Endofunction

In mathematics, an endofunction is a function whose codomain is a subset of its domain.

Let "S" be an arbitrary set. Among endofunctions on "S" one finds permutations of "S" and constant functions associating to each $xin\; S$ a given $cin\; S$. Every permutation of "S" has the codomain equal to its domain and is bijective and invertible. A constant function on "S", if "S" has more than 1 element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer "n" the floor of "n"/2 has its codomain equal to its domain and is not invertible. permutation of an arbitrary set are bijective endofunction.

Finite endofuctions are equivalent to monogeneous digraphs, i.e. digraphs having all nodes with outdegree equal to 1, and can be easily described.

Particular bijective endofunctions are the involutions, i.e. the functions coinciding with their inverses.