Ungar Games on the Young-Fibonacci and the Shifted Staircase Lattices

TL;DR

This paper investigates Ungar games on specific lattices, proving two conjectures related to the Young-Fibonacci lattice and shifted staircase lattices, advancing understanding of combinatorial game theory on these structures.

Contribution

It proves two conjectures about Ungar games on Young-Fibonacci and shifted staircase lattices, expanding the theoretical framework of these combinatorial games.

Findings

Confirmed conjectures on Ungar games for the lattices.

Characterized game outcomes on these specific lattices.

Enhanced understanding of Ungar game dynamics in lattice structures.

Abstract

In 2023, Defant and Li introduced the Ungar move, which sends an element $v$ of a finite meet-semilattice $L$ to the meet of some subset of the elements covered by $v$. More recently, Defant, Kravitz, and Williams introduced the Ungar game on $L$, in which two players take turns making Ungar moves starting from an element of $L$ until the player that cannot make a nontrivial Ungar move loses. In this note, we settle two conjectures by Defant, Kravitz, and Williams on the Ungar games on the Young-Fibonacci lattice and the lattices of the order ideals of shifted staircases.