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 \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.

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 x x^{+} 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 . \,
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