Some Extremal Values of the Number of Congruences of a Finite Lattice

We study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz' edli, we prove that the third, fourth and fifth largest numbers of congruences of an $n$–element lattice are: $5\cdot 2^{n-5}$ if $n\ge 5$, respectively $2^{n-3}$ and $7\cdot 2^{n-6}$ if $n\ge 6$. We also determine the structures of the $n$–element lattices having $5\cdot 2^{n-5}$, respectively $2^{n-3}$ congruences, along with the structures of their congruence lattices.