发表机构
Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究部分置换的扩展性质，证明$\u03a0$-扩展性质与$\u03a0$-LERF等价，并利用EP问题和$\u03a0$-解刻画其存在性，推广了已知结果，给出了无幂零解的例子，并完全刻画了2-弱游荡问题的解情况。
AI 中文摘要
受有限锦标赛的EPPA问题启发，我们考虑了部分置换的各种扩展性质。我们证明，对于任何素数集合$\u03a0$，$\u03a0$-扩展性质等价于$\u03a0$-LERF。作为已知结果的一个推论，奇数扩展性质等价于有限锦标赛的EPPA。为了研究$\u03a0$-扩展性质，我们使用EP问题和$\u03a0$-解的概念重新表述该性质。然后我们证明，EP问题存在$\u03a0$-解完全取决于其基本群。特别地，当基本群是平凡群或循环群时，EP问题对任何$\u03a0$都有$\u03a0$-解。这些结果推广了Huang、Pawliuk、Sabok和Wise [HPSW19]的一些已知结果。我们给出了没有幂零解的EP问题的例子；它们证明了对于任何素数$p$，$p$-扩展性质和$p$-LERF都不成立。然后我们考虑一类特殊的基本群具有两个生成元的EP问题。我们证明任何1-弱游荡问题对任何$\u03a0$都有$\u03a0$-解。对于2-弱游荡问题，我们证明它们都有奇数解，并且我们完全刻画了那些没有幂零解的问题。
英文摘要
Motivated by the EPPA problem for finite tournaments,we consider various extension properties for partial permutations. We show that for any set $Π$ of prime numbers, the $Π$-extension property is equivalent to the $Π$-LERF. As a consequence of known results, the odd-extension property is then equivalent to the EPPA for finite tournaments. To study the $Π$-extension property, we reformate the property using the concepts of EP problems and $Π$-solutions. We then show that the existence of a $Π$-solution for an EP problem depends entirely on its fundamental group. In particular, when the fundamental group is trivial or cyclic, the EP problem has a $Π$-solution for any $Π$. These extend some known results of Huang, Pawliuk, Sabok and Wise [HPSW19]. We give examples of EP problems without nilpotent-solutions; they witness that the $p$-extension property and the $p$-LERF fail for any prime $p$. Then we consider a special kind of EP problems whose fundamental groups have two generators. We show that any 1-weakly wandering problem has a $Π$-solution for any $Π$. For 2-weakly wandering problems, we show that they all have odd-solutions and we completely characterize those without nilpotent-solutions.
