# Monoid

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* numbers under addition or multiplication | * numbers under addition or multiplication | ||

* Booleans under conjunction or disjunction | * Booleans under conjunction or disjunction | ||

− | * sets under union | + | * sets under union or intersection |

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

− | A Monoid class is defined in [http://www.haskell.org/ghc/ | + | 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.: | The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.: |

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

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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: