# Monoid

A monoid is a semigroup with an identity element. More formally, given a set M and an operation `+`

, we say that `(M, +)`

is a *monoid* if it satisfies the following properties for any x, y, z âˆˆ M:

*Closure*:`x + y`

âˆˆ M*Associativity*:`(x + y) + z = x + (y + z)`

*Identity*: There exists an e âˆˆ M such that`e + x = x + e = x`

We also say that *M is a monoid under +.*

## Examples of Monoids

- The natural numbers
*N*are monoids under addition *N*is a monoid under multiplication.- Strings form a monoid under concatenation (
`""`

is the identity element)

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