View source for Endomorphism

{{Short description|Self-self morphism}}

[[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|automorphism]].]]

In [[Mathematics|mathematics]], an '''endomorphism''' is a [[Morphism|morphism]] from a [[Philosophy:Mathematical object|mathematical object]] to itself.  An endomorphism that is also an [[Isomorphism|isomorphism]] is an [[Automorphism|automorphism]]. For example, an endomorphism of a [[Vector space|vector space]] {{math|''V''}} is a [[Linear map|linear map]] {{math|''f'': ''V'' → ''V''}}, and an endomorphism of a [[Group (mathematics)|group]] {{math|''G''}} is a [[Group homomorphism|group homomorphism]] {{math|''f'': ''G'' → ''G''}}. In general, we can talk about endomorphisms in any [[Category (mathematics)|category]]. In the [[Category of sets|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|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|automorphism]].  The set of all automorphisms is a [[Subset|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|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|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|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|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|concrete category]], especially for [[Vector space|vector space]]s, 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|topology]], [[Metric (mathematics)|metric]], ...), such operators can have properties like continuity, [[Bounded function|boundedness]], and so on.  More details should be found in the article about [[Operator theory|operator theory]].

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

Let {{math|''S''}} be an arbitrary set. Among endofunctions on {{math|''S''}} one finds [[Permutation|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|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|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==
* {{Citation| last=Jacobson| first=Nathan|  year=2009| title=Basic algebra| edition=2nd| volume = 1 | publisher=Dover| isbn = 978-0-486-47189-1}}

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

[[Category:Morphisms]]

{{Sourceattribution|Endomorphism}}