Congruences and trajectories in planar semimodular lattices

2014/06/02 by George Grätzer, Grätzer, George
Mathematics · #06C10 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:06C10

paper · pdf · doi:10.48550/arxiv.1406.0439

The topic of this paper was subsumed by the paper Congruences and prime-perspectivities in finite lattices

arxiv created 2014/10/09 · arxiv updated 2014/10/10

Abstract

A 1955 result of J.~Jakubí k states that for the prime intervals \fp and \fq of a finite lattice, \con\fp ≥ \con\fq iff \fp is congruence-projective to~\fq (via intervals of arbitrary size). The problem is how to determine whether \con\fp ≥ \con\fq involving only prime intervals. Two recent papers approached this problem in different ways. G. Czédli's used trajectories for slim rectangular lattices---a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to generalize Czédli's result to arbitrary slim, planar, semimodular lattices.

Related

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• Congruences in slim, planar, semimodular lattices: The Swing Lemma
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• Using the Swing Lemma and C₁-diagrams for congruences of planar semimodular lattices
• An extension theorem for planar semimodular lattices