Endomorphism
In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. So, 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, etc. In general, we can talk about endomorphisms in any category.
Given an object X in a category C and two endomorphisms f and g of X, the composite f O g is also an endomorphism of X. Since the identity map on X is also an endomorphism of X, we see that the set of all endomorphisms of X forms a monoid, denoted End_{C}(X) or just End(X) if the category is understood.
In many but not all situations it is possible to add endomorphisms, and the endomorphisms of a given object then form a ring, called the endomorphism ring of the object. This is true, for example, in the categories of abelian groups, modules, and vector spaces. In general it is true in all preadditive categories.
An endomorphism that is also an isomorphism is termed an automorphism. In the following diagram, the arrows denote implication.
automorphism | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \to} | isomorphism |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \downarrow} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \downarrow} | |
endomorphism | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \to} | (homo)morphism |
External links
- Category of Endomorphisms and Pseudomorphisms. Victor Porton. 2005. - Endomorphisms of a category (particularly of a category with partially ordered morphisms) are also objects of certain categories.
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