# Monoid

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In algebra, a monoid is a set equipped with a binary operation satisfying certain properties similar to but less stringent than those of a group. A motivating example of a monoid is the set of positive integers with multiplication as the operation.

Formally, a monoid is set M with a binary operation ${\displaystyle \star }$ satisfying the following conditions:

• M is closed under ${\displaystyle \star }$;
• The operation ${\displaystyle \star }$ is associative
• There is an identity element ${\displaystyle I\in M}$ such that
${\displaystyle I\star x=x=x\star I}$ for all x in M.

A commutative monoid is one which satisfies the further property that ${\displaystyle x\star y=y\star x}$ for all x and y in M. Commutative monoids are often written additively.

An element x of a monoid is invertible if there exists an element y such that ${\displaystyle x\star y=y\star x=I}$: this is the inverse element for x and may be written as ${\displaystyle x^{-1}}$: by associativity an element can have at most one inverse (note that ${\displaystyle x=y^{-1}}$ as well). The identity element is self-inverse and the product of invertible elements is invertible,

${\displaystyle (xy)^{-1}=y^{-1}x^{-1}\,}$

so the invertible elements form a group, the unit group of M.

A pseudoinverse for x is an element ${\displaystyle x^{+}}$ such that ${\displaystyle xx^{+}x=x}$. The inverse, if it exists, is a pseudoinverse, but a pseudoinverse may exists for a non-invertible element.

A submonoid of M is a subset S of M which contains the identity element I and is closed under the binary operation.

A monoid homomorphism f from monoid ${\displaystyle (M,{\star })}$ to ${\displaystyle (N,{\circ })}$ is a map from M to N satisfying

• ${\displaystyle f(x\star y)=f(x)\circ f(y)\,}$;
• ${\displaystyle f(I_{M})=I_{N}.\,}$

## Examples

• The non-negative integers under addition form a commutative monoid, with zero as identity element.
• The positive integers under multiplication form a commutative monoid, with one as identity element.
• The set of all maps from a set to itself forms a monoid, with function composition as the operation and the identity map as the identity element.
• Square matrices under matrix multiplication form a monoid, with the identity matrix as the identity element: this monoid is not in general commutative.
• Every group is a monoid, by "forgetting" the inverse operation.

## Cancellation property

A monoid satisfies the cancellation property if

${\displaystyle xz=yz\Rightarrow x=y\,}$ and
${\displaystyle zx=zy\Rightarrow x=y.\,}$

A monoid is a submonoid of a group if and only if it satisfies the cancellation property.

## Free monoid

The free monoid on a set G of generators is the set of all words on G, the finite sequences of elements of G, with the binary operation being concatenation (juxtaposition). The identity element is the empty (zero-length) word. The free monoid on one generator g may be identified with the monoid of non-negative integers

${\displaystyle n\leftrightarrow g^{n}=gg\cdots g.\,}$