DOCSLIB.ORG

John Von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
John Von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R CONTINUOUS GEOMETRY PRINCETON LANDMARKS IN MATHEMATICS AND PHYSICS Non-standard Analysis, by Abraham Robinson General Theory of Relativity, by P.A.M. Dirac Angular Momentum in Quantum Mechanics, by A. R. Edmonds Mathematical Foundations of Quantum Mechanics, by John von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar Riemannian Geometry, by Luther Pfahler Eisenhart The Classical Groups, by Hermann Weyl Topology from the Differentiable Viewpoint, by John W. Milnor Algebraic Theory of Numbers, by Hermann Weyl Continuous Geometry, by John von Neumann Linear Programming and Extensions, by George B. D antzig Operator Techniques in Atomic Spectroscopy, by Brian R. Judd CONTINUOUS GEOMETRY BY John von Neumann FOREWORD BY ISRAEL HALPERIN PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY Copyright © 1960 by Princeton University Press Copyright renewed © 1988 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex L.C. Card: 59-11084 ISBN 0-691-05893-8 Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources First printing, in the Princeton Landmarks in Mathematics and Physics series, 1998 http://pup.princeton.edu Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 FOREWORD This book reproduces the notes of lectures on Continuous Geometry given by John von Neumann at Princeton. Part I was given during the academic year 1935—36, and Parts II and III were given during the academic year 1936—37. The notes were prepared, while the lectures were in progress, by L. Roy Wilcox, and multigraphed copies were distributed by the Institute for Advanced Study. The supply was soon exhausted, and the notes have not been reproduced until now. In the present edition many slips in typing have been corrected, in Part I with the help of Wallace Givens. I have inserted a few editorial remarks and I have made a small number of changes in the text. These changes, together with some comments, are listed at the back of the book. Of these changes, only one is essential and it was authorized explicitly by von Neumann. Continuous geometry was invented by von Neumann in the fall of 1935. His previous work on rings of operators in Hilbert space, partly in collabo­ ration with F. J. Murray, had led to the discovery of a new mathematical structure which possessed a dimension function (cf. F. J. Murray and J. von Neumann, On Rings of Operators, Annals of Mathematics, vol. 37 (1936), page 172, Case (Hi)). The new structure had incidence properties resembling those of the system Ln (Ln denotes the lattice of all linear sub­ sets of an n —1 dimensional projective geometry), but its dimension func­ tion assumed as values all real numbers in the interval (0, 1). Von Neumann set out to formulate suitable axioms to characterise the new structure. It happened that just previously, K. Menger and G. Birk- hoff had characterised Ln by lattice-type axioms; in particular, Birkhoff had shown the structures Ln could be characterised as the complemented modular irreducible lattices which satisfy a chain condition. Von Neu­ mann dropped the chain condition and replaced it by two of its weak consequences: (i) order completeness of the lattice (Axiom II on page 1 of this book), and (ii) continuity of the lattice operations (Axiom III on page 2 of this book). Lattices which are complemented, modular, irreducible, satisfy (i) and (ii), but do not satisfy a chain condition, were called by von Neumann: continuous geometries (reducible continuous geometries were considered later, in Part III). It is easy to see that in a continuous geometry there can be no minimal element — that is, no atomic element or point. The structure previously discovered by Murray and von Neumann in their research on rings of operators was an example, the first, of a con­ tinuous geometry. Von Neumann’s first fundamental result was the construction, for an arbitrary continuous geometry, of a dimension function with values rang­ ing over the interval (0, 1). The construction was based on the definition: x and y are to be called equidimensional if x and y are in perspective rela­ tion, that is: for some w the lattice join and meet of x with w are identical with those of y with w. The essential difficulty is to prove that the per­ spective relation is transitive. Part I of this book discusses the axioms for continuous geometry and gives the construction of the dimension function. These results were sum­ marized in von Neumann’s note, Continuous Geometry, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 92—100. In a second note, Examples of Continuous