Lattices of Closed Quasiorders

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 \(|Y| \geq 3\), 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.