Editing Endomorphism

{{Short description|Self-self morphism}}
{{redirect|Endomorphic|the Sheldon body type|Somatotype and constitutional psychology}}
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{{Use shortened footnotes|date=May 2025}}
[[File:Orthogonal projection.svg|frame|right|[[Orthogonal projection]] onto a line, {{math|''m''}}, 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]] {{math|''V''}} is a [[linear map]] {{math|''f'': ''V'' → ''V''}}, and an endomorphism of a [[group (mathematics)|group]] {{math|''G''}} is a [[group homomorphism]] {{math|''f'': ''G'' → ''G''}}. In general, we can talk about endomorphisms in any [[Category (mathematics)|category]]. In the [[category of sets]], endomorphisms are [[Function (mathematics)|functions]] from a [[Set (mathematics)|set]] ''S'' to itself.

In any category, the [[function composition|composition]] of any two endomorphisms of {{math|''X''}} is again an endomorphism of {{math|''X''}}.  It follows that the set of all endomorphisms of {{math|''X''}} forms a [[monoid]], the [[full transformation monoid]], and denoted {{math|End(''X'')}} (or {{math|End{{sub|''C''}}(''X'')}} to emphasize the category {{math|''C''}}).

==Automorphisms==
{{main|Automorphism}}
An [[Inverse function|invertible]] endomorphism of {{math|''X''}} is called an [[automorphism]].  The set of all automorphisms is a [[subset]] of {{math|End(''X'')}} with a [[group (mathematics)|group]] structure, called the [[automorphism group]] of {{math|''X''}} and denoted {{math|Aut(''X'')}}.  In the following diagram, the arrows denote implication:
{| style="border:none;"
|-
| align="center" width="42%" | [[Automorphism]]
| align="center" width="16%" | ⇒
| align="center" width="42%" | [[Isomorphism]]
|-
| align="center" | ⇓
|
| align="center" | ⇓
|-
| align="center" | Endomorphism
| align="center" | ⇒
| align="center" | [[Homomorphism|(Homo)morphism]]
|}

==Endomorphism rings==
{{main|Endomorphism ring}}
Any two endomorphisms of an [[abelian group]], {{math|''A''}}, can be added together by the rule {{math|(''f'' + ''g'')(''a'') {{=}} ''f''(''a'') + ''g''(''a'')}}.  Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a [[ring (mathematics)|ring]] (the [[endomorphism ring]]).  For example, the set of endomorphisms of <math>\mathbb{Z}^n</math> is the ring of all {{math|''n'' × ''n''}} [[Matrix (mathematics)|matrices]] with [[integer]] entries.  The endomorphisms of a vector space or [[module (mathematics)|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 theory==
In any [[concrete category]], especially for [[vector space]]s, endomorphisms are maps from a set into itself, and may be interpreted as [[unary operator]]s on that set, [[action (group theory)|acting]] on the elements, and allowing the notion of element [[orbit (group theory)|orbit]]s to be defined, etc.

Depending on the additional structure defined for the category at hand ([[topology]], [[metric (mathematics)|metric]], ...), such operators can have properties like [[continuous function (topology)|continuity]], [[Bounded function|boundedness]], and so on.  More details should be found in the article about [[operator theory]].

==Endofunctions==
An '''endofunction''' is a function whose [[domain of a function|domain]] is equal to its [[codomain]]. A [[homomorphism|homomorphic]] endofunction is an endomorphism.

Let {{math|''S''}} be an arbitrary set. Among endofunctions on {{math|''S''}} one finds [[permutation]]s of {{math|''S''}} and constant functions associating to every {{math|''x''}} in {{math|''S''}} the same element {{math|''c''}} in {{math|''S''}}.  Every permutation of {{math|''S''}} has the codomain equal to its domain and is [[bijection|bijective]] and invertible. If {{math|''S''}} has more than one element, a constant function on {{math|''S''}} has an [[Image (mathematics)|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]] {{math|''n''}} the floor of {{math|''n''/2}} has its image equal to its codomain and is not invertible.

Finite endofunctions are equivalent to [[directed pseudoforest]]s. For sets of size {{math|''n''}} there are {{math|''n''{{sup|''n''}}}} endofunctions on the set.

Particular examples of bijective endofunctions are the [[involution (mathematics)|involution]]s; i.e., the functions coinciding with their inverses.

==See also==
*[[Adjoint endomorphism]]
*[[Epimorphism]] (surjective homomorphism)
*[[Frobenius endomorphism]]
*[[Monomorphism]] (injective homomorphism)

==Notes==
<references />

==References==
{{refbegin}}
* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}}
{{refend}}

==External links==
* {{springer|title=Endomorphism|id=p/e035600}}

[[Category:Morphisms]]