Congruences and trajectories in planar semimodular lattices

Abstract

A 1955 result of J.~Jakubí k states that for the prime intervals $\fp$ and $\fq$ of a finite lattice, $\con{\fp} \geq \con{\fq}$ if{}f $\fp$ is congruence-projective to~$\fq$ (\emph{via} intervals of arbitrary size). The problem is how to determine whether $\con{\fp} \geq \con{\fq}$ 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.

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