In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector 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, etc. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In general, we can talk about endomorphisms in any category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. In Mathematics, a composite function represents the application of one function to the results of another 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). In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation

An invertible endomorphism of X is called an automorphism. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself The set of all automorphisms is a subgroup of End(X), called the automorphism group of X and denoted Aut(X). In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself 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). In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Under this addition, the endomorphisms of an abelian group form a ring (the endomorphism ring). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object 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. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, specifically in Ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring. In Mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer

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. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a unary operation is an operation with only one Operand, i In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

Depending on the additional structure defined for the category at hand (topology, metric, . Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. . . ), such operators can have properties like continuity, boundedness, and so on. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function More details should be found in the article about operator theory. In Mathematics, operator theory is the branch of Functional analysis which deals with Bounded linear operators and their properties

See also

• morphism
• Frobenius endomorphism
• adjoint endomorphism

External links

• Category of Endomorphisms and Pseudomorphisms. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Commutative algebra and field theory, which are branches of Mathematics, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a In Mathematics, the adjoint endomorphism or adjoint action is an Endomorphism of Lie algebras that plays a fundamental role in the development Victor Porton. 2005. - Endomorphisms of a category (particularly of a category with partially ordered morphisms) are also objects of certain categories. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
• Endomorphism on PlanetMath