等变除法

摘要：

设C是非空有限集，Gamma是对称群S（C）的子群.给定一个双射f：A cross C to B cross C，Gamma-等变除法的问题是找到一个商双射h：A to B，关于f在S（A）cross S（B）cross Gamma的作用下可能具有的任何对称性。假设Gamma是完全抵消的，如果这对任何f都是可能的，并且如果A，B是有限的，那么Gamma是完全抵消的。Feldman和Propp证明了一个置换群是可消的，只要它有一个全局不动点。本文证明了一个置换群是完全可消的，只要它是平凡的。这揭示了这样一个事实，即所有避开选择公理的已知带余除法都依赖于固定C元素的排序。

英文摘要：

Let C be a non-empty finite set, and Gamma a subgroup of the symmetric group S(C). Given a bijection f:A cross C to B cross C, the problem of Gamma-equivariant division is to find a quotient bijection h:A to B respecting whatever symmetries f may have under the action of S(A) cross S(B) cross Gamma. Say that Gamma is fully cancelling if this is possible for any f, and finitely cancelling if it is possible providing A,B are finite. Feldman and Propp showed that a permutation group is finitely cancelling just if it has a globally fixed point. We show here that a permutation group is fully cancelling just if it is trivial. This sheds light on the fact that all known division algorithms that eschew the Axiom of Choice depend on fixing an ordering for the elements of C.

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。