Continuous Geometry (ebook)

Argumento de Continuous Geometry (ebook)

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.

"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

"This historic book should be in the hands of everyone interested in rings and projective geometry."--R. J. Smith, The Australian Journal of Science0Foreword 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