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 
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/
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