- Endomorphism
In

mathematics , an**endomorphism**is amorphism (orhomomorphism ) from a mathematical object to itself. For example, an endomorphism of avector space "V" is alinear map ƒ: "V" → "V" and an endomorphism of a group "G" is agroup homomorphism ƒ: "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 asubgroup of End("X"), called theautomorphism group of "X" and denoted Aut("X"). In the following diagram, the arrows denote implication: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 (theendomorphism ring ). For example, the set of endomorphisms of**Z**^{"n"}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 apreadditive category . The endomorphisms of a nonabelian group generate an algebraic structure known as anearring .**Operator theory**In any

concrete category , especially forvector space s, endomorphisms are maps from a set into itself, and may be interpreted asunary operator s 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 aboutoperator theory .**ee also***

morphism

*Frobenius endomorphism

*adjoint endomorphism **External links*** [

*http://www.mathematics21.org/pseudomorphisms-category.xml Category of Endomorphisms and Pseudomorphisms*] . [*http://www.mathematics21.org/ Victor Porton*] . 2005. - "Endomorphisms" of a category (particularly of a category with partially ordered morphisms) are also objects of certain categories.

*planetmath reference|id=7462|title=Endomorphism

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