Using the Swing Lemma and C1-diagrams for congruences of planar semimodular lattices
George Grätzer · Discussiones Mathematicae - General Algebra and Applications · 2023
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.The Swing Lemma and a standardized diagram type are used to give direct proofs of Czédli's four properties.