Geometries, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 101—108, von Neumann used a simple device to construct an extensive class of new con­ tinuous geometries. He started with finite dimensional vector spaces over an arbitrary but fixed division ring 3 and showed how to imbed an n- dimensional vector space in a ^-dimensional one with k ^ 2; a countable number of repetitions of this imbedding procedure, followed by a metric completion, gives a continuous geometry. This construction uses a coun­ table number of factors k, each ^ 2, but the final result is independent of the particular choice of the k. This was shown by von Neumann in a manuscript (still unpublished): Independence of From The Sequence v (MS written in 1936—37). If 3 is the ring of all complex numbers, the continuous geometry obtained by the imbedding procedure is non-isomorphic to, and simpler than, the continuous geometry obtained from a ring of operators. This non-isomorphism is established in von Neumann’s paper The Non-Iso­ morphism Of Certain Continuous Rings, Annals of Mathematics, vol. 67 (1958), pages 485—496 (MS actually written in 1936—37). The imbedding procedure, the new examples, and the non-isomorphism theorem are not mentioned in the present book. Von Neumann’s next fundamental result was a deep and technically remarkable generalization of the classical Hilbert - Veblen and Young coordinatization theorem. This classical theorem asserts that if n ^ 4, then the points of Ln can be coordinatized with homogeneous coordinates, vi the coordinates to be taken in some suitable division ring.®. Von Neumann first expressed this classical theorem in the following two equivalent forms which do not mention points explicitly and which coordinatize all the linear subsets in Ln: Ln, as a lattice, is isomorphic to (i) the lattice of all right ideals in ®n («®n denotes the ring of n-th order matrices with elements in Sf) and to (ii) the lattice of all right submodules of Q)n (<®n denotes the right module over ® of all w-tuples (a;1, • • *, xn) with all x* in ®). Then he established these coordinatization-isomorphism theorems for every complemented modular lattice of order n ^ 4 (this restriction is a lattice generalization of the condition n ^ 4 for Ln). But in formulating the general coordinatization theorem, von Neumann made some changes: first, the division ring ® was replaced by a more general ring © with the properties: © has an identity element and for each x in @, xyx = x for some y in © (such rings © were called by von Neumann: regular)] second­ ly, in the isomorphism with right ideals in ©n only principal right ideals were to be used; finally, in the isomorphism with right submodules of ©n only those right submodules were to be used which are spanned by a finite number of vectors. The properties of regular rings and the proof of the coordinatization- isomorphism theorem are the main content of Part II of this book. This material was summarized in two notes: On Regular Rings, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 707—713; and Algebraic Theory of Continuous Geometries, Proceedings of the National Academy of Sciences, vol. 23 (1937), pages 16—22. The coordinatization is carried out and the ring 3? = ©n (©n is a regular ring along with ©) is obtained, without using any of the following axioms: completeness of the lattice, continuity of the lattice operations, irreducibility. However, if these axioms do hold, the ring 91 has special properties: a rank function R(x) can be defined for all x in 91. This rank function leads to a natural metric in 91 and 91 is topologically complete with respect to this metric. Rings with a rank function were called by von Neumann: rank rings, and when complete with respect to the rank metric: continuous rings. Rank rings and continuous rings are studied in Chapters XVII and XVIII of this book. Chapter IV of Part II considers complemented modular lattices L of order n ^ 3. If L can be coordinatized by a regular ring 91 (the coordinati­ zation theorem shows that n ^ 4 is sufficient), then as von Neumann shows, vii every lattice isomorphism of L is generated by a ring isomorphism of 9t, and every dual-automorphism of L can be generated by an anti-automorph­ ism of 9T If the anti-automorphism of L is an orthogonalization, then the corresponding anti-automorphism of 91 is a Hermitian conjugation: x -> x*, which is non-singular in the sense that x*x = 0 implies x = 0. The continuous geometries obtained from rings of operators do, in fact, possess such orthogonalizations; the Hermitian conjugation in the co- ordinatizing continuous ring coincides with the operation of taking the Hermitian adjoint operator in Hilbert space.
