By Patrick Dehornoy

This is the award-winning monograph of the Sunyer i Balaguer Prize 1999. The booklet provides lately found connections among Artin’s braid teams and left self-distributive platforms, that are units built with a binary operation pleasing the identification x(yz) = (xy)(xz). even if now not a accomplished path, the exposition is self-contained, and lots of simple effects are tested. specifically, the 1st chapters contain a radical algebraic learn of Artin’s braid groups.

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The shift mapping is 1/2-Lipschitz by definition, hence continuous, and so is exponentiation, which is defined by composing product, shift, and inverse. As B∞ is countable, it is not complete with respect to the above topology. We shall define an extension EB∞ of B∞ by adding some limits points. 6. (tau) For p, q ≥ 0, we put τp,q = σp σp+1 ··· σ1 sh(τp,q−1 ) for q > 0, and τp,0 = 1. Thus, τp,q is the positive braid where the q strands initially at positions p + 1 to p + q cross over the first p strands: p q τp,q : .

Ii) Prove that, in the LD-system (B∞ , ∧), (∃b)(a∧b = c) is equivalent to a∧c = (a∧a)∧c. [Hint: Expand a∧b = c and deduce that c is left divisible by a in (B∞ , ∧) if and only if a−1 c sh(a) σ1−1 belongs to sh(B∞ ), hence, by (i), if and only if sh(a−1 c sh(a) σ1−1 ) commutes with σ1 . 25. (right division) (i) For p ≥ q, let σp,q = σp σp−1 ··· σq . −1 For b ∈ B∞ , prove that b ∈ Bn is equivalent to sh(b) = σn,1 b σn,1 . (ii) Assume b ∈ Bn , and p < n. Prove that b ∈ Bp holds if and only if b commutes with σn,p+1 .

4) holds. Then two words in BWn+ code for positively isotopic geometric braids if and only if they are ≡+ -equivalent. 19. (positive presentation) The monoid Bn+ admits—as a monoid—the presentation σ1 , . . , σn−1 ; σi σj = σj σi for |i − j| ≥ 2, σi σj σi = σj σi σj for |i − j| = 1 . 2: Braid Colourings 2. Braid Colourings The first connection between braids and self-distributivity is the existence of an action of braids on sequences of elements chosen in a set equipped with a left self-distributive operation.

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