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arXiv · 1406.0439

Congruences and trajectories in planar semimodular lattices

George Grätzer
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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.

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George Grätzer. 2014-06-02. Congruences and trajectories in planar semimodular lattices. https://arxiv.org/abs/1406.0439

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