Using the Swing Lemma and $\mathcal{C}_1$-diagrams for congruences of planar semimodular lattices

TL;DR

This paper uses the Swing Lemma and specialized diagrams to verify four properties of congruence lattices in slim, planar, semimodular lattices, advancing understanding of their structural characteristics.

Contribution

It applies the Swing Lemma and diagram techniques to confirm four new properties of congruence lattices in slim, planar, semimodular lattices, including the No Child Property.

Findings

Verified four properties of congruence lattices in slim, planar, semimodular lattices.

Confirmed the No Child Property for these lattices.

Utilized the Swing Lemma and standardized diagrams for proofs.

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~$\mathcal{P}$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in \mathcal{P}$ and let $z$ be a~maximal element of $\mathcal{P}$. If $x \neq y$ and $x, y \prec z$ in $\mathcal{P}$, then there is no element $u$ of $\mathcal{P}$ such that $u \prec x, y$ in $\mathcal{P}$.} We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Cz\'edli's, to verify his four properties.