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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.

MSC:

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

Citations:

Zbl 0803.51004
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References:

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