frames by means of Spec(L). [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]

By Gelfand-Neumark duality, the category C"Alg of commutative C"- algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category ubal of uniformly complete bounded Archimedean ℓ -algebras. Consequently, C"Alg is equivalent to ubal, and this equivalence can be described through complexification. In this article we study ubal within the larger category bal of bounded Archimedean l-algebras. We show that ubal is the smallest nontrivial reective subcategory of bal, and that ubal consists of exactly those objects in bal that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for bal. It follows that ubal is the unique nontrivial reective epicomplete subcategory of bal. We also show that each nontrivial reective subcategory of bal is both monoreective and epireective, and exhibit two other interesting reective subcategories of bal involving Gelfand rings and square closed rings. Dually, we show that Specker R-algebras are precisely the co-epicomplete objects in bal. We prove that the category spec of Specker R-algebras is a mono-coreective subcategory of bal that is co-epireective in a mono-coreective subcategory of bal consisting of what we term ℓ -clean rings, a version of clean rings adapted to the order- theoretic setting of bal. We conclude the article by discussing the import of our results in the setting of complex "-algebras through complexification. [ABSTRACT FROM AUTHOR]