Equivariant division

TL;DR

This paper characterizes the permutation groups that allow for equivariant division, showing that only trivial groups are fully cancelling, which explains the reliance on fixed orderings in division algorithms.

Contribution

It proves that only trivial permutation groups are fully cancelling, providing a fundamental limitation on equivariant division without the Axiom of Choice.

Findings

Only trivial groups are fully cancelling.

Finitely cancelling groups are those with a fixed point.

Division algorithms depend on fixing an ordering of elements.

Abstract

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.