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Endomorphism
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an endomorphism is a morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
 (or homomorphism
Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
) from a mathematical object to itself. For example, an endomorphism of a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 V is a linear map ƒ: V ? V, and an endomorphism of a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 G is a group homomorphism
Group homomorphism

In mathematics, given two group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that...
 ƒ: G ? G. In general, we can talk about endomorphisms in any category
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
. In the category of sets, endomorphisms are simply functions from a set S into itself.

In any category, the composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
 of any two endomorphisms of X is again an endomorphism of X.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an endomorphism is a morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
 (or homomorphism
Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
) from a mathematical object to itself. For example, an endomorphism of a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 V is a linear map ƒ: V ? V, and an endomorphism of a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 G is a group homomorphism
Group homomorphism

In mathematics, given two group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that...
 ƒ: G ? G. In general, we can talk about endomorphisms in any category
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
. In the category of sets, endomorphisms are simply functions from a set S into itself.

In any category, the composition
Function composition

In mathematics, a composite function represents the application of one function to the results of another. For instance, the functions and can be composed by first computing a f and then applying a function g to the output of f....
 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
Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element....
, denoted End(X) (or EndC(X) to emphasize the category C).

An invertible
Inverse element

In mathematics, the idea of inverse element generalises the concepts of additive inverse, in relation to addition, and Multiplicative inverse, in relation to multiplication....
 endomorphism of X is called an automorphism
Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map the object to itself while preserving all of its structure....
. The set of all automorphisms is a subgroup
Subgroup

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *....
 of End(X), called the automorphism group of X and denoted Aut(X). In the following diagram, the arrows denote implication:

Any two endomorphisms of an abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 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
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
 (the endomorphism ring
Endomorphism ring

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 or module
Module (mathematics)

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 scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
 also form a ring, as do the endomorphisms of any object in a preadditive category
Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched category over the monoidal category of abelian groups....
. The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring
Nearring

In mathematics, a near-ring is an abstract algebra algebraic structure similar to a ring but satisfying fewer properties. Near-rings arise naturally from Function on group s....
.

Operator theory


In any concrete category
Concrete category

In mathematics, a concrete category is commonly understood as a category whose objects are mathematical structure Set , whose morphisms are structure-preserving function s, and whose composition operation is function composition....
, especially for vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s, 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
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, metric
Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
, ...), such operators can have properties like continuity
Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
, boundedness
Boundedness

Boundedness or bounded may refer to:*Bounded set, a set of finite size, including a bounded poset, a partially ordered set which has both a greatest element and a least element...
, and so on. More details should be found in the article about operator theory
Operator theory

In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them....
.

See also

  • morphism
    Morphism

    In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
  • Frobenius endomorphism
    Frobenius endomorphism

    In commutative algebra and field theory , which are branches of mathematics, the Frobenius endomorphism is a special endomorphism of Ring with prime characteristic p, a class importantly including fields....
  • adjoint endomorphism
    Adjoint endomorphism

    In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups....


External links


  • . . 2005. - Endomorphisms of a category
    Category theory

    In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
     (particularly of a category with partially ordered morphisms
    Category theory

    In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
    ) are also objects
    Category theory

    In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
     of certain categories.