ORDERED ONEPOINT-COMPACTIFICATIONS, STABLY CONTINUOUS FRAMES AND TENSORS

Abstract

We investigate the structure of semilattices K ₀ (X) of all ordered compactifications of ordered topological spaces X with a one-point Nachbin-compactification. These semilattices and their isomorphic copies are called oc _l-semilattices. We give an abstract characterization of all oc _l-lattices by means of certain generalized stably continuous frames. A finite ordered set is shown to be a dual oc _l-semilattice iff it is a distributive tensor, that is, a 2-consistently complete meet-semilattice T whose principal ideals are distributive and which contains two disjoint elements t _o, t ₁ such that s = (t _o & s) V (t ₁ ∧ s) for all s ∈ T. More generally, we characterize those dual oc _l-semilattices which are finite unions of principal ideals.