Modern Projective GeometryProjective 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. |
Contents
| 1 | |
Projective Geometries and Projective Lattices 2265 | 25 |
Closure Spaces and Matroids | 55 |
Dimension Theory 81 8888 | 80 |
Geometries of degree n | 107 |
Morphisms of Projective Geometries | 127 |
Embeddings and QuotientMaps 157 | 156 |
Endomorphisms and the Desargues Property | 187 |
Morphisms and Semilinear Maps | 235 |
Duality 255 | 254 |
Related Categories | 275 |
Lattices of Closed Subspaces 301 | 300 |
Orthogonality | 318 |
List of Problems | 330 |
List of Symbols | 357 |
Homogeneous Coordinates | 215 |
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Common terms and phrases
A₁ affine geometry arguesian geometry associated atom axioms axis H b₁ bijection canonical projection choose closed subset closure operator closure space collinear complete lattice conditions are equivalent consider deduces defined Definition denote Desargues dimension dualized geometry element embedding endomorphism epimorphism example exchange property exists a unique finite following conditions G₁ G₂ geometry G geometry of degree H₁ H₂ hence Hint homogeneous coordinates homomorphism homothety hyperplane hypothesis implies inclusion injective intersection system isomorphism Ker g Let G linearly independent Mackey geometries map f map ƒ map g matroid meet-continuous modular monomorphism morphism g non-empty obtains orthogeometry partial map preceding lemma preceding proposition projective geometry projective lattices projective subgeometry Proof quotient rank remark satisfies semilinear map subspace of G suppose surjective Theorem third point topological vector space trivial V₁ V₂ vector subspace verify