# Semilattices of Ordered Compactifications

Research paper by Jürgen Reinhold

Indexed on: 01 Sep '97Published on: 01 Sep '97Published in: Order

#### 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 $$K_o ({\text{X}})$$.