On the Largest Numbers of Congruences of Finite Lattices

Abstract

We investigate the possible values of the numbers of congruences of finite lattices of an arbitrary but fixed cardinality. Motivated by a result of Freese and continuing Czedli’s recent work, we prove that the third, fourth and fifth largest numbers of congruences of an n–element lattice are: 5 ⋅ 2n− 5 if n ≥ 5, 2n− 3 and 7 ⋅ 2n− 6 if n ≥ 6, respectively. We also determine the structures of the n–element lattices having 5 ⋅ 2n− 5, 2n− 3, respectively 7 ⋅ 2n− 6 congruences, along with the structures of their congruence lattices.