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Zusatztext "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 Informationen zum Autor John von Neumann (1903-1957) was a Permanent Member of the Institute for Advanced Study in Princeton. Klappentext 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. Zusammenfassung 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. Inhaltsverzeichnis Foreword Independence Perspectivity and Projectivity. Fundamental Properties Perspectivity by Decomposition Distributivity. Equivalence of Perspectivity and Projectivity Properties of the Equivalence Classes Dimensionality Theory of Ideals and Coordinates in Projective Geometry Theory of Regular Rings Appendix 1 Appendix 2 Appendix 3 Order of a Lattice and of a Regular Ring Isomorphism Theorems Projective Isomorphisms in a Complemented Modular Lattice Definition of L-Numbers; Multiplication Appendix Addition of L-Numbers Appendix The Distributive Laws! Subtraction; and Proof that the L-Numbers form a Ring Appendix Relations Between the Lattice and its Auxiliary Ring Further Properties of the Auxiliary Ring of the Lattice Special Considerations. Statement of the Induction to be Proved Treatment of Case I Preliminary Lemmas for the Treatment of Case II Completion of Treatment of Case II. The Fundamental Theorem Perspectivities and Projectivities Inner Automorphisms Properties of Continuous Rings Rank-Rings and Characterization of Continuous Rings Center of a Continuous Geometry Appendix 1 Appendix 2 Transitivity of Perspectivity and Properties of Equivalence Classes Minimal Elements List of Changes from the 1935-37 Edition and comments on the text by Israel Halperin Index ...
