Using the Swing Lemma and Cz\'edli diagrams for congruences of 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.