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.

Inhalt

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.