Load more

Recommended publications
  • Semimodular Lattices Theory and Applications
    P1: SDY/SIL P2: SDY/SJS QC: SSH/SDY CB178/Stern CB178-FM March 3, 1999 9:46 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Semimodular Lattices Theory and Applications MANFRED STERN P1: SDY/SIL P2: SDY/SJS QC: SSH/SDY CB178/Stern CB178-FM March 3, 1999 9:46 PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia c Cambridge University Press 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United States of America Typeset in 10/13 Times Roman in LATEX2ε[TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Stern, Manfred. Semimodular lattices: theory and applications. / Manfred Stern. p. cm. – (Encyclopedia of mathematics and its applications ; v. 73) Includes bibliographical references and index. I. Title. II. Series. QA171.5.S743 1999 511.303 – dc21 98-44873 CIP ISBN 0 521 46105 7 hardback P1: SDY/SIL P2: SDY/SJS QC: SSH/SDY CB178/Stern CB178-FM March 3, 1999 9:46 Contents Preface page ix 1 From Boolean Algebras to Semimodular Lattices 1 1.1 Sources of Semimodularity
    [Show full text]
  • Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector
    Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector Francesco de Giovanni and Carmela Musella Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy and Yaroslav P. Sysak1 Institute of Mathematics, Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine Communicated by Gernot Stroth Received November 14, 2000 DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY 1. INTRODUCTION A subgroup of a group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime order. are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 . 1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A.
    [Show full text]
  • Meet Representations in Upper Continuous Modular Lattices James Elton Delany Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1966 Meet representations in upper continuous modular lattices James Elton Delany Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Delany, James Elton, "Meet representations in upper continuous modular lattices " (1966). Retrospective Theses and Dissertations. 2894. https://lib.dr.iastate.edu/rtd/2894 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microiihned exactly as received 66-10,417 DE LA NY, James Elton, 1941— MEET REPRESENTATIONS IN UPPER CONTINU­ OUS MODULAR LATTICES. Iowa State University of Science and Technology Ph.D., 1966 Mathematics University Microfilms, Inc.. Ann Arbor, Michigan MEET REPRESENTATIONS IN UPPER CONTINUOUS MODULAR LATTICES by James Elton Delany A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved: Signature was redacted for privacy. I _ , r Work Signature was redacted for privacy. Signature was redacted for privacy. Iowa State University of Science and Technology Ames, Iowa 1966 il TABLE OP CONTENTS Page 1. INTRODUCTION 1 2. TORSION ELEMENTS, TORSION FREE ELEMENTS, AND COVERING CONDITIONS 4 3. ATOMIC LATTICES AND UPPER CONTINUITY 8 4. COMPLETE JOIN HOMOMORPHISMS 16 5. TWO HOMOMORPHISMS 27 6.
    [Show full text]
  • This Is the Final Preprint Version of a Paper Which Appeared at Algebraic & Geometric Topology 17 (2017) 439-486
    This is the final preprint version of a paper which appeared at Algebraic & Geometric Topology 17 (2017) 439-486. The published version is accessible to subscribers at http://dx.doi.org/10.2140/agt.2017.17.439 SIMPLICIAL COMPLEXES WITH LATTICE STRUCTURES GEORGE M. BERGMAN Abstract. If L is a finite lattice, we show that there is a natural topological lattice structure on the geo- metric realization of its order complex ∆(L) (definition recalled below). Lattice-theoretically, the resulting object is a subdirect product of copies of L: We note properties of this construction and of some variants, and pose several questions. For M3 the 5-element nondistributive modular lattice, ∆(M3) is modular, but its underlying topological space does not admit a structure of distributive lattice, answering a question of Walter Taylor. We also describe a construction of \stitching together" a family of lattices along a common chain, and note how ∆(M3) can be regarded as an example of this construction. 1. A lattice structure on ∆(L) I came upon the construction studied here from a direction unrelated to the concept of order complex; so I will first motivate it in roughly the way I discovered it, then recall the order complex construction, which turns out to describe the topological structures of these lattices. 1.1. The construction. The motivation for this work comes from Walter Taylor's paper [20], which ex- amines questions of which topological spaces { in particular, which finite-dimensional simplicial complexes { admit various sorts of algebraic structure, including structures of lattice. An earlier version of that paper asked whether there exist spaces which admit structures of lattice, but not of distributive lattice.
    [Show full text]
  • The Variety of Modular Lattices Is Not Generated
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 255, November 1979 THE VARIETYOF MODULAR LATTICES IS NOT GENERATED BY ITS FINITE MEMBERS BY RALPH FREESE1 Abstract. This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given. We prove the result of the title by constructing a simple modular lattice of length six not in the variety generated by all finite modular lattices. This lattice can be generated by five elements and thus the free modular lattice on five generators, FM (5), is not residually finite. Our lattice is constructed using the technique of Hall and Dilworth [9] and is closely related to their third example. Let F and K be countably infinite fields of characteristics p and q, where p and q are distinct primes. Let Lp be the lattice of subspaces of a four-dimensional vector space over F, Lq the lattice of subspaces of a four-dimensional vector space over K. Two-dimen- sional quotients (i.e. intervals) in both lattices are always isomorphic to Mu (the two-dimensional lattice with « atoms). Thus Lp and Lq may be glued together over a two-dimensional quotient via [9], and this is our lattice. Notice that if F and K were finite fields we could not carry out the above construction since two-dimensional quotients of Lp would have/)" + 1 atoms and those of Lq would have qm + 1, for some n, m > 1.
