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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 a single associative binary operation and an identity element. You can think of a monoid as a semigroup where you designate an identity element, or as a group except you don't require elements have inverses.

Examples

  • The set of non-negative integers $\mathbf{N} = \{0,1,2,\dotsc\}$ is a monoid under the operation of addition, the identity element being $0$.

  • Any group is also a monoid; you just forget the fact that the elements happen to have inverses.

Further reading