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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,

$$\{ r \in M \mid \mbox{for some } x \in M, \Delta = x r \},$$

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

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

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.

The name was coined by Dehornoy and Paris to mark the work of F. A. Garside on the conjugacy problem for braid groups.

References

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, doi:10.1112/s0024611599012071

Garside, F.A. (1969), "The braid group and other groups", Q. J. Math., Oxf. II. Ser. 20: 235–254, 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", Ann .Sci. Ecole Norm. Sup. (4) 35 (2002) 267-306. MR 2003f:20067.
Matthieu Picantin, "Garside monoids vs divisibility monoids", Math. Structures Comput. Sci. 15 (2005) 231-242. MR 2006d:20102.

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