Alternating Normal Forms for Braids and Locally Garside Monoids

We describe new types of normal forms for braid monoids, Artin– Tits monoids, and, more generally, all monoids in which divisibility has some convenient lattice properties (“locally Garside monoids”). We show that, in the case of braids, one of these normal forms turns out to coincide with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the canonical well-order of B n in terms of that of B + n−1. The first aim of this paper is to improve our understanding of the well-order of positive braids and of the so-called Burckel normal form of [7], which after more than a decade remain mysterious objects [16]. Here this aim is achieved, at least partially, by giving a new, alternative construction of the Burckel normal form that makes the latter hopefully more natural, and, in any case, very easily computable. However, it turns out that the construction we describe below relies on a very general scheme for which many monoids are eligible, and we may hope for further applications beyond the case of braids. Following the seminal work of F.A.Garside [22], we know that braid monoids and, more generally, Artin–Tits monoids and Garside monoids that generalize them, can be equipped with a normal form, namely the so-called greedy normal form of [6, 1, 20, 32], which constructs for each element of the monoid a distinguished representative word in terms of some standard generators. The latter normal form is excellent both in theory and in practice in that it provides an automatic structure, and it is easily computable [21, 9, 13]. What we do in this paper is to construct a new type of normal form for braid monoids and their generalizations. Our construction keeps one of the ingredients of the (right) greedy normal form, namely considering the maximal right divisor that lies in some subset A of the considered monoid M , but, instead of taking for A the finite set of so-called simple elements, i.e., the divisors of the Garside element ∆, we choose A to be some standard parabolic submonoid M0 of M , i.e., the monoid generated by some subset I of the standard generating set SS. When I is a proper subset of SS, the submonoid M0 is a proper subset of M , and the construction stops after one step. However, by considering two parabolic submonoids M1, M0 which together generate M , we can obtain a well-defined, unique decomposition alternatively involving M1 and M0, according to a scheme that is usual in the case of an amalgamated product. By considering convenient families of submonoids, we can iterate the process and finally obtain a unique normal form for each element 1991 Mathematics Subject Classification. 20F36, 20M05, 06F05.