Congruences and trajectories in planar semimodular lattices

TL;DR

This paper generalizes a 1955 lattice theory result by using prime-projectivity and trajectories to determine congruence relations in slim, planar, semimodular lattices, extending previous special-case findings.

Contribution

It extends Czédli's trajectory-based approach from slim rectangular lattices to all slim, planar, semimodular lattices using prime-projectivity concepts.

Findings

Generalization of trajectory method to broader lattice classes

Characterization of congruence relations via prime-projectivity

Unified approach for finite lattice congruences

Abstract

A 1955 result of J.~Jakub\'i 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. Cz\'edli'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 Cz\'edli's result to arbitrary slim, planar, semimodular lattices.