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.
A Garside monoid is a monoid with the following properties: 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.
[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.
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