《悲惨世界》的结构与分类
The structure and classification of misère quotients
摘要：
一个\emph{bipartite monoid}是一个交换幺半群$\Q$连同一个标识子集$¶\subset \Q$。在本文中，我们研究了一类二部幺半群，称为\emph{misère quotients}，这是自然相关的公平组合游戏。本文介绍了一种基于$|¶| = 2$的Misère类元的结构理论，并给出了所有该类元直到同构的完整分类。一个结果是，如果$|¶| = 2$和$\Q$是有限的，那么$|\Q| = 2^n+2$或$2^n+4$。然后，我们开发的计算技术，枚举misère小的顺序，并应用它们来计数的数量的非同构的顺序最多18。我们还包括一个手动证明，有一个商的顺序~8。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
