The lattice of congruence lattices of algebras on a finite set

الأصل · EN

The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice E. We describe the atoms and coatoms. Each meet-irreducible element of E 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 E; in particular, we prove that E is tolerance-simple whenever |A|≥ 4.