The structure and classification of misère quotients

Summary

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.

An impartial combinatorial game Γ is a two-player game with no hidden information and no chance elements, in which both players have exactly the same moves available at all times. When Γ is played under the misère play convention, the player who makes the last move loses.

Thirty years ago, Conway [] showed that the misère-play combinatorics of such games are often frighteningly complicated. However, new techniques recently pioneered by Plambeck [2005] have reinvigorated the subject. At the core of these techniques is the misère quotient, a commutative monoid that encodes the additive structure of an impartial combinatorial game (or a set of such games). See [Siegel 2015] for a gentle introduction to misère quotients, and [Plambeck and Siegel 2008] for a more rigorous one; see [Plambeck 2009] for a survey of the theory.

The introduction of misère quotients opens up a fascinating new area of study: the investigation of their algebraic properties. Such investigations are intrinsically interesting, and also have the potential to reveal new insights into the misère-play structure of combinatorial games. In this paper, we introduce several new results that expose quite a bit of structure in misère quotients.