# Two-sided fundamental theorem of affine geometry

## TL;DR

This paper extends the fundamental theorem of affine geometry to include bijections that can interchange left and right affine subspaces, showing they still conform to the expected affine map composition.

## Contribution

It introduces a two-sided analogue of the affine geometry theorem, allowing for interchange of left and right subspaces while maintaining the affine map structure.

## Key findings

- Self-bijections can interchange left and right affine subspaces.
- Such maps are still compositions of affine maps and field automorphisms.
- The result generalizes the classical affine geometry theorem.

## Abstract

The fundamental theorem of affine geometry says that a self-bijection $f$ of a finite-dimensional affine space over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected type, namely $f$ is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that these maps again are of the expected type.