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
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 Dehorney and Paris[1] to mark the work of F. A. Garside on the conjugacy problem for braid groups.[2]
References
- Benson Farb, Problems on mapping class groups and related topics (Volume 74 of Proceedings of symposia in pure mathematics) AMS Bookstore, 2006, ISBN 0821838385, p.357
- Patrick Dehorney, "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.