Garside element
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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 \text{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 \text{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.
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 \text{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 \text{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.
References
Benson Farb, Problems on mapping class groups and related topics (Volume 74 of Proceedings of symposia in pure mathematics) AMS Bookstore, 2006, , p. 357 Patrick Dehornoy, Groupes de Garside, Annales Scientifiques de l'École Normale Supérieure (4) 35 (2002) 267-306. . Matthieu Picantin, "Garside monoids vs divisibility monoids", Math. Structures Comput. Sci. 15 (2005) 231-242. .
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