We construct a new monoid structure for Artin groups associated with finite
Coxeter systems. This monoid shares with the classical positive braid monoid a
crucial algebraic property: it is a Garside monoid. The analogy with the
classical construction indicates there is a ``dual'' way of studying Coxeter
systems, where the pair (W,S) is replaced by (W,T), with T the set of all
reflections. In the type A case, we recover the monoid constructed by
Birman-Ko-LeeComment: 42 pages. Major revision, many new result
