Congruences in slim, planar, semimodular lattices: The Swing Lemma

TL;DR

This paper introduces the Swing Lemma, a powerful tool for describing how congruences are generated by prime intervals specifically in slim, planar, semimodular lattices, building on earlier concepts of prime-perspectivity.

Contribution

It specializes the Prime-projectivity Lemma to a specific class of lattices, providing a new and more powerful description of congruence spreading in these structures.

Findings

The Swing Lemma offers a detailed description of congruence generation.

It extends the concept of prime-projectivity to slim, planar, semimodular lattices.

The lemma simplifies understanding of congruence relations in this lattice class.

Abstract

In an earlier paper, to describe how a congruence spreads from a prime interval to another in a finite lattice, I introduced the concept of prime-perspectivity and its transitive extension, prime-projectivity and proved the Prime-projectivity Lemma. In this paper, I specialize the Prime-projectivity Lemma to slim, planar, semimodular lattices to obtain the Swing Lemma, a very powerful description of the congruence generated by a prime interval in this special class of lattices.