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,
- [math]\displaystyle{ \{ r \in M \mid \text{for some } x \in M, \Delta = x r \}, }[/math]
is the same set as the set of all left divisors of Δ,
- [math]\displaystyle{ \{ \ell \in M \mid \text{for some } x \in M, \Delta = \ell x \}, }[/math]
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]
References
- ↑ 1.0 1.1 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, Frank A. (1969), "The braid group and other groups", Quarterly Journal of Mathematics, Second Series 20: 235–254, doi:10.1093/qmath/20.1.235, Bibcode: 1969QJMat..20..235G
- 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.
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