Morphisms of projective geometries and semilinear maps. (English) Zbl 0826.51002

The authors present a surprisingly simple new approach to the fundamental theorems of projective geometry, which also covers non-injective collineations.
Let \(G\) and \(G'\) be (the point sets of) two projective geometries. A morphism between \(G\) and \(G'\) is a mapping \(g : G \setminus E \to G'\), where \(E \subseteq G\), such that for every subspace \(F\) of \(G'\) the set \(E \cup g^{-1}(F)\) is a subspace of \(G\). If \(G\) is arguesian and \(H\) is a hyperplane of \(G\), then the set of all morphisms of \(G\) with axis \(H\) can be naturally given the structure of a projective geometry, in which \(G\) embeds as a hyperplane. This leads to a simple proof of the fact that \(G\) can be coordinatized by a vector space over a skewfield. In a similar spirit it is proved that every non-degenerate morphism, i.e. one whose image is not contained in a line, is induced by a semilinear map.
The non-degenerate morphisms in the sense of the authors coincide with the weak linear mappings studied by H. Havlicek [Mitt. Math. Semin. Giessen 215, 27-41 (1994; Zbl 0803.51004)], who also proved that the second fundamental theorem holds for them.
The paper is strongly recommended to everyone interested in the foundations of geometry.


51A05 General theory of linear incidence geometry and projective geometries
51A10 Homomorphism, automorphism and dualities in linear incidence geometry


Zbl 0803.51004
Full Text: DOI


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