endomorphismEndomorphism22005-10-29 16:51:522007-06-23 06:12:03Definitionendomorphismautomorphismmorphismhomomorphismtypes of morphisms% this is the default PlanetMath preamble. as your knowledge
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% define commands here\emph{Endomorphism} is such morphism (morphism is another \PMlinkescapetext{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}.