Monoid-like structure with unary operator?

Question

If there is a monoid-like algebraic structure but with unary operator instead of binary one?

$(A,f,I)$ a set $A$ closed under unary operator $f$ and identity element $I$ which is a fixpoint of $f$ such as $f(I)=I$.

Answer 1

The closest algebraic structure I can think of is a one-letter deterministic automaton. That is, $A$ is the set of states, $\{f\}$ is the alphabet, and the transition function is defined by $$ p \cdot f = q \space\text{ if }f(p) = q $$ A fixpoint (which might not be unique) is a state $q$ such that $q \cdot f = q$.

If you insist to have a monoid, take the transition monoid of this automaton.