On the Structure of Mis\`ere Impartial Games

TL;DR

This paper explores the algebraic structure of misère impartial games, providing new theoretical insights into their monoid and group properties, and offering methods to classify and count game variants.

Contribution

It introduces new results on the monoid's structure, the group of fractions, prime games, and includes proofs of classical theorems, advancing the theoretical understanding of misère impartial games.

Findings

The group of fractions of the monoid is almost torsion-free.

A method to calculate the number of distinct games born by day 7.

New results on the structure of prime games.

Abstract

We consider the abstract structure of the monoid M of mis\`ere impartial game values. Several new results are presented, including a proof that the group of fractions of M is almost torsion-free; a method of calculating the number of distinct games born by day 7; and some new results on the structure of prime games. Also included are proofs of a few older results due to Conway, such as the Cancellation Theorem, that are essential to the analysis but whose proofs are not readily available in the literature. Much of the work presented here was done jointly with John Conway and Dan Hoey, and I dedicate this paper to their memory.