Geodesic
Using the Swing Lemma and \( \mathcal{C}_1 \)-diagrams for congruences of planar semimodular lattices
Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 1, pp. 63-74

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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.
Keywords: rectangular lattice, slim planar semimodular lattice, congruence lattice 
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Grätzer, George. Using the Swing Lemma and \( \mathcal{C}_1 \)-diagrams for congruences of planar semimodular lattices. Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 1, pp. 63-74. http://geodesic.mathdoc.fr/item/DMGAA_2023_43_1_a5/