Modern projective geometry / by Claude-Alain Faure and Alfred Frölicher.

Format:

Book

Author/Creator:

Faure, Claude-Alain.

Contributor:

Frölicher, Alfred.

Alumni and Friends Memorial Book Fund.

Series:

Mathematics and its applications (Kluwer Academic Publishers) ; v. 521.

Mathematics and its applications ; v. 521

Language:

English

Subjects (All):

Geometry, Projective.

Physical Description:

xvii, 363 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, [2000]

Contents:

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.

Notes:

Includes bibliographical references (pages [347]-355) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0792365259

OCLC:

44669784