Lattices in Chip-Firing

TL;DR

This paper investigates the structure of move posets in chip-firing games, revealing their role as join-irreducibles and applying this understanding to line configurations and labeled chip-firing.

Contribution

It establishes that the move poset forms the join-irreducibles of configuration posets and extends analysis to chip-firing on a line, answering a question of Propp.

Findings

Move poset forms join-irreducibles of configuration poset

Graph augmentation technique used in analysis

Application to labeled chip-firing on a line

Abstract

We analyze the poset of moves in chip-firing, as defined by Klivans and Liscio. Answering a question of Propp, we show that the move poset forms the join-irreducibles of the poset of configurations. The proof involves a graph augmentation and an analysis of configurations in which only one firing move is available. We then use this framework to analyze the problem of chip-firing on a line, where the move poset is relevant to the problem of labeled chip-firing.