Using the Swing Lemma and ${\mathcal{C}}_1$-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. Czédli 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 $x\ne y$ 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 Czédli's, to verify his four properties.

arXiv admin note: substantial text overlap with arXiv:2104.13444

References in corpus (1)

• A new property of congruence lattices of slim, planar, semimodular lattices