Continuous Geometry

Front Cover
Princeton University Press, 1960 - Mathematics - 299 pages
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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 historic book should be in the hands of everyone interested in rings and projective geometry."--R. J. Smith, The Australian Journal of Science"Much in this book is still of great value, partly because it cannot be found elsewhere ... partly because of the very clear and comprehensible presentation. This makes the book valuable for a first study of continuous geometry as well as for research in this field."--F. D. Veldkamp, Nieuw Archief voor Wiskunde
  

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Contents

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
PART II
61
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

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 LNumbers Multiplication
130
Addition of LNumbers
136
The Distributive Laws Subtraction and Proof that the LNumbers Form a Ring
151
Properties of Continuous Rings
222
RankRings and Characterization of Continuous Rings
231
PART III
239
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 193537 Edition and comments on the text
283
Index
297

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References to this book

Spinors in Hilbert Space
Roger Plymen
No preview available - 1994
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About the author (1960)

John von Neumann was a Hungarian-born American mathematician with a minor interest in economics. He earned a degree in mathematics from the University of Budapest and one in chemical engineering from the Hochschule in Zurich. He moved to the United States in 1930 and three years later joined the Institute for Advanced Study at Princeton, where he was the youngest member of the small but select group that included Albert Einstein (see Vol. 5) and Enrico Fermi (see Vol. 5). Von Neumann founded the branch of economics known as game theory. He published a seminal paper on the subject in 1928 and later collaborated with Oskar Morgenstern in developing and expanding it. Game theory analyzes individual choice in situations of risk, where the participants do not control or know the probability distributions of all variables on which the outcome of their acts depends. In 1944, von Neumann and Morgenstern published their classic Theory of Games and Economic Behavior, which achieved immediate prominence because it dealt with bargaining strategies under conditions of conflict. Game theory was applied to a number of situations, such as wage negotiations between unions and management. It was also popular with the Pentagon, which used it to model superpower confrontations that reflected outcomes based on the selection of various military options. Game theory was only one of von Neumann's many interests. In 1943 he worked on the Manhattan Project and was instrumental in developing the atomic bomb. He also helped develop ENIAC, the first electronic computer. Because of his work with the atomic bomb, he received a presidential appointment to the Atomic Energy Commission in 1955. In 1957, at the age of 53, his illustrious career ended when he died tragically of bone cancer.

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