# User Contributed Dictionary

### Noun

- The assimilation of surrounding rock by an intrusive igneous rock
- A homomorphism that maps a system to itself

#### Translations

- Croatian: endomorfizam
- Swedish: endomorfi , endomorfism

# Extensive Definition

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, 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
EndC(X) to emphasize the category C).

An invertible
endomorphism of X is called an automorphism. The set of
all automorphisms is a subgroup 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 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, module,
ring, or algebra
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.

## 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.

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 about operator
theory.

## 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.

endomorphism in Catalan: Endomorfisme

endomorphism in German: Endomorphismus

endomorphism in French: Endomorphisme

endomorphism in Italian: Endomorfismo

endomorphism in Dutch: Endomorfisme

endomorphism in Polish: Endomorfizm

endomorphism in Russian: Эндоморфизм

endomorphism in Serbian:
Ендоморфизам