In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others. The groups are named after Emil Artin, due to his early work on braid groups in the 1920s to 1940s, and Jacques Tits who developed the theory of a more general class of groups in the 1960s. Definition An Artin–Tits presentation is a group presentation where is a (usually finite) set of generators and is a set of Artin–Tits relations, namely relations of the form for distinct in , where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators . An Artin–Tits group is a group that admits an Artin–Tits presentation. Likewise, an Artin–Tits monoid is a monoid that, as a monoid, admits an Artin–Tits presentation. Alternatively, an Artin–Tits group can be specified by the set of generators and, for every in , the natural number that is the length of the words and such that is the relation connecting and , if any. By convention, one puts when there is no relation . Formally, if we define to denote an alternating product of and of length , beginning with — so that , , etc. — the Artin–Tits relations take the form The integers can be organized into a symmetric matrix, known as the Coxeter matrix of the group. If is an Artin–Tits presentation of an Artin–Tits group , the quotient of obtained by adding the relation for each of is a Coxeter group. Conversely, if is a Coxeter group presented by reflections and the relations are removed, the extension thus obtained is an Artin–Tits group. For instance, the Coxeter group associated with the -strand braid group is the symmetric group of all permutations of . Examples is the free group based on ; here for all . is the free abelian group based on ; here for all . is the braid group on strands; here for , and for . General properties Artin–Tits monoids are eligible for Garside methods based on the investigation of their divisibility relations, and are well understood: Artin–Tits monoids are cancellative, and they admit greatest common divisors and conditional least common multiples (a