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Braids and Self-Distributivity

  • Book
  • © 2000

Overview

Authors:
  1. Patrick Dehornoy

    1. Laboratoire SDAD Mathématiques, Université Campus 2, Caen, France

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Part of the book series: Progress in Mathematics (PM, volume 192)

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Table of contents (13 chapters)

  1. Front Matter

    Pages N1-xix
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  2. Ordering the Braids

    1. Front Matter

      Pages 1-1
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    2. Braids vs. Self-Distributive Systems

      • Patrick Dehornoy
      Pages 3-46

    3. Word Reversing

      • Patrick Dehornoy
      Pages 47-96

    4. The Braid Order

      • Patrick Dehornoy
      Pages 97-140

    5. The Order on Positive Braids

      • Patrick Dehornoy
      Pages 141-174

    6. Orders on Free LD-systems

      • Patrick Dehornoy
      Pages 177-233

    7. Normal Forms

      • Patrick Dehornoy
      Pages 235-283

  3. Free LD-systems

    1. Front Matter

      Pages 175-175
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    2. The Geometry Monoid

      • Patrick Dehornoy
      Pages 285-330

    3. The Group of Left Self-Distributivity

      • Patrick Dehornoy
      Pages 331-384

    4. Progressive Expansions

      • Patrick Dehornoy
      Pages 385-442

  4. Other LD-Systems

    1. Front Matter

      Pages 443-443
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    2. More LD-Systems

      • Patrick Dehornoy
      Pages 445-487

    3. LD-Monoids

      • Patrick Dehornoy
      Pages 489-535

    4. Elementary Embeddings

      • Patrick Dehornoy
      Pages 537-570

    5. More about the Laver Tables

      • Patrick Dehornoy
      Pages 571-601

  5. Back Matter

    Pages 603-623
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Keywords

About this book

The aim of this book is to present recently discovered connections between Artin's braid groups En and left self-distributive systems (also called LD­ systems), which are sets equipped with a binary operation satisfying the left self-distributivity identity x(yz) = (xy)(xz). (LD) Such connections appeared in set theory in the 1980s and led to the discovery in 1991 of a left invariant linear order on the braid groups. Braids and self-distributivity have been studied for a long time. Braid groups were introduced in the 1930s by E. Artin, and they have played an increas­ ing role in mathematics in view of their connection with many fields, such as knot theory, algebraic combinatorics, quantum groups and the Yang-Baxter equation, etc. LD-systems have also been considered for several decades: early examples are mentioned in the beginning of the 20th century, and the first general results can be traced back to Belousov in the 1960s. The existence of a connection between braids and left self-distributivity has been observed and used in low dimensional topology for more than twenty years, in particular in work by Joyce, Brieskorn, Kauffman and their students. Brieskorn mentions that the connection is already implicit in (Hurwitz 1891). The results we shall concentrate on here rely on a new approach developed in the late 1980s and originating from set theory.

Reviews

"In this book…P. Dehornoy has accomplished with remarkable success the task of presenting the area of interaction where Artin’s braid groups, left self-distributive systems (LD-systems) and set theory come together in a rigorous and clear manner…The exposition is self-contained and there are no prerequisites. A number of basic results about braid groups, self-distributive algebras, and set theory are provided."

--Mathematical Reviews

Authors and Affiliations

  • Laboratoire SDAD Mathématiques, Université Campus 2, Caen, France

    Patrick Dehornoy

Bibliographic Information

  • Book Title: Braids and Self-Distributivity

  • Authors: Patrick Dehornoy

  • Series Title: Progress in Mathematics

  • DOI: https://doi.org/10.1007/978-3-0348-8442-6

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 2000

  • Hardcover ISBN: 978-3-7643-6343-7Published: 01 July 2000

  • Softcover ISBN: 978-3-0348-9568-2Published: 23 October 2012

  • eBook ISBN: 978-3-0348-8442-6Published: 06 December 2012

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: XIX, 623

  • Topics: Topology

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