Composition Monoids
Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain. Then one can form long, potentially complicated chains of these functions composed together, such as f ∘ f ∘ g ∘ f. Such long chains have the algebraic structure of a monoid, called transformation monoid or composition monoid. In general, composition monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup on X.
If the functions are bijective, then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions.
The set of all bijective functions f: X → X form a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group.
Read more about this topic: Function Composition
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