Semilattices of ordered compactifications (Q1267553)

Let \(X\) be a completely regular ordered space and let \({\mathcal K}_0(X)\) be the complete join semilattice of all (non-equivalent) ordered compactifications of \(X\). If \(Y\) is a topological space containing \(X\) as a subspace, let \(Q(Y)_{X,\leq}\) be the complete meet semilattice of all closed quasiorders on \(Y\) which restrict on \(X\) to \(\leq\), and restricted to \((Y-X)^2\) are antisymmetric. The author shows that an ordered set is isomorphic to some \({\mathcal K}_0(X)\) if and only if it is dually isomorphic to \(Q(Y)_{X,\leq}\) for some \(Y\). He shows when a complete lattice is an oc-lattice, that is, has this property, and he gives a complete list of all non-distributive oc-semilattices with \(\leq 8\) elements.