Lattices with many congruences are planar

TL;DR

This paper proves that finite lattices with more than 2^{n-5} congruences are necessarily planar, establishing a sharp boundary between lattice structure and planarity.

Contribution

It introduces a precise threshold for the number of congruences that guarantees lattice planarity, and demonstrates the sharpness of this bound with explicit examples.

Findings

Lattices with > 2^{n-5} congruences are planar.

Existence of non-planar lattices with exactly 2^{n-5} congruences.

The bound is proven to be sharp for all n ≥ 8.

Abstract

Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences.