Lattices with congruence densities larger than $3/32$

TL;DR

This paper characterizes finite lattices with high congruence density, specifically those exceeding 3/32, revealing structural properties and enumerating possible congruence counts for larger lattices.

Contribution

It extends previous bounds on congruence densities from 1/8 to 3/32, providing a complete description of lattices exceeding this threshold and their structural implications.

Findings

Lattices with cd(L)>3/32 have equal numbers of join- and meet-irreducible elements.

The bound 3/32 is sharp, as shown by a specific 6-element lattice.

The paper determines the largest possible number of congruences for certain lattice sizes.

Abstract

By a 1997 result of R. Freese, an $n$-element lattice has at most $2^{n-1}$ congruences. This motivates us to define the congruence density cd$(L)$ of a finite $n$-element lattice as $|$Con$(L)|/2^{n-1}$, where $|$Con$(L)|$ is the number of elements of the congruence lattice Con$(L)$ of $L$. We prove that whenever $L$ is a finite lattice with cd$(L)>3/32$, then $L$ has the same number of join-irreducible and meet-irreducible elements. This result is sharp, since there exists a six-element lattice $R_6$ with cd$(R_6)=3/32$ but fewer join-irreducible than meet-irreducible elements. By R. Freese, C. Mure\c{s}an, J. Kulin, and the present author's results, lattices with congruence densities larger than $1/8$ have already been described. Here we decrease the lower threshold from $1/8$ to $3/32$. That is, we describe all finite lattices $L$ such that cd$(L)>3/32$. As a corollary, we give the $k$th largest number of congruences of $n$-element lattices for $n>8$ and $k\in\{n+1, n+2, n+3,n+4\}$.