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Two-sided fundamental theorem of affine geometry

math. arxiv. Cornell University, 2017. No. 1702.07701.
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.