The strange algebra of combinatorial games

TL;DR

This paper introduces an algebraic framework for analyzing various combinatorial games, unifying classical and recent game theories through the construction of quotient monoids, enabling deeper structural understanding.

Contribution

It develops a unified algebraic approach to analyze different classes of combinatorial games using quotient monoids, extending classical and misere game theories.

Findings

Unified algebraic framework for combinatorial games

Application of quotient monoids to multiple game classes

Enhanced understanding of game structures

Abstract

We present an algebraic framework for the analysis of combinatorial games. This framework embraces the classical theory of partizan games as well as a number of misere games, comply-constrain games, and card games that have been studied more recently. It focuses on the construction of the quotient monoid of a game, an idea that has been successively applied to several classes of games.