selfmappings

Selfmappings, or endofunctions, are functions that map a set to itself. If X is a set, a self-mapping is a function f: X -> X. The collection of all self-mappings on X is denoted X^X, and under function composition it forms a monoid with the identity map id_X as the unit element. When X has additional structure, such as a group, ring, or vector space, a self-mapping may be studied as an endomorphism if it preserves that structure.

A self-mapping can be classified by its effect on X: it may be injective, surjective, or bijective. In finite sets of size n, there are n^n distinct self-mappings. Special examples include the identity map, which fixes every element, and constant maps, which send all elements to a fixed y in X. A permutation is a bijective self-map; its inverse is also a self-map.

Dynamics: iterating a self-mapping yields the orbit of each element, producing fixed points (points with f(x)=x) and, in many cases, cycles and eventually periodic behavior. The study of self-maps therefore intersects with combinatorics, algebra, and dynamical systems.

Applications: self-mappings model state transitions in deterministic systems, automata, and various algebraic constructions through endomorphisms. In category theory, endomorphisms are morphisms from an object to itself, and their set forms the endomorphism monoid End(X).