# Alternation diameter of a product object

## TL;DR

This paper investigates the minimal number of coordinate-altering permutations needed to generate any permutation in various mathematical categories, establishing optimal bounds for finite sets, vector spaces, and topological spaces.

## Contribution

It proves the optimal alternation diameter for permutations on finite sets, vector spaces, and provides a counterexample in topological spaces, extending previous results to new contexts.

## Key findings

- Finite sets have an alternation diameter of three.
- Vector spaces also have an alternation diameter of three.
- In topological spaces, the alternation diameter is not finite, demonstrated by a specific homeomorphism.

## Abstract

We prove that every permutation of a Cartesian product of two finite sets can be written as a composition of three permutations, the first of which only modifies the left projection, the second only the right projection, and the third again only the left projection, and three alternations is indeed the optimal number. We show that for two countably infinite sets, the corresponding optimal number of alternations, called the alternation diameter, is four. The notion of alternation diameter can be defined in any category. In the category of finite-dimensional vector spaces, the diameter is also three. For the category of topological spaces, we exhibit a single self-homeomorphism of the plane which is not generated by finitely many alternations of homeomorphisms that only change one coordinate. The results on finite sets and vector spaces were previously known in the context of memoryless computation.