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{{About|the mathematical concept|the '''endomorphic''' body type|Somatotype}}
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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:&nbsp;''V''&nbsp;→&nbsp;''V'', and an endomorphism of a [[group (mathematics)|group]] ''G'' is a [[group homomorphism]] ƒ:&nbsp;''G''&nbsp;→&nbsp;''G''. In general, we can talk about endomorphisms in any [[category theory|category]]. In the category of [[Set (mathematics)|sets]], endomorphisms are simply functions from a set ''S'' into itself.
 
In any category, the [[function composition|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 End<sub>''C''</sub>(''X'') to emphasize the category ''C'').
 
An [[inverse element|invertible]] endomorphism of ''X'' is called an [[automorphism]]. The set of all automorphisms is a [[subset]] of End(''X'') with a [[group (mathematics)|group]] structure, called the [[automorphism group]] of ''X'' and denoted Aut(''X''). In the following diagram, the arrows denote implication:
{| border="0"
|-
| align="center" width="42%" | [[automorphism]]
| align="center" width="16%" | <math>\Rightarrow</math>
| align="center" width="42%" | [[isomorphism]]
|-
| align="center" | <math>\Downarrow</math>
|
| align="center" | <math>\Downarrow</math>
|-
| align="center" | endomorphism
| align="center" | <math>\Rightarrow</math>
| align="center" | [[homomorphism|(homo)morphism]]
|}
 
Any two endomorphisms of an [[abelian group]] ''A'' can be added together by the rule (ƒ&nbsp;+&nbsp;''g'')(''a'')&nbsp;=&nbsp;ƒ(''a'')&nbsp;+&nbsp;''g''(''a''). Under this addition, the endomorphisms of an abelian group form a [[ring (mathematics)|ring]] (the [[endomorphism ring]]). For example, the set of endomorphisms of '''Z'''<sup>''n''</sup> is the ring of all ''n''&nbsp;×&nbsp;''n'' 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 [[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,<ref>Jacobson (2009), p. 162, Theorem 3.2.</ref> however there are rings which 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 to define the notion of [[orbit (group theory)|orbit]]s of elements, 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]], [[boundedness]], and so on. More details should be found in the article about [[operator theory]].
 
==Endofunctions in mathematics==
 
In [[mathematics]], an '''endofunction''' is a [[function (mathematics)|function]] whose [[codomain]] is equal to its [[domain of a function|domain]]. A [[homomorphism|homomorphic]] endofunction is an endomorphism.
 
Let ''S'' be an arbitrary set. Among endofunctions on ''S'' one finds [[permutation]]s of ''S'' and constant functions associating to each <math>x\in S</math> a given <math>c\in S</math>.  
Every permutation of ''S'' has the codomain equal to its domain and is [[bijection|bijective]] and invertible. A constant function on ''S'', if ''S'' has more than 1 element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer ''n'' the floor of ''n''/2 has its codomain equal to its domain and is not invertible.
 
Finite endofunctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to 1, and can be easily described. For sets of size ''n'', there are ''n''<sup>''n''</sup> endofunctions on the set.
 
Particular bijective endofunctions are the [[Involution (mathematics)|involution]]s, i.e. the functions coinciding with their inverses.
 
==Notes==
<references />
 
==See also==
*[[Adjoint endomorphism]]
*[[Frobenius endomorphism]]
 
==References==
* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189-1}}
 
==External links==
* {{springer|title=Endomorphism|id=p/e035600}}
*{{planetmath reference|id=7462|title=Endomorphism}}
 
[[Category:Morphisms]]

Latest revision as of 02:48, 22 December 2014

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