Homomorphisms of affine spaces
The authors compare two notions of homomorphism of affine spaces. Let \(P,P'\) be the set of points and \({\mathcal S},S'\) the set of hyperplanes of two affine spaces of dimension \(\geq 2\). A point-homomorphism is a map \(\varphi : P \to P'\) such that collinear points are mapped to collinear points. The stronger notion of a homomorphism requires a pair of maps \((\varphi, \psi)\), where \(\varphi\) is a point-homomorphism, and \(\psi : {\mathcal S} \to {\mathcal S}' \) has the following properties: for all \((p,H) \in P \times {\mathcal S}\) with \(p \in H\) we have \(\varphi (p) \in \psi (H)\), and the images of parallel hyperplanes are parallel. Generalizing known results for affine planes, the main theorems are: If a point-homomorphism \(\varphi\) is surjective, or preserves independence and parallelism, or \(\varphi (P)\) is contained in a hyperplane, then there exists a map \(\psi : {\mathcal S} \to {\mathcal S}'\) such that \((\varphi, \psi)\) is a homomorphism. If for a homomorphism \((\varphi, \psi)\) the image \(\varphi (P)\) is not contained in a hyperplane, then \(\varphi\) preserves independence and parallelism. The following statements about a homomorphism \((\varphi, \psi)\) are equivalent: (a) \(\varphi, \psi\) are both injective; (b) \(\psi\) is injective; (c) for all \((p,H) \in P \times {\mathcal S}\) with \(\varphi (p) \in \psi (H)\) we have \(p \in H\); (d) the image \(\varphi (P)\) is not contained in a hyperplane.
scientific article; zbMATH DE number 878386