We investigate the structure of semilatticesK0(X) of all ordered compactifications of ordered topological spaces X with a one-point Nachbin-compactification. These semilattices and their isomorphic copies are calledocl-semilattices. We give an abstract characterization of allocl-lattices by means of certain generalized stably continuous frames. A finite ordered set is shown to be a dualocl-semilattice iff it is a distributive tensor, that is, a 2-consistently complete meet-semilatticeTwhose principal ideals are distributive and which contains two disjoint elementsto,t1such thats= (to&s) V (t1∧s) for alls∈T. More generally, we characterize those dualocl-semilattices which are finite unions of principal ideals.