Preface ........................................................ ix
Introduction ................................................. xiii
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
1.8 Exercises ................................................. 21
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
2.8 Exercises ................................................. 49
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
3.7 Exercises ................................................. 75
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
4.8 Exercises ................................................ 103
Chapter 5. Geometries of degree n ............................ 107
5.1 Definition and examples .................................. 108
5.2 Degree of submatroids and quotient geometries ............ 110
5.3 Affine geometries ........................................ 112
5.4 Embedding of a geometry of degree 1 ...................... 117
5.5 Exercises ................................................ 121
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
6.7 Exercises ................................................ 151
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
7.7 Exercises ................................................ 182
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
8.6 Exercises ................................................ 209
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
9.7 Exercises ................................................ 230
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
10.6 Exercises ................................................ 249
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
11.6 Exercises ................................................ 269
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
12.7 Exercises ................................................ 295
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
13.6 Exercises ................................................ 318
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
14.6 Exercises ................................................ 340
List of Problems .............................................. 345
Bibliography .................................................. 347
List of Axioms ................................................ 357
List of Symbols ............................................... 358
Index ......................................................... 359
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