The term endomorphism derives from the Greek adverb *endon* ("inside")
and *morphosis* ("to form" or "to shape").

In algebra, an endomorphism of a group, module, ring, vector space, etc. is a homomorphism from one object to itself (with surjectivity
not required).

In ergodic theory, let be a set, a sigma-algebra on and a probability measure.
A map is called an endomorphism (or measure-preserving transformation)
if

1. is surjective,

2. is measurable,

3. for all , where denotes the pre-image
of .

An endomorphism is called ergodic if it is true that implies or 1, where .

## See also

Endomorphism Ring,

Measurable Function,

Morphism,

Sigma-Algebra,

Surjection
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## References

Lang, S. *Algebra, rev. 3rd ed.* New York: Springer-Verlag, 2002.## Referenced on Wolfram|Alpha

Endomorphism
## Cite this as:

Weisstein, Eric W. "Endomorphism." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Endomorphism.html

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