Garside element

In mathematics, a Garside element is an element of an algebraic structure such as a monoid that has several desirable properties.

Formally, if M is a monoid, then an element Δ of M is said to be a Garside element if the set of all right divisors of Δ,

${\displaystyle \{r\in M\mid {\text{for some }}x\in M,\Delta =xr\},}$

is the same set as the set of all left divisors of Δ,

${\displaystyle \{\ell \in M\mid {\text{for some }}x\in M,\Delta =\ell x\},}$

and this set generates M.

A Garside element is in general not unique: any power of a Garside element is again a Garside element.

Garside monoid and Garside group

A Garside monoid is a monoid with the following properties:

• Finitely generated and atomic;
• Cancellative;
• The partial order relations of divisibility are lattices;
• There exists a Garside element.

A Garside monoid satisfies the Ore condition for multiplicative sets and hence embeds in its group of fractions: such a group is a Garside group. A Garside group is biautomatic and hence has soluble word problem and conjugacy problem. Examples of such groups include braid groups and, more generally, Artin groups of finite Coxeter type.^[1]

The name was coined by Patrick Dehornoy and Luis Paris^[1] to mark the work on the conjugacy problem for braid groups of Frank Arnold Garside (1915–1988), a teacher at Magdalen College School, Oxford who served as Lord Mayor of Oxford in 1984–1985.^[2]