Modern projective geometry /
| Main Author: | |
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| Corporate Author: | |
| Other Authors: | , |
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht ; Boston :
Kluwer Academic Publishers,
[2000], ©2000
Dordrecht ; Boston : c2000 Dordrecht ; Boston : [2000] |
| Series: | Mathematics and its applications (Kluwer Academic Publishers) ;
v. 521 Mathematics and its applications (Kluwer Academic Publishers) v. 521 |
| Subjects: |
Table of Contents:
- Ch. 1. Fundamental Notions of Lattice Theory
- Ch. 2. Projective Geometries and Projective Lattices
- Ch. 3. Closure Spaces and Matroids
- Ch. 4. Dimension Theory
- Ch. 5. Geometries of degree n
- Ch. 6. Morphisms of Projective Geometries
- Ch. 7. Embeddings and Quotient-Maps
- Ch. 8. Endomorphisms and the Desargues Property
- Ch. 9. Homogeneous Coordinates
- Ch. 10. Morphisms and Semilinear Maps
- Ch. 11. Duality
- Ch. 12. Related Categories
- Ch. 13. Lattices of Closed Subspaces
- Ch. 14. Orthogonality
- Ch. 1 Fundamental Notions of Lattice Theory
- Ch. 2. Projective Geometries and Projective Lattices
- Ch. 3. Closure Spaces and Matroids
- Ch. 4. Dimension Theory
- Ch. 5. Geometries of degree n
- Ch. 6. Morphisms of Projective Geometries
- Ch. 7. Embeddings and Quotient-Maps
- Ch. 8. Endomorphisms and the Desargues Property
- Ch. 9. Homogeneous Coordinates
- Ch. 10. Morphisms and Semilinear Maps
- Ch. 11. Duality
- Ch. 12. Related Categories
- Ch. 13. Lattices of Closed Subspaces
- Ch. 14. Orthogonality.
- Chapter 1 Fundamental Notions of Lattice Theory 1
- 1.1 Introduction to lattices 1
- 1.2 Complete lattices 5
- 1.3 Atomic and atomistic lattices 7
- 1.4 Meet-continuous lattices 9
- 1.5 Modular and semimodular lattices 12
- 1.6 The maximal chain property 15
- 1.7 Complemented lattices 17
- Chapter 2. Projective Geometries and Projective Lattices 25
- 2.1 Definition and examples of projective geometries 26
- 2.2 A second system of axioms 30
- 2.3 Subspaces 34
- 2.4 The lattice L (G) of subspaces of G 36
- 2.5 Correspondence of projective geometries and projective lattices 40
- 2.6 Quotients by subspaces and isomorphism theorems 43
- 2.7 Decomposition into irreducible components 47
- Chapter 3. Closure Spaces and Matroids 55
- 3.1 Closure operators 56
- 3.2 Examples of matroids 59
- 3.3 Projective geometries as closure spaces 63
- 3.4 Complete atomistic lattices 67
- 3.5 Quotients by a closed subset 70
- 3.6 Isomorphism theorems 73
- Chapter 4. Dimension Theory 81
- 4.1 Independent subsets and bases 83
- 4.2 The rank of a subspace 86
- 4.3 General properties of the rank 89
- 4.4 The dimension theorem of degree n 92
- 4.5 Dimension theorems involving the corank 97
- 4.6 Applications to projective geometries 98
- 4.7 Matroids as sets with a rank function 100
- Chapter 5. Geometries of degree n 107
- 5.2 Degree of submatroids and quotient geometries 110
- 5.3 Affine geometries 112
- 5.4 Embedding of a geometry of degree 1 117
- Chapter 6. Morphisms of Projective Geometries 127
- 6.1 Partial maps 128
- 6.2 Definition, properties and examples of morphisms 133
- 6.3 Morphisms induced by semilinear maps 137
- 6.4 The category of projective geometries 141
- 6.5 Homomorphisms 143
- 6.6 Examples of homomorphisms 148
- Chapter 7. Embeddings and Quotient-Maps 157
- 7.1 Mono-sources and initial sources 158
- 7.2 Embeddings 163
- 7.3 Epi-sinks and final sinks 169
- 7.4 Quotient-maps 172
- 7.5 Complementary subpaces 177
- 7.6 Factorization of morphisms 179
- Chapter 8. Endomorphisms and the Desargues Property 187
- 8.1 Axis and center of an endomorphism 188
- 8.2 Endomorphisms with a given axis 191
- 8.3 Endomorphisms induced by a hyperplane-embedding 195
- 8.4 Arguesian geometries 197
- 8.5 Non-arguesian planes 204
- Chapter 9. Homogeneous Coordinates 215
- 9.1 The homothety fields of an arguesian geometry 216
- 9.2 Coordinates and hyperplane-embeddings 218
- 9.3 The fundamental theorem for homomorphisms 221
- 9.4 Uniqueness of the associated fields and vector spaces 224
- 9.5 Arguesian planes 226
- 9.6 The Pappus property 228
- Chapter 10. Morphisms and Semilinear Maps 235
- 10.1 The fundamental theorem 236
- 10.2 Semilinear maps and extensions of morphisms 238
- 10.3 The category of arguesian geometries 242
- 10.4 Points in general position 244
- 10.5 Projective subgeometries of an arguesian geometry 247
- Chapter 11. Duality 255
- 11.1 Duality for vector spaces 256
- 11.2 The dual geometry 258
- 11.3 Pairings, dualities and embedding into the bidual 261
- 11.4 The duality functor 264
- 11.5 Pairings and sesquilinear forms 267
- Chapter 12. Related Categories 275
- 12.1 The category of closure spaces 276
- 12.2 Galois connections and complete lattices 278
- 12.3 The category of complete atomistic lattices 281
- 12.4 Morphisms between affine geometries 284
- 12.5 Characterization of strong morphisms 287
- 12.6 Characterization of morphisms 291
- Chapter 13. Lattices of Closed Subspaces 301
- 13.1 Topological vector spaces 302
- 13.2 Mackey geometries 305
- 13.3 Continuous morphisms 308
- 13.4 Dualized geometries 310
- 13.5 Continuous homomorphisms 315
- Chapter 14. Orthogonality 323
- 14.1 Orthogeometries 324
- 14.2 Ortholattices and orthosystems 327
- 14.3 Orthogonal morphisms 330
- 14.4 The adjunction functor 334
- 14.5 Hilbertian geometries 337.