Quasiorder lattices of varieties

Abstract

The set ${{\mathrm{Quo}}}(\mathbf {A})$ of compatible quasiorders (reflexive and transitive relations) of an algebra $\mathbf {A}$ forms a lattice under inclusion, and the lattice ${{\mathrm{Con}}}(\mathbf {A})$ of congruences of $\mathbf {A}$ is a sublattice of ${{\mathrm{Quo}}}(\mathbf {A})$. We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice $\mathbf {M}_3$. We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.