In mathematics, a Garside element is an element of an

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...

such as a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

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 In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . A ...

; * The

partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

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 In abstract algebra, the conjugacy problem for a group ''G'' with a given presentation is the decision problem of determining, given two words ''x'' and ''y'' in ''G'', whether or not they represent conjugate elements of ''G''. That is, the prob ...

. Examples of such groups include

braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

s and, more generally,

Artin group Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun Harutyun ( hy, Հարություն and in Western Armenian Յարութիւն) also spelled Haroutioun, Harut ...

s of finite Coxeter type. The name was coined by

Patrick Dehornoy Patrick Dehornoy (11 September 1952 – 4 September 2019) was a mathematician at the University of Caen Normandy who worked on set theory and group theory. Early life and education Dehornoy was born on 11 September 1952 in Rouen, France. ...

and Luis Paris to mark the work on the conjugacy problem for braid groups of Frank Arnold Garside (1915–1988), a teacher at

Magdalen College School, Oxford Magdalen College School (MCS) is a public school (English independent day school) in Oxford, England, for boys aged seven to eighteen and for girls in the sixth form. It was founded by William Waynflete about 1480 as part of Magdalen Colle ...

who served as

Lord Mayor of Oxford Lord is an appellation for a person or deity who has authority, control, or power over others, acting as a master, chief, or ruler. The appellation can also denote certain persons who hold a title of the peerage in the United Kingdom, or a ...

in 1984–1985.

References

Benson Farb Benson Stanley Farb (born October 25, 1967) is an American mathematician at the University of Chicago. His research fields include geometric group theory and low-dimensional topology. Early life A native of Norristown, Pennsylvania, Farb ear ...

, ''Problems on mapping class groups and related topics'' (Volume 74 of Proceedings of symposia in pure mathematics) AMS Bookstore, 2006, , p. 357 *

, ''Groupes de Garside'',

Annales Scientifiques de l'École Normale Supérieure ''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in m ...

(4) 35 (2002) 267-306. . * Matthieu Picantin, "Garside monoids vs divisibility monoids", ''Math. Structures Comput. Sci.'' 15 (2005) 231-242. . Abstract algebra Semigroup theory {{Abstract-algebra-stub