There is an astonishing number of things in programming that are monoids:
- Numbers, addition, 0
- Numbers, multiplication, 1
- Lists, concatenation, [] (including strings)
- Sorted lists, merge with respect to a linear order, []
- Sets, union, {}
- Maps, left-biased or right-biased union, {}
- Maps K->V, union where Vs for same K get merged in some other monoid, {}
- For any M: Subsets of M, intersection, M
- Any lattice with an upper bound, minimum, upper bound;
symmetrically for a lower-bounded set
- If (S, *, u) is a monoid, then (A -> S, \f g x -> f x * g x, \x ->
u) is a monoid
- Product (a,b) and co-product (Either) of monoids
- Parsers, alternation, a parser that always fails
- etc.
The benefits of calling something a monoid arise from using
general-purpose structures operating on monoids:
- Finger trees http://apfelmus.nfshost.com/monoid-fingertree.html
- Aforementioned maps which merge values for a key in a given monoid
- Aforementioned monoids lifted to functions
- Monoidal folds (Data.Foldable)
- ...
2009/11/13 Magicloud Magiclouds
Hi, I have looked the concept of monoid and something related, but still, I do not know why we use it?
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