Using the Swing Lemma and Cz\'edli diagrams for congruences of planar semimodular lattices

TL;DR

This paper applies the Swing Lemma and Czédli diagrams to verify four properties of congruence lattices of slim, planar, semimodular lattices, including the No Child Property, advancing understanding of their structure.

Contribution

It introduces a method combining the Swing Lemma and Czédli diagrams to verify key properties of congruence lattices in slim, planar, semimodular lattices.

Findings

Verification of Czédli's four properties, including the No Child Property.

Demonstration of the effectiveness of the Swing Lemma in this context.

Enhanced understanding of the structure of congruence lattices in slim, planar, semimodular lattices.

Abstract

A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_3$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No Child Property}: \emph{Let~$P$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in P$ and let $z$ be a~maximal element of $P$. If $x \neq y$, $x, y \prec z$ in $P$, then there is no element $u$ of $P$ such that $u \prec x, y$ in $P$.} We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Cz\'edli's, to verify Cz\'edli's four properties.