# 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 | ||

+ | * 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 [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. | ||

+ | |||

+ | The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.: | ||

* 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] | ||

− | |||

* 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] | ||

* [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] | * [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] | ||

− | + | ||

+ | Generalizations of monoids feature in [[Category theory]], for example: | ||

* [http://www.cs.ru.nl/~heunen/publications/2006/arrows/arrows.pdf Arrows, like Monads, are Monoids] (PDF) | * [http://www.cs.ru.nl/~heunen/publications/2006/arrows/arrows.pdf Arrows, like Monads, are Monoids] (PDF) |

## Revision as of 13:56, 7 February 2012

*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: