Search

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 concept of a strong inclusion on a biframe [Sch93] to that of a proximity on a biframe, which is related to the concept of a strong bi-inclusion on a frame introduced in [PP12b]. We also generalize the concept of a bi-compactification of a biframe [Sch93] to that of a compactification of a biframe, and prove that the poset of compactifications of a biframe L is isomorphic to the poset of proximities on L. As a corollary, we obtain Schauerte's characterization of bi-compactifications of a biframe [Sch93]. In the spatial case this yields Blatter and Seever's characterization of compactifications of completely regular ordered spaces [BS76] and a characterization of bi-compactifications of completely regular bispaces. [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]