A Gelfand duality for compact pospaces

Abstract

Let X be a compact Hausdorff space. It is well known that X can be characterized by its ring of real continuous functions, by its set of regular open subsets or more simply by its set of open subsets. These characterizations lead to dualities between the category KHaus, of compact Hausdorff space and respectively the categories C -alg (or equivalently ubal), of commutative C -algebras, DeV of de Vries algebras and KRFrm of compact regular frames. Later, G.Bezhanishvili and J.Harding extended in [1] a part square to dualities between the categories StKSp of stably compact spaces, RPrFrm of regular proximity frames and StKFrm of stably compact frames.We thus get the square of dualities extended this way. StKSp RPrFrmKHaus DeV C -alg KRFrm StKFrmOur aim is to complete the outside triangle, looking for a category generalizing the Calgebras.Using the equivalences between StKSp and the category KPSp of compact po-spaces (see [4]), an essential fact, due to G.Hansoul in [5] leads us to consider a category of ordered semiring. Indeed, we can see that the Nachbin-Stone-Cech compactification of a completely regular ordered po-space X can be realized through its semi-ring of increasing, continuous and real, positive functions, denoted I(X, R + ). Following the definitions of G.Bezhanishvili, P.Morandi and B.Olberding in [2], we define the bounded Archimedean -semi-algebras this way. Definition 1.1. An -semi-ring is an algebra (A, +, ., 0, 1, ≤) with the following axioms :(a) (A, +, 0) and (A, ., 1) are commutative monoids.2. An -semi-ring A is bounded if for all a ∈ A, there is n ∈ N such that a ≤ n.1.3. An -semi-ring A is Archimedean if for all a, b, c, d ∈ A, whenever n.a + b ≤ n.c + d, then a ≤ c.

Towards a Gleason Cover for Compact Pospaces

“…We have completed the external network of equivalences and dualities started in [4] and [8], generalising to the "distributive setting" the duality between Gleason spaces and compact Hausdorff spaces of [3]. Hence, we obtain the following commutative diagram, where the arrowed lines represent adjunctions and the non-arrowed ones equivalences or dualities.…”

“…Indeed, Bezhanishvili and Harding extended in [4] the dualities and equivalences between KHaus, KrFrm and DeV to dualities and equivalence between the categories StKSp of stably compact spaces, StKFrm of stably compact frames and PrFrm of proximity frames. As for the duality between KHaus and C ⋆ -alg, a real version of the duality, given in [5], was extended in [8] to a duality between KPSp of compact pospaces and the category usbal of Stone semirings. We refer to [4] and [8] for the relevant definitions.…”

“…As for the duality between KHaus and C ⋆ -alg, a real version of the duality, given in [5], was extended in [8] to a duality between KPSp of compact pospaces and the category usbal of Stone semirings. We refer to [4] and [8] for the relevant definitions. The aim of this paper is to complete the extensions initiated in [4] and [8] to the category GlSp.…”

“…We refer to [4] and [8] for the relevant definitions. The aim of this paper is to complete the extensions initiated in [4] and [8] to the category GlSp. We point out that this extension process follows the same spirit as passing from Boolean algebras to distributive lattices, and from Stone spaces to Priestley spaces in the zero-dimensional setting (from the Boolean to the distributive setting as we shall often say in this paper).…”

A Unified Approach to Gelfand and de Vries Dualities

“…In [12, 13] both dualities were extended to completely regular spaces and their compactifications. In [21] Gelfand duality was generalized to the setting of compact ordered spaces studied by Nachbin [33], and in [25] a general categorical framework was developed that yields de Vries duality and its generalizations. However, as far as we know, there is no unifying approach to Gelfand and de Vries dualities.…”