Difference between revisions of "Monoid"

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A monoid is an algebraic structure with an associative binary operation that has an identity element. Examples include:

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:

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-A monoid is an algebraic structure with a single, associative binary operation and an identity element.
+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
+* functions from a type to itself, under composition
+The monoid interface enables a number of algorithms, including parallel algorithms and tree searches, e.g.:
-== See also ==
+* The [http://www.haskell.org/ghc/dist/current/docs/libraries/base/Data-Monoid.html Data.Monoid] module
 * An introduction: [http://sigfpe.blogspot.com/2009/01/haskell-monoids-and-their-uses.html Haskell Monoids and their Uses]
-* [http://www.haskell.org/ghc/docs/latest/html/libraries/base/Data-Monoid.html The Data.Monoid module]
 * 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]
-* [[Category theory]]
+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)