The structure and classification of mis\`ere quotients

Abstract

A \emph{bipartite monoid} is a commutative monoid $\Q$ together with an identified subset $\P \subset \Q$. In this paper we study a class of bipartite monoids, known as \emph{mis\`ere quotients}, that are naturally associated to impartial combinatorial games. We introduce a structure theory for mis\`ere quotients with $|\P| = 2$, and give a complete classification of all such quotients up to isomorphism. One consequence is that if $|\P| = 2$ and $\Q$ is finite, then $|\Q| = 2^n+2$ or $2^n+4$. We then develop computational techniques for enumerating mis\`ere quotients of small order, and apply them to count the number of non-isomorphic quotients of order at most~18. We also include a manual proof that there is exactly one quotient of order~8.