[[Abstract algebra MOC]]
# Monoid

A **monoid** $(M, \cdot)$ is a [[Semigroup]] with an identity element. #m/def/general
In general elements needn't have an inverse.

2. **Associative** $a\cdot (b\cdot c) = (a\cdot b) \cdot c$ for each $a,b,c \in M$
3. **Identity** there exists (provably unique) $e \in M$ such that $a\cdot e=e\cdot a=a$ for all $a \in M$

A monoid may be [[Oidification|generalized]] to a [[category]], which can be thought of as a typed monoid.
See also [[Monoid object]].

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#state/develop | #lang/en | #SemBr