Endomorphism

This article is about the mathematical concept. For the endomorphic body type, see Somatotype.

In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V and an endomorphism of a group G is a group homomorphism ƒ: G → G, etc. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are simply functions from a set S into 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, denoted End(X) (or End_C(X) to emphasize the category C).

An invertible endomorphism of X is called an automorphism. The set of all automorphisms is a subgroup of End(X), called the automorphism group of X and denoted Aut(X). In the following diagram, the arrows denote implication:

automorphism	$\Rightarrow$	isomorphism
$\Downarrow$		$\Downarrow$
endomorphism	$\Rightarrow$	(homo)morphism

Any two endomorphisms of an abelian group A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under this addition, the endomorphisms of an abelian group form a ring (the endomorphism ring). For example, the set of endomorphisms of Zⁿ is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space, module, ring, or algebra also form a ring, as do the endomorphisms of any object in a preadditive category. The endomorphisms of a nonabelian group generate an algebraic structure known as a nearring.

Operator theory

In any concrete category, especially for vector spaces, endomorphisms are maps from a set into itself, and may be interpreted as unary operators on that set, acting on the elements, and allowing to define the notion of orbits of elements, etc.

Depending on the additional structure defined for the category at hand (topology, metric, ...), such operators can have properties like continuity, boundedness, and so on. More details should be found in the article about operator theory.

See also

• morphism
• Frobenius endomorphism
• adjoint endomorphism

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.
• "Endomorphism". PlanetMath.
• Dr. Ronald Joe Record Ph.D. Dissertation "The Method Of Critical Curves For Discrete Dynamical Systems In Two Dimensions", June 1994, University of California at Santa Cruz, presents research in the study of noninvertible endomorphisms of the plane carried out by developing the method of critical curves and incorporating this theory in experimental digital simulations.

Software

• [1] Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, endo, for graphically exploring iterated endomorphisms of the plane. Lyapunov exponents can be calculated and displayed for a region of parameter space. Phase portraits can be constructed and histographic data displayed. Finally, critical curves and their iterates may be displayed (curves for which the determinant of the Jacobian is zero).The contents and manual pages of the mathrec software laboratory are also available.