Endomorphism

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: "V" → "V" and an endomorphism of a group "G" is a group homomorphism ƒ: "G" → "G", etc. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are simply functions from a set "S" into itself.

In any category, the 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"C"("X") to emphasize the category "C").

An invertible endomorphism of "X" is called an automorphism. The set of all automorphisms is a subgroup of End("X"), called the automorphism group of "X" and denoted Aut("X"). In the following diagram, the arrows denote implication:

Any two endomorphisms of an abelian group "A" can be added together by the rule (ƒ + "g")("a") = ƒ("a") + "g"("a"). Under this addition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of Z"n" is the ring of all "n" × "n" matrices with integer entries. The endomorphisms of a vector space, module, ring, or algebra 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.

Operator theory

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 to define the notion of orbits of elements, 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.

ee also

*morphism
*Frobenius endomorphism
*adjoint endomorphism

External links

* [http://www.mathematics21.org/pseudomorphisms-category.xml Category of Endomorphisms and Pseudomorphisms] . [http://www.mathematics21.org/ Victor Porton] . 2005. - "Endomorphisms" of a category (particularly of a category with partially ordered morphisms) are also objects of certain categories.
*planetmath reference|id=7462|title=Endomorphism


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  • endomorphism — [en΄dōmôr′fiz΄əm] n. a type of metamorphism in which intrusive igneous rock is changed by the action of the surrounding rock …   English World dictionary

  • endomorphism — endomorfizmas statusas T sritis fizika atitikmenys: angl. endomorphism vok. Endomorphismus, n rus. эндоморфизм, m pranc. endomorphisme, m …   Fizikos terminų žodynas

  • endomorphism — noun Date: 1909 a homomorphism that maps a mathematical set into itself compare isomorphism …   New Collegiate Dictionary

  • endomorphism — /en doh mawr fiz euhm, deuh/, n. 1. Petrol. a change brought about within the mass of an intrusive igneous rock. 2. Math. a homomorphism of a set into itself. [1950 55; ENDOMORPH(IC) + ISM] * * * …   Universalium

  • endomorphism — noun a) The assimilation of surrounding rock by an intrusive igneous rock b) A homomorphism that maps a system to itself …   Wiktionary

  • endomorphism — n. change which occurs within an endomorph (Petrology) …   English contemporary dictionary

  • endomorphism — en·do·mor·phism …   English syllables

  • endomorphism — en•do•mor•phism [[t]ˌɛn doʊˈmɔr fɪz əm, də [/t]] n. pet a change brought about within the mass of an intrusive igneous rock • Etymology: 1950–55 …   From formal English to slang

  • endomorphism — /ɛndoʊˈmɔfɪzəm/ (say endoh mawfizuhm) noun a change within the mass of an intrusive igneous rock brought about by the rock s own magma …   Australian English dictionary

  • endomorphism — I. ˌ ̷ ̷ ̷ ̷ˈ ̷ ̷ˌfizəm noun ( s) Etymology: International Scientific Vocabulary end + morphism; originally formed as French endomorphisme : a change (as in texture or in chemical composition by assimilation of foreign material) produced in an… …   Useful english dictionary

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