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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.[1]

The name was coined by Dehornoy and Paris[1] to mark the work of F. A. Garside on the conjugacy problem for braid groups.[2]

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.

Mathematics Encyclopedia