Editing Endomorphism (section)

==Endofunctions==
An '''endofunction''' is a function whose [[domain of a function|domain]] is equal to its [[codomain]]. A [[homomorphism|homomorphic]] endofunction is an endomorphism.

Let {{math|''S''}} be an arbitrary set. Among endofunctions on {{math|''S''}} one finds [[permutation]]s of {{math|''S''}} and constant functions associating to every {{math|''x''}} in {{math|''S''}} the same element {{math|''c''}} in {{math|''S''}}.  Every permutation of {{math|''S''}} has the codomain equal to its domain and is [[bijection|bijective]] and invertible. If {{math|''S''}} has more than one element, a constant function on {{math|''S''}} has an [[Image (mathematics)|image]] that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each [[natural number]] {{math|''n''}} the floor of {{math|''n''/2}} has its image equal to its codomain and is not invertible.

Finite endofunctions are equivalent to [[directed pseudoforest]]s. For sets of size {{math|''n''}} there are {{math|''n''{{sup|''n''}}}} endofunctions on the set.

Particular examples of bijective endofunctions are the [[involution (mathematics)|involution]]s; i.e., the functions coinciding with their inverses.