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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 a set M with a binary operation \star satisfying the following conditions:

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: x+y rather than x \star y.

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 monoid (N,{\circ}) is a map from M to N satisfying

[edit] Examples

[edit] 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.

[edit] 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 . \,
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