# Monoid

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Rimmington (Talk | contribs) m (fixed "Arrows, like Monads, are Monoids" link) |
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* numbers under addition or multiplication |
* numbers under addition or multiplication |
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* Booleans under conjunction or disjunction |
* Booleans under conjunction or disjunction |
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− | * sets under union |
+ | * sets under union or intersection |

* functions from a type to itself, under composition |
* 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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+ | |||

+ | 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.: |
The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.: |
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− | * The [http://www.haskell.org/ghc/dist/current/docs/libraries/base/Data-Monoid.html Data.Monoid] module |
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* An introduction: [http://sigfpe.blogspot.com/2009/01/haskell-monoids-and-their-uses.html Haskell Monoids and their Uses] |
* An introduction: [http://sigfpe.blogspot.com/2009/01/haskell-monoids-and-their-uses.html Haskell Monoids and their Uses] |
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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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Generalizations of monoids feature in [[Category theory]], for example: |
Generalizations of monoids feature in [[Category theory]], for example: |
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− | * [http://www.cs.ru.nl/~heunen/publications/2006/arrows/arrows.pdf Arrows, like Monads, are Monoids] (PDF) |
+ | * [http://www.researchgate.net/publication/235540658_Arrows_like_Monads_are_Monoids/file/d912f511ccdf2c1016.pdf Arrows, like Monads, are Monoids] (PDF) |

## Latest 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: