Finite semilattices with many congruences

TL;DR

This paper investigates the possible sizes of congruence lattices of n-element semilattices, identifying the four largest sizes for large n and describing the semilattices that realize these sizes.

Contribution

It determines the four largest congruence lattice sizes for large n-element semilattices and characterizes the semilattices that achieve these sizes.

Findings

Identified the four largest numbers in NCSL(n) for large n.

Described the specific n-element semilattices that realize these largest sizes.

Established conditions under which these sizes are attained.

Abstract

For an integer $n\geq 2$, 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$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.