  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.