    [Show full text]
  • DISTRIBUTIVE LATTICES FACT 1: for Any Lattice <A,≤>: 1 and 2 and 3 and 4 Hold in <A,≤>: the Distributive Inequal
    DISTRIBUTIVE LATTICES FACT 1: For any lattice <A,≤>: 1 and 2 and 3 and 4 hold in <A,≤>: The distributive inequalities: 1. for every a,b,c ∈ A: (a ∧ b) ∨ (a ∧ c) ≤ a ∧ (b ∨ c) 2. for every a,b,c ∈ A: a ∨ (b ∧ c) ≤ (a ∨ b) ∧ (a ∨ c) 3. for every a,b,c ∈ A: (a ∧ b) ∨ (b ∧ c) ∨ (a ∧ c) ≤ (a ∨ b) ∧ (b ∨ c) ∧ (a ∨ c) The modular inequality: 4. for every a,b,c ∈ A: (a ∧ b) ∨ (a ∧ c) ≤ a ∧ (b ∨ (a ∧ c)) FACT 2: For any lattice <A,≤>: 5 holds in <A,≤> iff 6 holds in <A,≤> iff 7 holds in <A,≤>: 5. for every a,b,c ∈ A: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) 6. for every a,b,c ∈ A: a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) 7. for every a,b,c ∈ A: a ∨ (b ∧ c) ≤ b ∧ (a ∨ c). A lattice <A,≤> is distributive iff 5. holds. FACT 3: For any lattice <A,≤>: 8 holds in <A,≤> iff 9 holds in <A,≤>: 8. for every a,b,c ∈ A:(a ∧ b) ∨ (a ∧ c) = a ∧ (b ∨ (a ∧ c)) 9. for every a,b,c ∈ A: if a ≤ b then a ∨ (b ∧ c) = b ∧ (a ∨ c) A lattice <A,≤> is modular iff 8. holds. FACT 4: Every distributive lattice is modular. Namely, let <A,≤> be distributive and let a,b,c ∈ A and let a ≤ b. a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) [by distributivity] = b ∧ (a ∨ c) [since a ∨ b =b]. The pentagon: The diamond: o 1 o 1 o z o y x o o y o z o x o 0 o 0 THEOREM 5: A lattice is modular iff the pentagon cannot be embedded in it.
    [Show full text]
  • From the Collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ
    From the collections of the Seeley G. Mudd Manuscript Library, Princeton, NJ These documents can only be used for educational and research purposes (“Fair use”) as per U.S. Copyright law (text below). By accessing this file, all users agree that their use falls within fair use as defined by the copyright law. They further agree to request permission of the Princeton University Library (and pay any fees, if applicable) if they plan to publish, broadcast, or otherwise disseminate this material. This includes all forms of electronic distribution. Inquiries about this material can be directed to: Seeley G. Mudd Manuscript Library 65 Olden Street Princeton, NJ 08540 609-258-6345 609-258-3385 (fax) [email protected] U.S. Copyright law test The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or other reproduction is not to be “used for any purpose other than private study, scholarship or research.” If a user makes a request for, or later uses, a photocopy or other reproduction for purposes in excess of “fair use,” that user may be liable for copyright infringement. The Princeton Mathematics Community in the 1930s Transcript Number 27 (PMC27) © The Trustees of Princeton University, 1985 ROBERT SINGLETON (with ALBERT TUCKER) This is an interview of Robert Singleton at the Cromwell Inn on 6 June 1984. The interviewer is Albert Tucker.
