Endomorphism
From Wikipedia, the free encyclopedia
In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are simply functions from a set S into itself.
In any category, the composition of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid, denoted End(X) (or EndC(X) to emphasize the category C).
An invertible endomorphism of X is called an automorphism. The set of all automorphisms is a subgroup of End(X), called the automorphism group of X and denoted Aut(X). In the following diagram, the arrows denote implication:
| automorphism |  |
isomorphism |
 |
|
|
| endomorphism | |
(homo)morphism |
Any two endomorphisms of an abelian group A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under this addition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of Zn is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group, however there are rings which are not the endomorphism ring of any abelian group.
[edit] Operator theory
In any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits of elements, etc.
Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory.
[edit] See also
[edit] External links
- Category of Endomorphisms and Pseudomorphisms. Victor Porton. 2005. - Endomorphisms of a category (particularly of a category with partially ordered morphisms) are also objects of certain categories.
- Endomorphism on PlanetMath
