Congruences and trajectories in planar semimodular lattices

Abstract

A 1955 result of J. Jakubik 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. Czedli’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 Czedli’s result and generalize it to arbitrary slim, planar, semimodular lattices.