Unary operation ($f:S\to S$)

A unary operation on a set $S$ is a function $f:S\to S$ taking one input and returning an element of the same set. Examples include the additive inverse $n\mapsto -n$ on the integers and inversion $g\mapsto g^{-1}$ in a group.

Unary operations can be composed and iterated. The identity function and all maps $S\to S$ form a monoid under function composition. For negation, $f\circ f=\mathrm{id}$; for a successor map on the nonnegative integers, iteration instead produces an unbounded sequence.

An operation's identities are extra data. Being unary alone says nothing about injectivity, surjectivity or whether repeated application returns to the starting element.