Difference between revisions of "Monoid"
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| − | A monoid is an algebraic structure with |
+ | A monoid is an algebraic structure with an associative binary operation that has an identity element. Examples include: |
| + | * lists under concatenation |
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| + | * numbers under addition or multiplication |
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| + | * Booleans under conjunction or disjunction |
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| + | * sets under union or intersection |
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| + | * functions from a type to itself, under composition |
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| + | Note that in most of these cases the operation is also commutative, but it need not be; concatenation and function composition are not commutative. |
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| − | == See also == |
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| − | + | A Monoid class is defined in [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-Monoid.html Data.Monoid], and used in [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-Foldable.html Data.Foldable] and in the Writer monad. |
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| + | The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.: |
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* The blog article [http://apfelmus.nfshost.com/monoid-fingertree.html Monoids and Finger Trees] |
* The blog article [http://apfelmus.nfshost.com/monoid-fingertree.html Monoids and Finger Trees] |
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| + | * [http://haskell.org/sitewiki/images/6/6a/TMR-Issue11.pdf Monad.Reader issue 11, "How to Refold a Map."] (PDF), and a [http://haskell.org/haskellwiki/The_Monad.Reader/Discuss_Issue11 follow up] |
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| − | * [[Category theory]] |
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| − | * [http://www.cs.ru.nl/~heunen/publications/2006/arrows/arrows.pdf Arrows, like Monads, are Monoids] (PDF) |
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| + | Generalizations of monoids feature in [[Category theory]], for example: |
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| + | * [http://www.researchgate.net/publication/235540658_Arrows_like_Monads_are_Monoids/file/d912f511ccdf2c1016.pdf Arrows, like Monads, are Monoids] (PDF) |
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Revision as of 09:41, 18 February 2014
This article is a stub. You can help by expanding it.
A monoid is an algebraic structure with an associative binary operation that has an identity element. Examples include:
- lists under concatenation
- numbers under addition or multiplication
- Booleans under conjunction or disjunction
- sets under union or intersection
- functions from a type to itself, under composition
Note that in most of these cases the operation is also commutative, but it need not be; concatenation and function composition are not commutative.
A Monoid class is defined in Data.Monoid, and used in Data.Foldable and in the Writer monad.
The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.:
- An introduction: Haskell Monoids and their Uses
- The blog article Monoids and Finger Trees
- Monad.Reader issue 11, "How to Refold a Map." (PDF), and a follow up
Generalizations of monoids feature in Category theory, for example: