Lesson plan
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.
Article summary
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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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