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 X be a set, F a sigma-algebra on X and m a probability measure. A map T:X->X is called an endomorphism (or measure-preserving transformation) if

1. T is surjective,

2. T is measurable,

3. m(T^(-1)A)=m(A) for all A in F, where T^(-1)(A) denotes the pre-image of A.

An endomorphism is called ergodic if it is true that T^(-1)A=A implies m(A)=0 or 1, where T^(-1)A={x in X:T(x) in A}.

See also

Endomorphism Ring, Measurable Function, Morphism, Sigma-Algebra, Surjection

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Lang, S. Algebra, rev. 3rd ed. New York: Springer-Verlag, 2002.

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Cite this as:

Weisstein, Eric W. "Endomorphism." From MathWorld--A Wolfram Web Resource.

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