Conjugation of injections by permutations

TL;DR

This paper characterizes when a third injective map can be expressed as a conjugation of two given injective maps by permutations, extending previous results to uncountable sets.

Contribution

It provides a necessary and sufficient condition for representing a map as a conjugation of two injective maps by permutations, generalizing earlier work to uncountable sets.

Findings

Characterization of conjugation conditions for injective maps

Extension of results to uncountable sets

Generalization of Droste and Ore's results

Abstract

Let X be a countably infinite set, and let f, g, and h be any three injective self-maps of X, each having at least one infinite cycle. (For instance, this holds if f, g, and h are not bijections.) We show that there are permutations a and b of X such that h=afa^{-1}bgb^{-1} if and only if |X\Xf|+|X\Xg|=|X\Xh|. We also prove a version of this statement that holds for infinite sets X that are not necessarily countable. This generalizes results of Droste and Ore about permutations.