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