Finite semilattices with many congruences

Abstract

For an integer $n\ge 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\ge 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.

14 pages, 4 figures