The structure and classificationof misère quotients

A bipartite monoid is a commutative monoid Q together with an identified subset P ⊂ Q. In this paper we study a class of bipartite monoids, known as misère quotients, that are naturally associated to impartial combinatorial games. We introduce a structure theory for misère 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| = 2n + 2 or 2n + 4. We then develop computational techniques for enumerating misère quotients of small order, and apply them to count the number of nonisomorphic quotients of order at most 18. We also include a manual proof that there is exactly one quotient of order 8.