The dual braid monoid

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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-Lee