关于左死局的分解
On factorisations of Left dead ends
AI总结：
本文研究偏局劣势理论中左死局幺半群的代数结构，证明宇宙由左端点决定，揭示其既约、偏可消去及FF-幺半群性质，给出分解长度界限与原子寻找工具，并发现具有唯一分解的游戏族及素元$\overline{1}$。
摘要：
输掉游戏是困难的。Larsson、Nowakowski 和 Santos 以及 Siegel 的近期工作，为偏局劣势理论开辟了诱人的研究方向。劣势幺半群的代数结构目前尚未被充分理解。一个自然的研究对象是游戏的宇宙，即满足各种闭包性质的游戏集合。我们证明了一个宇宙由其左端点决定，从而激发了对这类游戏的研究，Siegel 也曾特别针对死局宇宙指出这一点。我们通过研究左死局幺半群的原子结构来启动这项研究，表明它是既约的、偏可消去的，并且是一个 FF-幺半群。我们给出了分解长度的精确界限，并构建了寻找原子的工具。一些结果需要引入一个称为柔性的新概念，它衡量游戏到整数的距离。我们还发现了具有唯一分解的大型游戏族；这包括发现了一个素元：$\overline{1}$。
英文摘要：
Losing a game is difficult. Recent work of Larsson, Nowakowski, and Santos, as well as that of Siegel, has opened up enticing lines of research into partizan misère theory. The algebraic structure of misère monoids is not yet well understood. A natural object to study is a universe of games, which is a set of games satisfying various closure properties. We prove that a universe is determined by its Left ends, thus motivating a study of such games, which was also noted by Siegel for dead ending universes in particular. We initiate this study by investigating the atomic structure of the monoid of Left dead ends, showing that it is reduced, pocancellative, and an FF-monoid. We give sharp bounds on the lengths of factorisations, and also build tools to find atoms. Some results require the introduction of a new concept called flexibility, which is a measure of the distance from a game to integers. We also discover large families of games admitting unique factorisations; this includes the uncovering of a prime element: $\overline{1}$.
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。
