A Ring structure on the Class of Combinatorial Games

TL;DR

This paper introduces a new equivalence relation on combinatorial games, demonstrating that under this relation, the class of games forms a ring structure, extending Conway's group framework.

Contribution

It defines a finer equivalence relation than Conway's, enabling the combinatorial games to form a ring rather than just a group.

Findings

Games form a ring under the new equivalence relation

Conway product is well-defined on these equivalence classes

Potential for other relations on the class of combinatorial games

Abstract

J. Conway defined useful operations on the Class of combinatorial games and also introduced a notion of equivalence between games. Conway showed that, under his equivalence, games form a Group. However, Conway product is not well defined on equivalence classes of arbitrary games (though it is well defined for surreals). We consider an equivalence relation finer than Conway's and show that under such a relation combinatorial games actually form a Ring. We hint to other possible relations on the Class of combinatorial games.