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Inabstract algebra, an endomorphism is ahomomorphism from a mathematical object to itself.[1] More generally incategory theory, an endomorphism is amorphism from acategory of objects to itself.[2] An endomorphism that is also anisomorphism is anautomorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of agroup G is agroup homomorphism f: G → G.
In general, we can talk about endomorphisms in anycategory. In thecategory of sets, endomorphisms arefunctions from asetS to itself.
In any category, thecomposition of any two endomorphisms ofX is again an endomorphism ofX. It follows that the set of all endomorphisms ofX forms amonoid, thefull transformation monoid, and denotedEnd(X) (orEndC(X) to emphasize the categoryC).[citation needed]
Aninvertible endomorphism ofX is called anautomorphism. The set of all automorphisms is asubset ofEnd(X) with agroup structure, called theautomorphism group ofX and denotedAut(X). In the following diagram, the arrows denote implication:
| Automorphism | ⇒ | Isomorphism |
| ⇓ | ⇓ | |
| Endomorphism | ⇒ | (Homo)morphism |
Any two endomorphisms of anabelian group,A, can be added together by the rule(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 aring (theendomorphism ring). For example, the set of endomorphisms of is the ring of alln ×nmatrices withinteger entries. The endomorphisms of a vector space ormodule also form a ring, as do the endomorphisms of any object in apreadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as anear-ring. Every ring with one is the endomorphism ring of itsregular module, and so is a subring of an endomorphism ring of an abelian group;[3] however there are rings that are not the endomorphism ring of any abelian group.
In anyconcrete category, especially forvector spaces, endomorphisms are maps from a set into itself, and may be interpreted asunary operators on that set,acting on the elements, and allowing the notion of elementorbits to be defined, etc.
Depending on the additional structure defined for the category at hand (topology,metric, ...), such operators can have properties likecontinuity,boundedness, and so on. More details should be found in the article aboutoperator theory.
Anendofunction is a function whosedomain is equal to itscodomain. Ahomomorphic endofunction is an endomorphism.
LetS be an arbitrary set. Among endofunctions onS one findspermutations ofS and constant functions associating to everyx inS the same elementc inS. Every permutation ofS has the codomain equal to its domain and isbijective and invertible. IfS has more than one element, a constant function onS has animage that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to eachnatural numbern the floor ofn/2 has its image equal to its codomain and is not invertible.
Finite endofunctions are equivalent todirected pseudoforests. For sets of sizen there arenn endofunctions on the set.
Particular examples of bijective endofunctions are theinvolutions; i.e., the functions coinciding with their inverses.