Zur Existenz und Fortsetzbarkeit von o-Epimorphismen angeordneter projektiver Ebenen. (On the existence and extension of o-epimorphisms of ordered projective planes.)
Let \(\phi\) be an order-preserving epimorphisms of a projective plane \({\mathfrak F}\) lying in an ordered projective plane \({\mathfrak E}\). Let U, V be points and a, b, c, d, w lines of \({\mathfrak F}\) with \(U^{\phi}\neq V^{\phi}\), \(U\in a,b,w\) and \(V\in c,d,w\). The set \({\mathcal V}=(a,b,c,d)\) of all points \(P\in {\mathfrak E}\), where a and b separate P and V, and c and d separate P and U, is called a quadrangle with respect to U and V. \({\mathcal V}\) is called admissible for a point \(P\in {\mathcal E}\) if there exists a quadrangle \({\mathcal V}'=(a',b',c',d')\) with respect to U and V such that \(P\in {\mathcal V}'\subset {\mathcal V}\) and the images of a, b, c, d, a', b', c', d' and w under \(\phi\) are pairwise distinct. Let \({\mathcal V}_{U,V}(P)\) denote the set of quadrangles with respect to U and V that are admissible for P. The authors prove that \(\phi\) can be extended to an order-preserving epimorphism of \({\mathfrak E}\), if the `Vierecksbedingung' is fulfilled, i.e., if for all points of P of \({\mathfrak E}\), U,V,W of \({\mathfrak F}\) with \({\mathcal V}_{U,W}(P)\neq \emptyset\), and for all \({\mathcal V}\in {\mathcal V}_{U,V}(P)\) there exists a quadrangle \({\mathcal W}\in {\mathcal V}_{U,W}(P)\) such that \({\mathcal W}\subset {\mathcal V}.\) Although, even in the desarguesian case, this condition is not necessary, the paper under consideration provides us with strong and deep results on the existence of order-preserving epimorphisms in arbitrary projective planes: In a second part of their work, the authors show that the `Vierecksbedingung' is satified whenever \({\mathfrak F}\) is order-dense in \({\mathfrak E}\) or whenever \(\phi\) (\({\mathfrak F})\) is an Archimedean ordered plane. The latter allows for instance, to carry over the classical construction of natural places (valuations) of ordered fields to all planes admitting an Archimedean subplane, and thus to all ordered translation planes.
scientific article; zbMATH DE number 4124472