Lattices with congruence densities larger than $3/32$

TL;DR

This work introduces the congruence density $cd(L)$ for finite lattices and proves a sharp threshold: when $cd(L)> frac{3}{32}$, a finite lattice $L$ has balanced join- versus meet-irreducible structure and a symmetric distribution of covering relations. It develops a structural classification of lattices with $cd(L)> frac{3}{32}$, based on the core lattice $\textup{Cor}(L)$ and canonical glued-sum decompositions, and identifies eight core configurations that realize the threshold. The authors provide tools linking the congruence lattice $\textup{Con}(L)$ to edge- and interval-structures via congruence-determining edge sets, enabling a concrete catalog of possible cores (including $M_3$, $N_{5,5}$, and glued sums of $B_4$ and circ-type lattices). Consequences include explicit values for the $k$th largest congruence counts $\textup{lcd}(n,k)$ for $n\ge 8$, and the results offer a path for refining the density threshold and understanding maximal congruence patterns in finite lattices.

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. Mureşan, 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 $k\in\{n+1, n+2, n+3,n+4\}$.

Figures (11)

• Figure 1: Some lattices occurring in the paper
• Figure 2: Illustrations for the canonical glued sum decompositions
• Figure 3: The core lattices in Theorem \ref{['thm:main']}(c\ref{['th3']})
• Figure 4: $N_{5,5}$ and illustrations for the proof of Lemma \ref{['lemma-3a-cha']}
• Figure 5: The core lattices in Theorem \ref{['thm:main']}(c\ref{['th6']})

Theorems & Definitions (40)

• Theorem 1.1
• Lemma 2.1: Czédli czg864
• Theorem 2.2: Main Theorem
• Lemma 2.3
• Corollary 2.4
• Lemma 3.1
• proof
• Lemma 3.2: Grätzer GGprimpers
• Lemma 3.3
• proof