Table of Contents

Foreword

Foundations and Elementary Properties 1

Independence 8

Perspectivity and Projectivity. Fundamental Properties 16

Perspectivity by Decomposition 24

Distributivity. Equivalence of Perspectivity and Projectivity 32

Properties of the Equivalence Classes 42

Dimensionality 54

Theory of Ideals and Coordinates in Projective Geometry 63

Theory of Regular Rings 69

Appendix 1 82

Appendix 2 84

Appendix 3 90

Order of a Lattice and of a Regular Ring 93

Isomorphism Theorems 103

Projective Isomorphisms in a Complemented Modular Lattice 117

Definition of L-Numbers; Multiplication 130

Appendix 133

Addition of L-Numbers 136

Appendix 148

The Distributive Laws, Subtraction; and Proof that the L-Numbers form a Ring 151

Appendix 158

Relations Between the Lattice and its Auxiliary Ring 160

Further Properties of the Auxiliary Ring of the Lattice 168

Special Considerations. Statement of the Induction to be Proved 177

Treatment of Case I 191

Preliminary Lemmas for the Treatment of Case II 197

Completion of Treatment of Case II. The Fundamental Theorem 199

Perspectivities and Projectivities 209

Inner Automorphisms 217

Properties of Continuous Rings 222

Rank-Rings and Characterization of Continuous Rings 231

Center of a Continuous Geometry 240

Appendix 1 245

Appendix 2 259

Transitivity of Perspectivity and Properties of Equivalence Classes 264

Minimal Elements 277

List of Changes from the 1935-37 Edition and comments on the text by Israel Halperin 283

Index 297