Endfunktion

Endfunktion, in mathematics often referred to as an endofunction in English, is a function whose domain and codomain are the same set. Formally, an Endfunktion on a set X is any map f: X → X. The collection of all endofunctions on X is denoted End(X).

Endofunctions form a monoid under composition: the operation is function composition, the neutral element is the identity function id_X, and associativity follows from the associativity of composition. For finite sets with |X| = n, the number of endofunctions is n^n, since each of the n elements can be mapped independently to any of the n elements.

A common way to study endofunctions is via their functional graph: represent X as vertices and draw an edge x → f(x) for every x ∈ X. In such a graph every vertex has out-degree 1. Each connected component contains exactly one directed cycle, with trees feeding into the cycle. This structure yields insights into fixed points, periodic points, and dynamical behavior under iteration of f.

Endofunctions are a special case of endomorphisms in a more general sense: for a given mathematical structure, endomorphisms are structure-preserving maps from the object to itself. On sets, End(X) consists of all functions X → X; on algebraic structures such as groups or vector spaces, endomorphisms are homomorphisms or linear maps from the structure to itself.

Applications appear in computer science and mathematics, including modeling state updates in deterministic automata and analyzing fixed points, cycles, and convergence in discrete dynamical systems. See also Endomorphism, Automorphism, Permutation, and Functional graph.