Quasiorder lattices of varieties
The set \(\mathrm{Quo}(A)\) of compatible quasiorders of an algebra \(A\) forms a lattice under inclusion and the congruence lattice \(\mathrm{Con}(A)\) is its sublattice. It is proved that a locally finite variety is congruence distributive (modular) if and only if it is quasiorder distributive (modular). It is shown that the same does not hold for meet semi-distributivity. It is known that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of the congruence lattice isomorphic to \(M_3\). The authors prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for infinite algebras even in the variety of semilattices.
scientific article; zbMATH DE number 6904423
scientific article