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'endomorphism'
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| Title of object: |
endomorphism |
| Canonical Name: |
Endomorphism2 |
| Type: |
Definition |
| Created on: |
2005-10-29 16:51:52 |
| Modified on: |
2005-12-06 11:32:45 |
| Classification: |
msc:18A05, msc:18A20 |
| Keywords: |
morphism, homomorphism, types of morphisms |
| Defines: |
Endomorphism, Automorphism |
Preamble:
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Content:
\emph{Endomorphism} is such morphism (morphism is an other term for homomorphism) whose source and destination are the same object.
That is a morphism $f$ is \emph{endomorphism}, when $\mathrm{Src}f=\mathrm{Dst}f=A$ where $A$ is some object (e.g. $A$ may be an abstract algebra). Then one can say, the object of endomorphism $f$ is $A$.
In the most general case endomorphisms are encountered in category theory. As a special case of this endomorphisms are also encountered in abstract algebra.
A morphism which is both an endomorphism and an isomorphism is called \emph{automorphism}.
The sets of endomorphisms and automorphisms for an object $A$ of a category are often denoted correspondingly as $\mathrm{End}(A)$ and $\mathrm{Aut}(A)$ or sometimes as $\mathrm{end}(A)$ and $\mathrm{aut}(A)$.
\emph{Endomorphisms} also can be considered as objects of \PMlinkname{category of intermorphisms}{PseudomorphismsAndIntermorphisms} and (if the set of morphisms of our category is preordered) also of \PMlinkname{category of pseudomorphisms}{PseudomorphismsAndIntermorphisms}. |
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