Geodesic
Congruences and Trajectories in Planar Semimodular Lattices
Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 1, pp. 131-142

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A 1955 result of J. Jakubík states that for the prime intervals 𝔭 and 𝔮 of a finite lattice, con(𝔭) ≥ con(𝔮) iff 𝔭 is congruence-projective to 𝔮 (via intervals of arbitrary size). The problem is how to determine whether con(𝔭) ≥ con(𝔮) 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 reprove Czédli’s result and generalize it to arbitrary slim, planar, semimodular lattices.
Keywords: semimodular lattice, planar lattice, slim lattice, rectangular lattice, congruence, trajectory, prime interval 
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Grätzer, G. Congruences and Trajectories in Planar Semimodular Lattices. Discussiones Mathematicae. General Algebra and Applications, Tome 38 (2018) no. 1, pp. 131-142. http://geodesic.mathdoc.fr/item/DMGAA_2018_38_1_a9/