The lattice of congruence lattices of algebras on a finite set
Jakub\'ikov\'a-Studenovsk\'a, Danica, P\"oschel, Reinhard, Radeleczki, S\'andor · arXiv (Cornell University) · 2016
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 $|A|\geq 4$.