Alternating normal forms for braids and locally Garside monoids

Patrick Dehornoy (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We describe new types of normal forms for braid monoids, Artin-Tits monoids, and, more generally, for 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 coincides 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 standard ordering (``Dehornoy order'') of Bn in terms of that of B(n-1), and to a quadratic upper bound for the complexity of this ordering.

Mots clés: braid group; braid ordering; tree; Garside monoid; Artin--Tits monoid; locally Garside monoid

Domaines: Théorie des groupes [math.GR]

Patrick Dehornoy. Alternating normal forms for braids and locally Garside monoids. 2008. ⟨hal-00132277v3⟩