Isomorphism Theorems for Impartial Combinatorial Games

TL;DR

This paper develops a categorical framework for impartial combinatorial games, demonstrating that classical algebraic isomorphism theorems apply, and introduces quotient constructions for game analysis.

Contribution

It introduces the category of optiongraphs and proves that universal algebra isomorphism theorems hold within this framework, enabling new simplification methods.

Findings

Outcomes and nim-values are preserved under option-preserving maps

Universal algebra isomorphism theorems are valid in the category of optiongraphs

Quotient optiongraphs facilitate game analysis

Abstract

We introduce the category of optiongraphs and option-preserving maps as a model to study impartial combinatorial games. Outcomes, remoteness, and extended nim-values are preserved under option-preserving maps. We show that the four isomorphism theorems from universal algebra are valid in this category. Quotient optiongraphs, including the minimum quotient, provide simplifications that can help in the analysis of games.