    [Show full text]
  • Modular Lattices
    Wednesday 1/30/08 Modular Lattices Definition: A lattice L is modular if for every x; y; z 2 L with x ≤ z, (1) x _ (y ^ z) = (x _ y) ^ z: (Note: For all lattices, if x ≤ z, then x _ (y ^ z) ≤ (x _ y) ^ z.) Some basic facts and examples: 1. Every sublattice of a modular lattice is modular. Also, if L is distributive and x ≤ z 2 L, then x _ (y ^ z) = (x ^ z) _ (y ^ z) = (x _ y) ^ z; so L is modular. 2. L is modular if and only if L∗ is modular. Unlike the corresponding statement for distributivity, this is completely trivial, because the definition of modularity is invariant under dualization. 3. N5 is not modular. With the labeling below, we have a ≤ b, but a _ (c ^ b) = a _ 0^ = a; (a _ c) ^ b = 1^ ^ b = b: b c a ∼ 4. M5 = Π3 is modular. However, Π4 is not modular (exercise). Modular lattices tend to come up in algebraic settings: • Subspaces of a vector space • Subgroups of a group • R-submodules of an R-module E.g., if X; Y; Z are subspaces of a vector space V with X ⊆ Z, then the modularity condition says that X + (Y \ Z) = (X + Y ) \ Z: Proposition 1. Let L be a lattice. TFAE: 1. L is modular. 2. For all x; y; z 2 L, if x 2 [y ^ z; z], then x = (x _ y) ^ z. 2∗. For all x; y; z 2 L, if x 2 [y; y _ z], then x = (x ^ z) _ y.
    [Show full text]
  • Andrew S. Toms Curriculum Vitae
    Andrew S. Toms Curriculum Vitae Contact Information Personal Information Department of Mathematics Born March 12, 1975, Montreal, PQ Purdue University Married with two sons 150 N. University St., West Lafayette, IN Canadian and British citizen 47907-2067 US Permanent Resident Telephone: (765) 494-1901 [email protected] Education 1999–2002 Ph.D. University of Toronto 1997–1999 M.Sc. University of Toronto 1993–1997 B.Sc.H. Queen’s University Employment 2013–2018 University Faculty Scholar, Purdue University 2013–present Professor, Purdue University 2010–2013 Associate Professor, Purdue University 2008–2010 Associate Professor, York University 2006–2008 Assistant Professor, York University 2004–2006 Assistant Professor, University of New Brunswick 2003–2004 NSERC Postdoctoral Fellow, Copenhagen University Honours and Awards 1. Purdue University Faculty Scholar, 2013-2018 2. AMS Centennial Fellowship 2011-12 3. CMS Gilbert de Beauregard Robinson Award (with Wilhelm Winter) 4. Israel Halperin Prize 20101 5. Canada Research Chair Nomination, 2009 (declined) 6. Ontario Early Researcher Award2, 2008–2013 7. NSERC Postdoctoral Fellowship, 2003–2004 8. Daniel B. DeLury Teaching Award, University of Toronto, 2003 9. Israel Halperin Graduate Award, University of Toronto, 2001–2002 10. NSERC Postgraduate Scholarship, 1997–2001 11. Governor General’s Medal3, Queen’s University, 1997 12. Prince of Wales Prize4, Queen’s University, 1997 1Awarded every five years to the mathematician from the Canadian community within 10 years of his/her Ph.D. deemed to have made the greatest contribution to operator algebras. 2Comparable an NSF CAREER Award in terms of funds to researcher. 3Awarded to the graduate with the highest aggregate GPA.
    [Show full text]
  • Dimensional Functions Over Partially Ordered Sets
    Intro Dimensional functions over posets Application I Application II Dimensional functions over partially ordered sets V.N.Remeslennikov, E. Frenkel May 30, 2013 1 / 38 Intro Dimensional functions over posets Application I Application II Plan The notion of a dimensional function over a partially ordered set was introduced by V. N. Remeslennikov in 2012. Outline of the talk: Part I. Definition and fundamental results on dimensional functions, (based on the paper of V. N. Remeslennikov and A. N. Rybalov “Dimensional functions over posets”); Part II. 1st application: Definition of dimension for arbitrary algebraic systems; Part III. 2nd application: Definition of dimension for regular subsets of free groups (L. Frenkel and V. N. Remeslennikov “Dimensional functions for regular subsets of free groups”, work in progress). 2 / 38 Intro Dimensional functions over posets Application I Application II Partially ordered sets Definition A partial order is a binary relation ≤ over a set M such that ∀a ∈ M a ≤ a (reflexivity); ∀a, b ∈ M a ≤ b and b ≤ a implies a = b (antisymmetry); ∀a, b, c ∈ M a ≤ b and b ≤ c implies a ≤ c (transitivity). Definition A set M with a partial order is called a partially ordered set (poset). 3 / 38 Intro Dimensional functions over posets Application I Application II Linearly ordered abelian groups Definition A set A equipped with addition + and a linear order ≤ is called linearly ordered abelian group if 1. A, + is an abelian group; 2. A, ≤ is a linearly ordered set; 3. ∀a, b, c ∈ A a ≤ b implies a + c ≤ b + c. Definition The semigroup A+ of all nonnegative elements of A is defined by A+ = {a ∈ A | 0 ≤ a}.
