Congruence structure of planar semimodular lattices: The General Swing Lemma

TL;DR

This paper generalizes the Swing Lemma, a key result describing how congruences propagate in slim planar semimodular lattices, to a broader class of planar semimodular lattices, enhancing understanding of their structure.

Contribution

The paper extends the Swing Lemma from slim to all planar semimodular lattices, broadening its applicability in lattice theory.

Findings

Generalized Swing Lemma to all planar semimodular lattices

Provided new insights into congruence propagation in these lattices

Enhanced theoretical framework for lattice congruences

Abstract

The Swing Lemma of the second author describes how a congruence spreads from a prime interval to another in a slim (having no $M_3$ sublattice), planar, semimodular lattice. We generalize the Swing Lemma to planar semimodular lattices.