# 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,

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

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[edit]

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]}

## References[edit]

- ^
^{a}^{b}Dehornoy, Patrick; Paris, Luis (1999), "Gaussian groups and Garside groups, two generalisations of Artin groups",*Proceedings of the London Mathematical Society*,**79**(3): 569–604, CiteSeerX 10.1.1.595.739, doi:10.1112/s0024611599012071 **^**Garside, Frank A. (1969), "The braid group and other groups",*Quarterly Journal of Mathematics*, Second Series,**20**: 235–254, Bibcode:1969QJMat..20..235G, doi:10.1093/qmath/20.1.235

- Benson Farb,
*Problems on mapping class groups and related topics*(Volume 74 of Proceedings of symposia in pure mathematics) AMS Bookstore, 2006, ISBN 0-8218-3838-5, p. 357 - Patrick Dehornoy,
*Groupes de Garside*, Annales Scientifiques de l'École Normale Supérieure (4)**35**(2002) 267-306. MR2003f:20067. - Matthieu Picantin, "Garside monoids vs divisibility monoids",
*Math. Structures Comput. Sci.***15**(2005) 231-242. MR2006d:20102.