{
  "id": "math/0703070",
  "version": "v1",
  "published": "2007-03-02T19:59:04.000Z",
  "updated": "2007-03-02T19:59:04.000Z",
  "title": "The structure and classification of misère quotients",
  "authors": [
    "Aaron N. Siegel"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-02T19:59:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "91A46"
    ],
    "keywords": [
      "misère quotients",
      "impartial combinatorial games",
      "non-isomorphic quotients",
      "small order",
      "computational techniques"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3070S"
    }
  }
}