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.
