Endomorphism

An endomorphism is a morphism from an object to itself in a given mathematical context. In more concrete terms, it is a structure-preserving map from a structure to the same type of structure.

In algebra, endomorphisms are homomorphisms from an object to itself. For example, a group endomorphism is a map f: G → G that preserves the group operation; a linear endomorphism of a vector space V over a field F is a linear map f: V → V; a ring endomorphism is a map f: R → R that preserves both addition and multiplication (and, for unital rings, the multiplicative identity).

The set of all endomorphisms of a given object often carries algebraic structure. For a vector space V, End(V) denotes the endomorphism ring: it is equipped with addition (pointwise) and composition as multiplication. It contains the zero map (additive identity) and the identity map (multiplicative identity). In many contexts End(V) is noncommutative, and its elements can be viewed as linear transformations on V.

An endomorphism is invertible precisely when it is an automorphism. The automorphisms of an object G form a group Aut(G) under composition; for a finite-dimensional vector space, Aut(V) is the general linear group GL(V). Endomorphisms need not be invertible, but those that are form a subgroup of the endomorphism monoid.

In category theory, End(X) denotes the set of endomorphisms of an object X, forming a monoid under composition. Endomorphisms are central to areas such as module theory, representation theory, and the study of symmetry, providing a framework for self-mimicking structure-preserving maps.