Endomorphism

Endomorphism

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Endomorphism

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 ?: V ? V and an endomorphism of a group G is a group homomorphism ?: G ? G, etc. 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 End_C(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	\Rightarrow	isomorphism
\Downarrow	\Downarrow
endomorphism	\Rightarrow	(homo)morphism

Any two endomorphisms of an abelian group A can be added together by the rule (? + g)(a) = ?(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, module, ring, or algebra 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.

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.