The lattice of congruence lattices of algebras on a finite set

Abstract

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

arXiv comment22 pages, 8 figures, some typos corrected, minor changes for better presentation, Remark 7.8 added