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
Famous quotes containing the word composition:
“There is singularly nothing that makes a difference a difference in beginning and in the middle and in ending except that each generation has something different at which they are all looking. By this I mean so simply that anybody knows it that composition is the difference which makes each and all of them then different from other generations and this is what makes everything different otherwise they are all alike and everybody knows it because everybody says it.”
—Gertrude Stein (1874–1946)