Accumulation points of congruence densities of finite lattices

TL;DR

This paper investigates the set of congruence densities of finite lattices within a variety, revealing their order-theoretic structure and characterizing modularity through accumulation points.

Contribution

It establishes the structure of congruence density sets as countably infinite dually well-ordered monoids and characterizes modularity via accumulation points.

Findings

SCD(𝓦) is a countably infinite dually well-ordered monoid.

The set of accumulation points of SCD(𝓦) is either a singleton or countably infinite.

Modularity is characterized by SCD(𝓥(K)) having a single accumulation point.

Abstract

Let $\mathcal W$ be a nontrivial variety of lattices, and let $L$ be a finite lattice in $\mathcal W$. The congruence density of $L$ with respect to $\mathcal W$ is the number of congruences of $L$ divided by the maximum number of congruences of $|L|$-element lattices belonging to $\mathcal W$. We prove that, with respect to the order and multiplication of the real numbers, the set SCD$(\mathcal W)$ of congruence densities of finite members of $\mathcal W$ as well as its topological closure are countably infinite dually well-ordered monoids. We also prove that the set of accumulation points of SCD$(\mathcal W)$ is either a singleton or it is countably infinite; furthermore, it is a singleton if and only if $\mathcal W$ is a subvariety of the variety of modular lattices. This gives a complicated characterization of modularity: a non-singleton lattice $K$ is modular if and only if SCD$(\mathcal V(K))$, where $\mathcal V(K)$ denotes the variety generated by $K$, has only one accumulation point. The class $\mathcal S$ of semimodular lattices is not a variety, but SCD$(\mathcal S)$ is still meaningful; we prove that SCD$(\mathcal S)$ has exactly one accumulation point.