The structure and classification of misère quotients
Aaron N. Siegel · arXiv (Cornell University) · 2007
A \emph{bipartite monoid} is a commutative monoid $\Q$ together with an identified subset $¶\subset \Q$. In this paper we study a class of bipartite monoids, known as \emph{misère quotients}, that are naturally associated to impartial combinatorial games. We introduce a structure theory for misère quotients with $|¶| = 2$, and give a complete classification of all such quotients up to isomorphism. One consequence is that if $|¶| = 2$ and $\Q$ is finite, then $|\Q| = 2^n+2$ or $2^n+4$. We then develop computational techniques for enumerating misère 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.