Semilattices of ordered compactifications

We investigate the structure of the complete join-semilattice K-0(X) of all (non-equivalent) ordered compactifications of a completely regular ordered space X. We show that an ordered set is an oc-semilattice, that is, isomorphic to some X,(X), if and only if it is dually isomorphic to the system Q(Y)(X), (less than or equal to) of all closed quasiorders rho on a compact space Y = (Y, tau) which induce a given order less than or equal to on a subset X of Y and for which the relation rho\(Y\X)(2) is antisymmetric. It turns out that the complete lattices of the form K-0(X) are, up to isomorphism, exactly the duals of intervals in the closure systems GZ(Y) of all closed quasiorders on compact spaces Y. For finite oc-semilattices, we give a purely order-theoretical description. In particular, we show that a finite lattice is isomorphic to some K-0(X) if and only if it is dually isomorphic to an interval in the lattice Q(Y) of all quasiorders on a finite set Y. In connection with very recent investigations of lattices of the form Q(Y) and Q(Y) and their intervals we gain from these representation theorems substantial insights into the structure of the semilattices K-0(X).