Editing Function composition (section)

==Composition monoids==
{{main|Transformation monoid}}
Suppose one has two (or more) functions {{math|''f'': ''X'' → ''X'',}} {{math|''g'': ''X'' → ''X''}} having the same domain and codomain; these are often called ''[[Transformation (function)|transformations]]''. Then one can form chains of transformations composed together, such as {{math|''f'' ∘ ''f'' ∘ ''g'' ∘ ''f''}}. Such chains have the [[algebraic structure]] of a [[monoid]], called a ''[[transformation monoid]]'' or (much more seldom) a ''composition monoid''.  In general, transformation monoids can have remarkably complicated structure. One particular notable example is the [[de Rham curve]]. The set of ''all'' functions {{math|''f'': ''X'' → ''X''}} is called the [[full transformation semigroup]]<ref name="Hollings_2014"/> or ''symmetric semigroup''<ref name="Grillet_1995"/> on&nbsp;{{mvar|X}}. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.<ref name="Dömösi-Nehaniv_2005"/>)

[[File:SVG skew and rotation.svg|thumb|Composition of a [[shear mapping]] <small>(red)</small> and a clockwise rotation by 45° <small>(green)</small>. On the left is the original object. Above is shear, then rotate. Below is rotate, then shear.]]
If the given transformations are [[bijective]] (and thus invertible), then the set of all possible combinations of these functions forms a [[transformation group]] (also known as a [[permutation group]]); and one says that the group is [[group generator|generated]] by these functions.

The set of all bijective functions {{math|''f'': ''X'' → ''X''}} (called [[permutation]]s) forms a group with respect to function composition. This is the [[symmetric group]], also sometimes called the ''composition group''. A fundamental result in group theory, [[Cayley's theorem]], essentially says that any group is in fact just a subgroup of a symmetric group ([[up to]] isomorphism).<ref name="Carter_2009"/>

In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a [[regular semigroup]].<ref name="Ganyushkin-Mazorchuk_2008"/>