Braids and Self-Distributivity

ISBN-13: 9783034895682 / Angielski / Miękka / 2012 / 623 str.

Patrick Dehornoy

Opis

This is the award-winning monograph of the Sunyer i Balaguer Prize 1999. The book presents recently discovered connections between Artin's braid groups and left self-distributive systems, which are sets equipped with a binary operation satisfying the identity x(yz) = (xy)(xz). Although not a comprehensive course, the exposition is self-contained, and many basic results are established. In particular, the first chapters include a thorough algebraic study of Artin's braid groups.

Recenzje

"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

Spis treści

A: Ordering the Braids.- I. Braids vs. Self-Distributive Systems.- I.1 Braid Groups.- I.2 Braid Colourings.- I.3 A Self-Distributive Operation on Braids.- I.4 Extended Braids.- I.5 Notes.- II. Word Reversing.- II.1 Complemented Presentations.- II.2 Coherent Complements.- II.3 Garside Elements.- II.4 The Case of Braids.- II.5 Double Reversing.- II.6 Notes.- III. The Braid Order.- III.1 More about Braid Colourings.- III.2 The Linear Ordering of Braids.- III.3 Handle Reduction.- III.4 Alternative Definitions.- III.5 Notes.- IV. The Order on Positive Braids.- IV.1 The Case of Three Strands.- IV.2 The General Case.- IV.3 Applications.- IV.4 Notes.- IV.Appendix: Rank Tables.- B: Free LD-systems.- V. Orders on Free LD-systems.- V.1 Free Systems.- V.2 The Absorption Property.- V.3 The Confluence Property.- V.4 The Comparison Property.- V.5 Canonical Orders.- V.6 Applications.- V.7 Notes.- VI. Normal Forms.- VI.1 Terms as Trees.- VI.2 The Cuts of a Term.- VI.3 The O-Normal Form.- VI.4 The Right Normal Form.- VI.5 Applications.- VI.6 Notes.- VI.Appendix: The First Codes.- VII. The Geometry Monoid.- VII.1 Left Self-Distributivity Operators.- VII.2 Relations in the Geometry Monoid.- VII.3 Syntactic Proofs.- VII.4 Cuts and Collapsing.- VII.5 Notes.- VIII. The Group of Left Self-Distributivity.- VIII.1 The Group GLD and the Monoid MLD.- VIII.2 The Blueprint of a Term.- VIII.3 Order Properties in GLD.- VIII.4 Parabolic Subgroups.- VIII.5 Simple Elements in MLD.- VIII.6 Notes.- IX. Progressive Expansions.- IX.1 The Polish Algorithm.- IX.2 The Content of a Term.- IX.3 Perfect Terms.- IX.4 Convergence Results.- IX.5 Effective Decompositions.- IX.6 The Embedding Conjecture.- IX.7 Notes.- IX.Appendix: Other Algebraic Identities.- C: Other LD-Systems.- X. More LD-Systems.- X.1 The Laver Tables.- X.2 Finite Monogenic LD-systems.- X.3 Multi-LD-Systems.- X.4 Idempotent LD-systems.- X.5 Two-Sided Self-Distributivity.- X.6 Notes.- X.Appendix: The First Laver Tables.- XI. LD-Monoids.- XI.1 LD-Monoids.- XI.2 Completion of an LD-System.- XI.3 Free LD-Monoids.- XI.4 Orders on Free LD-Monoids.- XI.5 Extended Braids.- XI.6 Notes.- XII. Elementary Embeddings.- XII.1 Large Cardinals.- XII.2 Elementary Embeddings.- XII.3 Operations on Elementary Embeddings.- XII.4 Finite Quotients.- XII.5 Notes.- XIII. More about the Laver Tables.- XIII.1 A Dictionary.- XIII.2 Computation of Rows.- XIII.3 Fast Growing Functions.- XIII.4 Lower Bounds.- XIII.5 Notes.- List of Symbols.