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Dehornoy P., Digne F. Godelle E., Kramer D., Michel J. Foundations of Garside Theory

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Dehornoy P., Digne F. Godelle E., Kramer D., Michel J. Foundations of Garside Theory
European Mathematical Society, 2015. — 710 p. — (EMS Tracts in Mathematics 22) — ISBN: 3037191392

This text is a monograph on algebra, with connections to geometry and low-dimensional topology. It mainly involves groups, monoids, and categories, and aims to provide a unified treatment for those situations in which one can find distinguished decompositions by iteratively extracting a maximal fragment lying in a prescribed family. Initiated in 1969 by F. A. Garside in the case of Artin's braid groups, this approach led to interesting results in a number of cases, the central notion being what the authors call a Garside family. The study is far from complete, and the purpose of this book is to present the current state of the theory and to invite further research. The book has two parts: In Part A, the bases of a general theory, including many easy examples, are developed. In Part B, various more sophisticated examples are specifically addressed. To make the content accessible to a wide audience of nonspecialists, the book's exposition is essentially self-contained and very few prerequisites are needed. In particular, it should be easy to use this as a textbook both for Garside theory and for the more specialized topics investigated in Part B: Artin-Tits groups, Deligne-Lusztig varieties, groups of algebraic laws, ordered groups, and structure groups of set-theoretic solutions of the Yang-Baxter equation. The first part of the book can be used as the basis for a graduate or advanced undergraduate course. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Contents
General theory
Some examples
Preliminaries
Normal decompositions
Garside families
Bounded Garside families
Germs
Subcategories
Conjugacy
Braids
Deligne–Lusztig varieties
Left self-distributivity
Ordered groups
Set-theoretic solutions of Yang–Baxter equation
More examples
Appendixes
Groupoids of fractions
Working with presented categories
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