Endomorphism

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This article is about the mathematical concept. For the endomorphic body type, see Somatotype.

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. In the category of sets, endomorphisms are simply maps from a set S into itself.

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.

An endomorphism that is also an isomorphism is termed an automorphism. In the following diagram, the arrows denote implication:

automorphism	$\to$	isomorphism
$\downarrow$		$\downarrow$
endomorphism	$\to$	(homo)morphism

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.

[edit] 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.