## 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

Zbl 0803.51004
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### References:

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