  Abstract. We address the conjecture which states that an intersection of
parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove
that the conjecture is equivalent to a, a priori, weaker conjecture. We also
prove the conjecture in a specific case. Along the way, we provide short and
almost self-contain algebraical proofs of several classical results on
Artin-Tits groups, such as those of Van der Lek on intersection of standard
parabolic subgroups.
