Congruences and Trajectories in Planar Semimodular Lattices

A 1955 result of J. Jakubík states that for the prime intervals \(\mathfrak{p}\) and \(\mathfrak{q}\) of a finite lattice, \(con(\mathfrak{p}) ≥ con(\mathfrak{q})\) iff \(\mathfrak{p}\) is congruence-projective to \(\mathfrak{q}\) (via intervals of arbitrary size). The problem is how to determine whether \(con(\mathfrak{p}) ≥ con(\mathfrak{q})\) 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.