Editing Monoid (section)

{{short description|Algebraic structure with an associative operation and an identity element}}
{{for|monoid objects in category theory|Monoid (category theory)}}
{{Distinguish|Monad (disambiguation){{!}}Monad}}
{{Algebraic structures |group}}
[[File:Algebraic structures - magma to group.svg|thumb|Algebraic structures between [[Magma (algebra)|magmas]] and [[Group (mathematics)|groups]]. For example, monoids are [[semigroup]]s with identity.]]

In [[abstract algebra]], a '''monoid''' is a set equipped with an [[associative]] [[binary operation]] and an [[identity element]]. For example, the nonnegative [[integer]]s with addition form a monoid, the identity element being {{math|0}}.

Monoids are [[semigroup]]s with identity.  Such [[algebraic structure]]s occur in several branches of mathematics. 

The functions from a set into itself form a monoid with respect to function composition. More generally, in [[category theory]], the morphisms of an [[object (category theory)|object]] to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. 

In [[computer science]] and [[computer programming]], the set of [[string (computer science)|strings]] built from a given set of [[Character (computing)|characters]] is a [[free monoid]]. [[Transition monoid]]s and [[syntactic monoid]]s are used in describing [[finite-state machine]]s. [[Trace monoid]]s and [[history monoid]]s provide a foundation for [[process calculi]] and [[concurrent computing]].

In [[theoretical computer science]], the study of monoids is fundamental for [[automata theory]] ([[Krohn–Rhodes theory]]), and [[formal language theory]] ([[star height problem]]).

See [[semigroup]] for the history of the subject, and some other general properties of monoids.