least common multiple exists whenever a common multiple does). If is an Artin–Tits monoid, and if is the associated Coxeter group, there is a (set-theoretic) section of into , and every element of admits a distinguished decomposition as a sequence of elements in the image of ("greedy normal form"). Very few results are known for general Artin–Tits groups. In particular, the following basic questions remain open in the general case: – solving the word and conjugacy problems — which are conjectured to be decidable, – determining torsion — which is conjectured to be trivial, – determining the center — which is conjectured to be trivial or monogenic in the case when the group is not a direct product ("irreducible case"), – determining the cohomology — in particular solving the conjecture, i.e., finding an acyclic complex whose fundamental group is the considered group. Partial results involving particular subfamilies are gathered below. Among the few known general results, one can mention: Artin–Tits groups are infinite countable. In an Artin–Tits group , the only relation connecting the squares of the elements of is if is in (John Crisp and Luis Paris ). For every Artin–Tits presentation , the Artin–Tits monoid presented by embeds in the Artin–Tits group presented by (Paris). Every (finitely generated) Artin–Tits monoid admits a finite Garside family (Matthew Dyer and Christophe Hohlweg). As a consequence, the existence of common right-multiples in Artin–Tits monoids is decidable, and reduction of multifractions is effective. Particular classes of Artin–Tits groups Several important classes of Artin groups can be defined in terms of the properties of the Coxeter matrix. Artin–Tits groups of spherical type An Artin–Tits group is said to be of spherical type if the associated Coxeter group is finite — the alternative terminology "Artin–Tits group of finite type" is to be avoided, because of its ambiguity: a "finite type group" is just one that admits a finite generating set. Recall that a complete classification is known, the 'irreducible types' being labeled as the infinite series , , , and six exceptional groups , , , , , and . In the case of a spherical Artin–Tits group, the group is a group of fractions for the monoid, making the study much easier. Every above-mentioned problem is solved in the positive for spherical Artin–Tits groups: the word and conjugacy problems are decidable, their torsion is trivial, the center is monogenic in the irreducible case,