Quasiorders and sublattices of distributive lattices
The author studies the lattice \(\text{Sub}_{01}(L)\) of all \((0,1)\)-sublattices of a distributive lattice \(L\), using certain compatible quasiorders on the Priestley space of \(L\). The main theorems describe \(\text{Sub}_{01}(L)\) in terms of special quasiorders on the Priestley space of \(L\) and characterize the covering relation in \(\text{Sub}_{01}(L)\) in terms of critical pairs. Special emphasis is laid on the case where \(L\) is finite, where epic sublattices, Frattini sublattices and covers are considered. The author demonstrates that quasiorders may serve as a concept that is suitable to unify the many different representations of sublattices of \(L\) that are found in the literature.
scientific article; zbMATH DE number 1785650