Based on von Neumann's lecture notes, this book 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.

ForewordFoundations and Elementary Properties1Independence8Perspectivity and Projectivity. Fundamental Properties16Perspectivity by Decomposition24Distributivity. Equivalence of Perspectivity and Projectivity32Properties of the Equivalence Classes42Dimensionality54Theory of Ideals and Coordinates in Projective Geometry63Theory of Regular Rings69Appendix 182Appendix 284Appendix 390Order of a Lattice and of a Regular Ring93Isomorphism Theorems103Projective Isomorphisms in a Complemented Modular Lattice117Definition of L-Numbers; Multiplication130Appendix133Addition of L-Numbers136Appendix148The Distributive Laws, Subtraction; and Proof that the L-Numbers form a Ring151Appendix158Relations Between the Lattice and its Auxiliary Ring160Further Properties of the Auxiliary Ring of the Lattice168Special Considerations. Statement of the Induction to be Proved177Treatment of Case I191Preliminary Lemmas for the Treatment of Case II197Completion of Treatment of Case II. The Fundamental Theorem199Perspectivities and Projectivities209Inner Automorphisms217Properties of Continuous Rings222Rank-Rings and Characterization of Continuous Rings231Center of a Continuous Geometry240Appendix 1245Appendix 2259Transitivity of Perspectivity and Properties of Equivalence Classes264Minimal Elements277List of Changes from the 1935-37 Edition and comments on the text by Israel Halperin283Index297