Modern Projective Geometry

1. Front Matter

   Pages i-xvii

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2. Fundamental Notions of Lattice Theory

   • Claude-Alain Faure, Alfred Frölicher

   Pages 1-24

3. Projective Geometries and Projective Lattices

   • Claude-Alain Faure, Alfred Frölicher

   Pages 25-53

4. Closure Spaces and Matroids

   • Claude-Alain Faure, Alfred Frölicher

   Pages 55-79

5. Dimension Theory

   • Claude-Alain Faure, Alfred Frölicher

   Pages 81-106

6. Geometries of Degree n

   • Claude-Alain Faure, Alfred Frölicher

   Pages 107-125

7. Morphisms of Projective Geometries

   • Claude-Alain Faure, Alfred Frölicher

   Pages 127-155

8. Embeddings and Quotient-Maps

   • Claude-Alain Faure, Alfred Frölicher

   Pages 157-186

9. Endomorphisms and the Desargues Property

   • Claude-Alain Faure, Alfred Frölicher

   Pages 187-213

10. Homogeneous Coordinates

    • Claude-Alain Faure, Alfred Frölicher

    Pages 215-234

11. Morphisms and Semilinear Maps

    • Claude-Alain Faure, Alfred Frölicher

    Pages 235-253

12. Duality

    • Claude-Alain Faure, Alfred Frölicher

    Pages 255-273

13. Related Categories

    • Claude-Alain Faure, Alfred Frölicher

    Pages 275-299

14. Lattices of Closed Subspaces

    • Claude-Alain Faure, Alfred Frölicher

    Pages 301-322

15. Orthogonality

    • Claude-Alain Faure, Alfred Frölicher

    Pages 323-344

16. Back Matter

    Pages 345-363

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Projective geometry is a very classical part of mathematics and one might think that the subject is completely explored and that there is nothing new to be added. But it seems that there exists no book on projective geometry which provides a systematic treatment of morphisms. We intend to fill this gap. It is in this sense that the present monograph can be called modern. The reason why morphisms have not been studied much earlier is probably the fact that they are in general partial maps between the point sets G and G, noted ' 9 : G -- ~ G', i.e. maps 9 : D -4 G' whose domain Dom 9 := D is a subset of G. We give two simple examples of partial maps which ought to be morphisms. The first example is purely geometric. Let E, F be complementary subspaces of a projective geometry G. If x E G \ E, then g(x) := (E V x) n F (where E V x is the subspace generated by E U {x}) is a unique point of F, i.e. one obtains a map 9 : G \ E -4 F. As special case, if E = {z} is a singleton and F a hyperplane with z tf. F, then g: G \ {z} -4 F is the projection with center z of G onto F.

• Lattice
• Volume
• matroid
• projective geometry
• quantum mechanics
• matrix theory
• combinatorics

• University of Geneva, Geneva, Switzerland

  Claude-Alain Faure, Alfred Frölicher

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