Braids described as an orthogonal rewriting system Paul-Andr´eMelli`es Laboratoire Preuves, Programmes, Syst`emes CNRS { Universit´eParis Denis Diderot August 12, 2009 In honour of Roel de Vrijer, on the occasion of his 60th birthday Foreword During my postdoctoral stay at the Vrije Universiteit of Amsterdam in 1995, I gave an alternative proof of a folklore theorem on braids, established by Garside at the end of the 1960's. My ambition at the time was to estab- lish the result by applying the traditional tools of Rewriting Theory, and more specifically the theory of residuals for orthogonal rewriting systems formulated by Huet and L´evyat the end of the 1970's. I liked the idea that confluence could be established by the same means on braids and on the λ- calculus. A few years later, it appeared that this stream of ideas was strongly related to the notion of Garside monoid introduced by Dehornoy [5, 6] at about the same time, this opening a nice and fruitful bridge between Algebra and Rewriting Theory. My interest in confluence for braids was raised in the first place by the Amsterdam group in Rewriting Theory, an enthousiastic group of students and young researchers federated around Jan Willem Klop and Roel de Vrijer under the nickname of TeReSe. Here, I take the opportunity of Roel's birthday to publish fifteen years later the rough notes written at the time. I have chosen to keep the notes just as dry as they were, and improved only marginally the original version. The interested reader will find a much clearer exposition of this material in Jan Willem Klop, Vincent van Oostrom and Roel de Vrijer's lecture notes on braids [7]. 1 However, let me briefly indicate the basic intuitions which guided that work. The monoid of positive braids on n + 1 strands is traditionally pre- sented by a set of n generators σ1; ··· ; σn together with the relation σi · σi+1 · σi = σi+1 · σi · σi+1 (1) for every 1 ≤ i ≤ n and the relation σi · σj = σj · σi for every pair i; j whose difference j − i is not equal to 1 or −1. One should think of σi as the operation of permuting the i-th strand of the braid over the (i + 1)-th strand. In particular, Equation (1) reflects the equality up to topological deformation of the two braid diagrams below: i i + 1 i + 2 i i + 1 i + 2 = i i + 1 i + 2 i i + 1 i + 2 From the point of view of Rewriting Theory, Equation (1) means that the following diagram commutes: σi+1 · / · σi σi · (2) σi+1 σi+1 σi · / · / · At this point, one would like to think of the path σi · σi+1 as the residual of the permutation σi after the permutation σi+1, by analogy with the λ- calculus where the rewriting path u1 ·u2 defines the residual of the β-redex u 2 after the β-redex v in the confluence diagram v (λx.xx)(Ia) / (Ia)(Ia)· u1 u a(Ia) I = λy:y u2 v0 (λx.xx)a / aa This very preliminary analysis indicates already that the class of generators σ1; ··· ; σn is not closed under the residual relation we wish to construct. For that reason, the class of generators σ1; ··· ; σn should be replaced by a larger class of generators, closed by the residual relation we have in mind. It appears that a nice and convenient class of generators to consider for that purpose is the whole set of (n + 1)! permutations on n + 1 strands. Every generator σi is then identified with the transposition (i; i + 1) which exchanges the strands i and i + 1. Now, imagine that we are given a total order A on the n + 1 strands. In that case, one may see a permutation σ on these strands as a total order B defined as follows: 8i; j; iBj () σ(i)Aσ(j): It is important here that the two points of view on the redex (seen as a permutation, or as a total order) are equivalent. In particular, the pair of total orders A and B induce a partial order (called their distance) defined as [A; B] = B \ Aop where A∗ denotes the dual of A, that is, the opposite of the complementary relation. This partial order [A; B] should be really understood as the per- mutation σ formulated in the language of order theory. In particular, it is possible to recover B from A and [A; B] by adding them: B = A + [A; B]: This provides yet another alternative way to define a redex between braids, this time as a partial order U included in Aop and such that the relation A + U 3 defines a total order. Reformulating the permutation σ as the partial order [A; B] is the only way I found at the time to define the residual U[V ] of a redex U = [A; B] after a redex V = [A; C]. It appears then that this residual U[V ] is inherently related to the transitive closure of the union U [V of the two partial orders. This phenomenon is nicely illustrated by the local confluence diagram (2) instantiated at i = 1 and n = 3, which becomes (3<2) 123 / 132 (2<1) (3<1;2<1) (3<1;3<2) 213 / 321 where the vertices are total orders on the set f1; 2; 3g of strands, with the notation abc standing for the total order a < b < c, and where the edges A ! B are the redexes U, seen as the partial orders [A; B]. In particular: (3 < 2) = [123; 132] (3 < 1; 2 < 1) = [132; 321] Notice that the union of the two redexes σ1 = (2 < 1) and σ2 = (3 < 2) is the transitive closure (3 < 2 < 1) of the set-theoretic union of (2 < 1) and (3 < 2). This is precisely this observation which convinced me to treat the permutations σ as partial orders U. On the other hand, my feeling when I wrote this paper fifteen years ago was that the construction of residuals was far too complicated to be adapted to other rewriting situations of interest. This is precisely what motivated my later paper [9] where the same residual construction on braids is achieved by much simpler means. The trick is to revisit the traditional theory of resid- uals developed by Huet and L´evy[1], and to distinguish there an atomic and a molecular notion of computation { called redexes and treks respec- tively. By way of illustration, in the case of braids, the transpositions σi are the redexes, while the general permutations σ are the treks. This idea of distinguishing between various levels of atomicity of computation is a nice outcome of the early work on braids exposed below. Among other virtues, this refined theory of residuals offers a uniform framework in order to es- tablish confluence for braids and for the associativity law underlying the 4 coherence theorem for monoidal categories established by MacLane. The residual theory also clarifies a longstanding question in Rewriting Theory, about the conceptual nature of families and extraction in the work by L´evy on optimal reductions in the λ-calculus [8]. Plan of the paper. The seven opening sections deliver a pretty dry and technical account of the basic properties of addition (Section 1) and the relationship with distance (Sections 2 and 3), duality (Section 4), transitive relations (Section 5), scopic relations (Section 6) and transitive closure (Section 7). The residual system for braids is then described in the very last section (Section 8) of these notes. For that reason, we advise the reader to start from this last section, and to proceed in a call-by-need fashion, by reading the opening sections whenever this appears necessary. Abstract. We show that braids can be described as an or- thogonal system in the sense of [1, 2]. Their computation is therefore concluent, which was already shown by Garside [4]. 1 An addition on binary relations We introduce a notion of addition on binary relations. Given a binary relation A on a set X, we note Ac the complement (or negation) of A, Aop its reverse and A∗ = (Aop)c its dual. Definition 1 (addition) The addition of two binary relations A and B on the set X is defined as the binary relation: A + B = (A \ B∗) [ B: Lemma 2 (associativity) Suppose that A, B and C are binary relations on the set X. Then, (A + B) + C = A + (B + C): Proof: 5 (A + B) + C = ((A + B) \ C∗) [ C = (((A \ B∗) [ B) \ C∗) [ C = (((A \ B∗) [ B) \ C∗) [ C = (A \ B∗ \ C∗) [ (B \ C∗) [ C = (A \ B∗ \ C∗) [ (B + C). A + (B + C) = (A \ (B + C)∗) [ (B + C) = (A \ ((B \ C∗) [ C)∗) [ (B + C) = (A \ (B∗ [ C) \ C∗) [ (B + C) = (A \ B∗ \ C∗) [ (A \ C \ C∗) [ (B + C) We have to check that (A \ C \ C∗) ⊆ (A \ C \ C∗) [ (B + C), a fact which follows from C ⊆ (B + C). @ We remark that A + B may be also defined as A + B = (A [ B) \ (A [ A∗): We define Ab = A [ A∗. Lemma 3 (commutativity) Suppose that A and B are binary relations on the set X. The two following assertions are equivalent: • A + B = B + A, • A ⊆ Bb and B ⊆ Ab. Proof: Let A and B be binary relations on the set X. By definition, we know that A ⊆ B + A and B ⊆ A + B. Moreover, A + B can be computed as (A [ B) \ (B [ B∗), a fact which implies that A + B ⊆ A [ B and by symmetry that B +A ⊆ A[B.