On a property of congruence lattices of slim, planar, semimodular lattices

TL;DR

This paper simplifies the proof of the Three-pendant Three-crown Property for congruence lattices of slim, planar, semimodular lattices by using the Swing Lemma, making the proof more elementary and concise.

Contribution

It provides a shorter, elementary proof of the 3P3C property using the Swing Lemma, replacing a lengthy previous proof.

Findings

3P3C property verified with Swing Lemma

Elementary proof approach established

Simplifies understanding of congruence lattices

Abstract

In a 2121 paper with G\'abor Cz\'edli, we introduced and verified the Three-pendant Three-crown Property, 3P3C, for congruence lattices of slim, planar, semimodular lattices. The proof is very long; in part, because it relies on Cz\'edli's 2021 paper on lamps. This paper verifies 3P3C using the Swing Lemma, an elementary and short approach.