Patrick Dehornoy

Patrick Dehornoy

Livres / Books

La théorie des ensembles

Introduction à une théorie de l'infini

Parution octobre 2017 chez Calvage et Mounet
Collection "Tableau noir", xx+650 pages

  • Avant-propos, fichier pdf
  • Partie A. Théorie élémentaire
    - I. Le type "ensemble", fichier pdf
    - II. Les ordinaux
    - III. Le système de Zermelo-Fraenkel
    - IV. L'axiome du choix
    - V. Les cardinaux
  • Partie B. Un peu de logique mathématique
    - VI. Logique propositionnel
    - VII. Logique du premier ordre
    - VIII. Théorèmes de limitation, fichier pdf
    - IX. Théorie descriptive des ensembles
  • Partie C. Théorie axiomatique des ensembles
    - X. Modèles de ZF
    - XI. Les ensembles constructibles
    - XII. La méthode du forcing
    - XIII. Les grands cardinaux (I)
    - XIV. Les grands cardinaux (II), fichier pdf
    - XV. La détermination projective
    - XVI. Un bilan mitigé, fichier pdf

Foundations of Garside Theory


François Digne, Eddy Godelle, Daan Krammer, and Jean Michel

EMS Tracts in Mathematics, volume 22, xviii + 690 pages
European Mathematical Society, 2015

EMS Monograph Award 2014

The Garside structure of braids consists of the algebraic properties underlying their decompositions into fractions and the associated normal forms. It turns out that similar structures occur in various frameworks. The aim of the text is to elaborate a unified theory for such structures and, more generally, to develop what can be called Garside methods in the many situations of algebra, geometry, and low-dimensional topology where they are relevant.

The pdf of the final text is freely accessible here: pdf file.

A few addenda (corrections, skipped proofs, solutions to selected exercises) are available here: pdf file, as well as on arXiv:1412.5299.

Ordering Braids


Ivan Dynnikov, Dale Rolfsen, and Bert Wiest

Surveys and Monographs vol. 148; ix + 321 pages
American Mathematical Society (2008)

The present volume follows a book, "Why Are Braids Orderable?", written by the same authors and published in 2002 by the Société Mathématique de France. We emphasize that this is not a new edition of that book. Although this book contains most of the material in the previous book, it also contains a considerable amount of new material. In addition, much of the original text has been completely rewritten, with a view to making it more readable and up-to-date.

Introduction + contents: pdf file (16 pages, 140 Ko).

Why Are Braids Orderable?


Ivan Dynnikov, Dale Rolfsen, and Bert Wiest

Panoramas et synthèses n. 14; xiii + 192 pages
Société Mathématique de France (2002)

In the decade since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several quite different methods have been applied to understand this phenomenon. This book is an account of those techniques, including self-distributive algebra, finite trees, combinatorial group theory, mapping class groups, laminations and hyperbolic geometry.

Introduction + contents: pdf file

Braids and Self Distributivity

Progress in Mathematics, volume 192; xvi + 624 pages
Birkhauser (2000)

Ferran Sunyer i Balaguer Prize 1999

The aim of this text is to give a first synthesis of recent works that connect Artin's braid groups and left self-distributive algebra, defined as the study of those algebraic systems that involve a binary operation satisfying the left self-distributivity identity x(yz)=(xy)(xz). The emphasis is put on the geometric features, as illustrated in the slogan: "The geometry of braids is a projection of the geometry of left self-distributivity". The text is an introduction to four objects, which had never been considered twelve years ago, but which have such simple definitions and such rich properties that they seem to deserve some attention. These objects are:

  • the free left self-distributive system of rank 1, with its canonical ordering, its normal forms, and its distinguished realizations, one as exponentiation of braids, and one as the iterations of an elementary embedding in set theory;
  • the canonical linear ordering of braids, with its several equivalent characterizations, the comparison algorithms and the well-ordering of positive braids;
  • the finite self-distributive Laver tables A_n, with the mysterious asymptotic behaviour of their periods that is known only assuming an unprovable large cardinal axiom;
  • the group that describes the geometry of self-distributivity, with its connection with braids that explains both the existence of braid exponentiation and of the linear ordering of braids.

Table of contents, preface, introduction of the chapters: pdf file

Mathématiques de l'informatique

Cours et exercices corrigés
Birkhauser (2000)

Collection Sciences Sup; xiii + 302 pages
Dunod (2000)

Centré sur les notions de calcul et de définition, ce cours est une introduction à l'étude des structures mathématiques sous-jacentes à l'informatique. Les principaux développements concernent les automates, les langages algébriques, la calculabilité effective et la complexité des algorithmes, la logique booléenne et les logiques du premier ordre.
Cent cinquante exercices d'application et de complément sont proposés, dont plus de la moitié avec un corrigé rédigé.

Complexité et décidabilité

Collection Mathématiques et applications; 208 pages
Birkhauser (2000)

Springer (1993)

Cet ouvrage présente, d'une facon concise mais avec des démonstrations complètes qui ne supposent aucune connaissance antérieure du sujet, un certain nombre de résultats fondamentaux de la théorie de le complexité des algorithmes en liaison avec la logique.