Fundamental theorem of geometry without the 1-to-1 assumption

Abstract:

It is proved that any mapping of an $n$-dimensional affine space over a division ring $\mathbb {D}$ onto itself which maps every line into a line is semi-affine, if $n\in \{2,3,\dots \}$ and $\mathbb {D}\ne \mathbb {Z}_{2}$. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also – if the reader is interested in $\mathbb {D}$ other than $\mathbb {R}$, $\mathbb {Z}_{p}$, or $\mathbb {C}$ – division rings.