Endomorphism

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Orthogonal projection onto a line, Template:Math, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism.

In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space Template:Math is a linear map Template:Math, and an endomorphism of a group Template:Math is a group homomorphism Template:Math. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.

In any category, the composition of any two endomorphisms of Template:Math is again an endomorphism of Template:Math. It follows that the set of all endomorphisms of Template:Math forms a monoid, the full transformation monoid, and denoted Template:Math (or Template:Math to emphasize the category Template:Math).

AutomorphismsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An invertible endomorphism of Template:Math is called an automorphism. The set of all automorphisms is a subset of Template:Math with a group structure, called the automorphism group of Template:Math and denoted Template:Math. In the following diagram, the arrows denote implication:

Automorphism	⇒	Isomorphism
⇓		⇓
Endomorphism	⇒	(Homo)morphism

Endomorphism ringsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Any two endomorphisms of an abelian group, Template:Math, can be added together by the rule Template:Math. Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of <math>\mathbb{Z}^n</math> is the ring of all Template:Math 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 near-ring. 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;<ref>Jacobson (2009), p. 162, Theorem 3.2.</ref> however there are rings that are not the endomorphism ring of any abelian group.

Operator theoryEdit

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 the notion of element orbits to be defined, 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.

EndofunctionsEdit

An endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism.

Let Template:Math be an arbitrary set. Among endofunctions on Template:Math one finds permutations of Template:Math and constant functions associating to every Template:Math in Template:Math the same element Template:Math in Template:Math. Every permutation of Template:Math has the codomain equal to its domain and is bijective and invertible. If Template:Math has more than one element, a constant function on Template:Math has an image that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each natural number Template:Math the floor of Template:Math has its image equal to its codomain and is not invertible.