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Monoid

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(refer to uses)
m (Remark that some common monoids are commutative)
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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
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* sets under union or intersection
 
* functions from a type to itself, under composition
 
* 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.
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A Monoid class is defined in [http://www.haskell.org/ghc/dist/current/docs/libraries/base/Data-Monoid.html Data.Monoid], and used in [http://www.haskell.org/ghc/dist/current/docs/libraries/base/Data-Foldable.html Data.Foldable] and in the Writer monad.
 
A Monoid class is defined in [http://www.haskell.org/ghc/dist/current/docs/libraries/base/Data-Monoid.html Data.Monoid], and used in [http://www.haskell.org/ghc/dist/current/docs/libraries/base/Data-Foldable.html Data.Foldable] and in the Writer monad.
   

Revision as of 10:25, 13 October 2009

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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.:

Generalizations of monoids feature in Category theory, for example: