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:
“Boswell, when he speaks of his Life of Johnson, calls it my magnum opus, but it may more properly be called his opera, for it is truly a composition founded on a true story, in which there is a hero with a number of subordinate characters, and an alternate succession of recitative and airs of various tone and effect, all however in delightful animation.”
—James Boswell (1740–1795)