2007/03/02 by Aaron N. Siegel, Siegel, Aaron N.
Computer Science · Mathematics · #91A46 #Artificial Intelligence in Games #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO #msc:91A46
paper · pdf · doi:10.48550/arxiv.math/0703070
23 pages
arxiv created 2007/03/02 · openalex publication_date 2007/03/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A bipartite monoid is a commutative monoid \Q together with an identified subset ¶⊂ \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 |¶| = 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| = 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 non-isomorphic quotients of order at most~18. We also include a manual proof that there is exactly one quotient of order~8.