Swing lattice game and a short proof of the swing lemma for planar semimodular lattices

TL;DR

This paper presents a shorter proof of the swing lemma for planar semimodular lattices and introduces an online game called Swing lattice game inspired by the lemma and pinball mechanics.

Contribution

It provides a more concise proof of the swing lemma and introduces a novel online game based on lattice theory concepts.

Findings

Shorter proof of the swing lemma for planar semimodular lattices

Development of the Swing lattice game inspired by the lemma

Availability of a computer program implementing the game

Abstract

The swing lemma, due to G. Gr\"atzer for slim semimodular lattices and extended by G. Cz\'edli and G. Gr\"atzer for all planar semimodular lattices, describes the congruence generated by a prime interval in an efficient way. Here we present a new proof for this lemma, which is shorter than the earlier two. Also, motivated by the swing lemma and mechanical pinball games with flippers, we construct an online game called Swing lattice game. A computer program realizing this game is available from the authors' websites.