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

Abstract

A planar semimodular lattice K is slim if 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 No Child Property: Let P be the ordered set of join-irreducible congruences of K. Let x, y, z ∈ P and let z be a maximal element of P. If x = y and x, y ≺ z in P, then there is no element u of P such that u ≺ x, y in P.We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Czédli's, to verify his four properties.