The lattice of congruence lattices of algebras on a finite set

TL;DR

This paper investigates the structure of the lattice formed by all congruence lattices of algebras on a finite set, characterizing atoms, coatoms, and certain meet-irreducibles, and establishing properties like tolerance-simplicity for larger sets.

Contribution

It provides a complete characterization of atoms, coatoms, and specific meet-irreducible elements in the lattice of congruence lattices on finite sets, and proves new properties such as tolerance-simplicity.

Findings

Characterization of atoms and coatoms in the lattice

Complete description of meet-irreducible elements determined by unary mappings

Proof that the lattice is tolerance-simple for sets of size at least 4

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 $|A|\geq 4$.