CONTENTS
List of Contributors = ⅸ
Series Editor's Statement = xi
Foreword by Gian-Carlo Rota = xiii
Preface = xv
Chapter 1 Examples and Basic Concepts / Henry Crapo = 1
1.1 Examples from Linear Algebra and Projective Geometry = 1
1.2 Further Algebraic Examples = 12
1.3 Combinatorial Examples = 16
1.4 Structure and Related Geometries = 25
Exercises = 27
Chapter 2 Axiom Systems / Giorgio Nicoletti ; Neil White = 29
2.1 Basis Axioms = 30
2.2 Other Families of Subsets = 32
2.3 Closure and Rank = 38
2.4 Combinatorial Geometries and Infinite Matroids = 42
Exercises = 43
References = 44
Chapter 3 Lattices / Ulrich Faigle = 45
3.1 Posets and Lattices = 46
3.2 Modularity = 49
3.3 Semimodular Lattices of Finite Length = 51
3.4 Geometric Lattices = 53
3.5 Decomposition of Geometric Lattices = 56
3.6 Projective Geometry and Modular Geometric Lattices = 58
Exercises = 60
References = 61
Chapter 4 Basis-Exchange Properties / Joseph P. S. Kung = 62
4.1 Bracket Identities and Basis-Exchange Properties = 62
4.2 The Exchange Graph = 64
4.3 Multiple and Alternating Exchanges = 66
Historical Notes = 69
Exercises = 69
References = 73
Chapter 5 Orthogonality / Henry Crapo = 76
5.1 Introduction = 76
5.2 Orthogonal Geometries = 77
5.3 Vector Geometries and Function-Space Geometries = 81
5.4 Orthogonality of Vector Geometries = 85
5.5 Orthogonality of Simplicial Geometries = 87
5.6 Orthogonality of Planar Graphic Geometries = 90
5.7 Research Problem : Orthogonality between Other Pairs of Simplicial Geometries = 91
5.8 The Orthogonal of a Structure Geometry = 94
References = 96
Chapter 6 Graphs and Series-Parallel Networks / James Oxiey = 97
6.1 Polygon Matroids, Bond Matroids, and Planar Graphs = 98
6.2 Connectivity for Graphs and Matroids = 107
6.3 Whitney's 2-Isomorphism Theorem = 110
6.4 Series-Parallel Networks = 116
Exercises = 120
References = 125
Chapter 7 Constructions / Thomas Brylawski = 127
7.1 Introduction = 127
7.2 Isthmuses and Loops = 128
7.3 Deletions, Submatroids, and Extensions = 130
7.4 Contractions, Minors, and Lifts = 138
7.5 Truncations, Lifts, and Matroid Bracing = 162
7.6 Direct Sum and Its Generalizations = 173
7.7 Lower Truncations = 193
7.8 Index of Constructions = 201
Exercises = 209
References = 222
Chapter 8 Strong Maps / Joseph P. S. Kung = 224
8.1 Minors and Strong Maps = 224
8.2 The Factorization Theorem = 230
8.3 Elementary Quotient Maps = 237
8.4 Further Topics = 240
Historical Notes = 242
Exercises = 243
References = 252
Chapter 9 Weak Maps / Joseph P. S. Kung ; Hien Q. Nguyen = 254
9.1 The Weak Order = 254
9.2 Weak Cuts = 256
9.3 Rank-preserving Weak Maps = 260
9.4 Simple Weak Maps of Binary Matroids = 262
Historical Notes = 267
Exercises = 268
References = 270
Chapter 10 Semimodular Functions / Hien Q. Nguyen = 272
10.1 General Properties of Semimodular Functions = 273
10.2 Expansions and Dilworth's Embedding = 275
10.3 Reductions = 282
10.4 Applications of Expansions and Reductions = 289
Historical Notes = 296
References = 297
Appendix of Matroid Cryptomorphisms / Thomas Brylawski = 298
Axiomatizations for the Matroid M(E) = 300
Cryptomorphisms = 304
Prototypical Examples = 305
Special Cryptomorphisms Characterizing Binary Matroids = 310
Index = 313
