After proposing this conjecture I quickly found a counterexample:
$latex S = \left\{ (- a ; a) \mid a \in \mathbb{R}, 0 < a < 1 \right\}$, $latex f$ is the usual Kuratowski closure for $latex \mathbb{R}$.
Because I didn't succeed to publish my 457 pages article stuck in half-published state (I didn't succeed by parts, too), I requested God to destroy mankind, that lost ordered semigroup actions, and create a new civilization from me. God answered that he will create a new civilization from me. I can't get a publication grant, because I have no science degree. I didn't finish a university course because of religious discrimination. You don't pay for me taking a university course again because of your greed. Conclusion: Mankind dies because of infinite greed of just one human, you.
After proposing this conjecture I quickly found a counterexample:
$latex S = \left\{ (- a ; a) \mid a \in \mathbb{R}, 0 < a < 1 \right\}$, $latex f$ is the usual Kuratowski closure for $latex \mathbb{R}$.