Endomorphism

In abstract algebra, an endomorphism is a homomorphism from a mathematical object to itself.[1] More in general, in category theory, an endomorphism is a morphism from a category of objects to itself.[2] An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G.

In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to 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, the full transformation monoid, and denoted End(X) (or EndC(X) to emphasize the category C).

  1. ^ Lang. Algebra. p. 10.
  2. ^ Lang. Algebra. p. 54.
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