Lattices of Closed Quasiorders

Abstract

Motivated by questions about semilattices of ordered compactifications, we study the structure of the lattice $\mathcal{Q}(Y)$ of all closed quasiorders on a (compact) Hausdorff space $Y$. For example, we show that the meets of coatoms are precisely those quasiorders which make the underlying space totally order-disconnected.
We describe the covering relation of such lattices and characterize “modular” and “semimodular” elements. In particular, we show that the closed equivalence relations on $Y$ are precisely those upper semimodular elements of $\mathcal{Q}(Y)$ which are not coatoms, and for $|Y|\ge 3$, they are just the joins of bi-semimodular elements.
As a consequence of these results, two compact spaces are homeomorphic if and only if their lattices of closed quasiorders are isomorphic.