In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. So, for example, an endomorphism of a vector space V is a linear map f : V → V and an endomorphism of a group G is a group homomorphism f : G → G, etc. In general, we can talk about endomorphisms in any category. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... // Homomorphism for beginners Homomorphism is one of the fundamental concepts in abstract algebra. ... Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... Jump to: navigation, search In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

Given an object X in a category C and two endomorphisms f and g of X, the composite f O g is also an endomorphism of X. Since the identity map on X is also an endomorphism of X, we see that the set of all endomorphisms of X forms a monoid, denoted End_C(X) or just End(X) if the category is understood. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...

In many but not all situations it is possible to add endomorphisms, and the endomorphisms of a given object then form a ring, called the endomorphism ring of the object. This is true, for example, in the categories of abelian groups, modules, and vector spaces. In general it is true in all preadditive categories. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In abstract algebra, a module is a generalization of a vector space. ... Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...

An endomorphism that is also an isomorphism is termed an automorphism. In the following diagram, the arrows denote implication. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

automorphism		isomorphism
endomorphism		(homo)morphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... // Homomorphism for beginners Homomorphism is one of the fundamental concepts in abstract algebra. ...

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.