    [Show full text]
  • On the Equational Theory of Projection Lattices of Finite Von Neumann
    The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000 ON THE EQUATIONAL THEORY OF PROJECTION LATTICES OF FINITE VON-NEUMANN FACTORS CHRISTIAN HERRMANN Abstract. For a finite von-Neumann algebra factor M, the projections form a modular ortholattice L(M) . We show that the equational theory of L(M) coincides with that of some resp. all L(Cn×n) and is decidable. In contrast, the uniform word problem for the variety generated by all L(Cn×n) is shown to be undecidable. §1. Introduction. Projection lattices L(M) of finite von-Neumann algebra factors M are continuous orthocomplemented modular lattices and have been considered as logics resp. geometries of quantum meachnics cf. [25]. In the finite dimensional case, the correspondence between irreducible lattices and algebras, to wit the matrix rings Cn×n, has been completely clarified by Birkhoff and von Neumann [5]. Combining this with Tarski’s [27] decidability result for the reals and elementary geometry, decidability of the first order theory of L(M) for a finite dimensional factor M has been observed by Dunn, Hagge, Moss, and Wang [7]. The infinite dimensional case has been studied by von Neumann and Murray in the landmark series of papers on ‘Rings of Operators’ [23], von Neumann’s lectures on ‘Continuous Geometry’ [28], and in the treatment of traces resp. transition probabilities in a ring resp. lattice-theoretic framework [20, 29]. The key to an algebraic treatment is the coordinatization of L(M) by a ∗- regular ring U(M) derived from M and having the same projections: L(M) is isomorphic to the lattice of principal right ideals of U(M) (cf.
    [Show full text]
  • Andrew S. Toms Curriculum Vitae
    Andrew S. Toms Curriculum Vitae Contact Information Personal Information Department of Mathematics Born March 12, 1975, Montreal, PQ Purdue University Married with two sons 150 N. University St., West Lafayette, IN Canadian and British citizen 47907-2067 US Permanent Resident Telephone: (765) 494-1901 [email protected] Education 1999–2002 Ph.D. University of Toronto 1997–1999 M.Sc. University of Toronto 1993–1997 B.Sc.H. Queen’s University Employment 2010–present Associate Professor, Purdue University 2008–2010 Associate Professor, York University 2006–2008 Assistant Professor, York University 2004–2006 Assistant Professor, University of New Brunswick 2003–2004 NSERC Postdoctoral Fellow, Copenhagen University Honours and Awards 1. AMS Centennial Fellowship 2011-12 2. Gilbert de Beauregard Robinson Award (with Wilhelm Winter)1 3. Israel Halperin Prize2 4. Canada Research Chair Nomination, 2009 (declined) 5. Ontario Early Researcher Award3, 2008–2013 6. NSERC Postdoctoral Fellowship, 2003–2004 7. Daniel B. DeLury Teaching Award, University of Toronto, 2003 8. Israel Halperin Graduate Award, University of Toronto, 2001–2002 9. NSERC Postgraduate Scholarship, 1997–2001 10. Governor General’s Medal4, Queen’s University, 1997 11. Prince of Wales Prize5, Queen’s University, 1997 1Awarded bi-annually for the best paper to have appeared in the Canadian Journal of Mathematics. 2Awarded every five years to the mathematician from the Canadian community within 10 years of his/her Ph.D. deemed to have made the greatest contribution to operator algebras. I am the first recipient in ten years— the committee did not award the prize in 2005. 3Comparable in frequency and about 70% of the value of a NSF CAREER Award in terms of funds to researcher.
    [Show full text]