Monoid

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 $\star$ satisfying the following conditions:

• M is closed under $\star$;
• The operation $\star$ is associative
• There is an identity element $I\in M$ such that

$I\star x=x=x\star I$ for all x in M.

A commutative monoid is one which satisfies the further property that $x\star y=y\star x$ 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 $x\star y=y\star x=I$: this is the inverse element for x and may be written as $x^{-1}$: by associativity an element can have at most one inverse (note that $x=y^{-1}$ as well). The identity element is self-inverse and the product of invertible elements is invertible,

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

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

A pseudoinverse for x is an element $x^{+}$ such that $xx^{+}x=x$. 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 $(M,{\star })$ to $(N,{\circ })$ is a map from M to N satisfying

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

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

$xz=yz\Rightarrow x=y\,$ and

$zx=zy\Rightarrow x=y.\,$

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

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