The lattice of quasiorder lattices of algebras on a finite set

Abstract

The quasiorders of an algebra (A, F) constitute a common generalization of its congruences and compatible partial orders. The quasiorder lattices of all algebras defined on a fixed set A ordered by inclusion form a complete lattice \({\mathcal{L}}\). The paper is devoted to the study of this lattice \({\mathcal{L}}\). We describe its join-irreducible elements and its coatoms. Each meet-irreducible element of \({\mathcal{L}}\) being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterizations, we deduce several properties of the lattice \({\mathcal{L}}\); in particular, we prove that \({\mathcal{L}}\) is always tolerance-simple.