Continuous Geometry
1st Edition
Published on 24. October 2016
312 pages
978-1-4008-8395-0 (ISBN)
Description
In his work on rings of operators in Hilbert space, John von Neumann discovered a new mathematical structure that resembled the lattice system Ln. In characterizing its properties, von Neumann founded the field of continuous geometry.
This book, based on von Neumann's lecture notes, begins with the development of the axioms of continuous geometry, dimension theory, and--for the irreducible case--the function D(a). The properties of regular rings are then discussed, and a variety of results are presented for lattices that are continuous geometries, for which irreducibility is not assumed. For students and researchers interested in ring theory or projective geometries, this book is required reading.
This book, based on von Neumann's lecture notes, begins with the development of the axioms of continuous geometry, dimension theory, and--for the irreducible case--the function D(a). The properties of regular rings are then discussed, and a variety of results are presented for lattices that are continuous geometries, for which irreducibility is not assumed. For students and researchers interested in ring theory or projective geometries, this book is required reading.
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05/1998
Princeton University Press
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John von Neumann (1903-1957) was a Permanent Member of the Institute for Advanced Study in Princeton.
Content
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
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
