Semilattices of Ordered Compactifications

Abstract

We investigate the structure of the complete join-semilattice $K_o ({\text{X}})$ of all (non-equivalent) ordered compactifications ofa completely regular ordered space X. We show that anordered set is an oc-semilattice, that is, isomorphic to some $K_o ({\text{X}})$, if and only if it is dually isomorphic to thesystem $\mathcal{Q}\left( {\text{Y}} \right)_{X, \leqslant }$ of all closed quasiorders ρ on a compact space ${\text{Y = (}}Y,\tau {\text{)}}$ whichinduce a given order ≤ on a subset X of Y and for which therelation $\rho \backslash (Y\backslash X)^2$ is antisymmetric. Itturns out that the complete lattices of the form $K_o ({\text{X}})$ are, up to isomorphism, exactly the duals ofintervals in the closure systems $\mathcal{Q}\left( {\text{Y}} \right)$ of allclosed quasiorders on compact spaces Y. For finiteoc-semilattices, we give a purely order-theoretical description. Inparticular, we show that a finite lattice is isomorphic to some $K_o ({\text{X}})$ if and only if it is dually isomorphic to aninterval in the lattice $\mathcal{Q}\left( Y \right)$ of all quasiorders on afinite set Y. In connection with very recent investigations of lattices ofthe form $\mathcal{Q}\left( {\text{Y}} \right)$ and $\mathcal{Q}\left( Y \right)$ andtheir intervals we gain from these representation theorems substantialinsights into the structure of the semilattices $K_o ({\text{X}})$.