related topics
{math, number, function}
{math, energy, light}

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 ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. 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 EndC(X) to emphasize the category C).

An invertible endomorphism of X is called an automorphism. The set of all automorphisms is a subset of End(X) with a group structure, 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 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 (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 or module 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. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group[1], however there are rings which are not the endomorphism ring of any abelian group.


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.

Full article ▸

related documents
Sum rule in differentiation
Column vector
Abelian category
Elias delta coding
Brun's constant
Inverse functions and differentiation
Category of sets
Complete category
Kleene star
Additive function
Random sequence
Double precision
Hamiltonian path problem
Characteristic subgroup
Mutual recursion
Identity matrix
AVL tree
Row and column spaces
Baire category theorem
Lagged Fibonacci generator
Sophie Germain prime
Up to
Axiom of union
Normal morphism
Blum Blum Shub