Fix a topological space $X$, and write $\iota$ and $\kappa$ for (respectively) the interior and closure operators associated with $X$. Hence $\{\iota,\kappa\}$ is a subset of the monoid of endofunctions of the powerset of $X$. Let $M_X$ denote the submonoid generated by this subset.
Question. Do any non-trivial equations (i.e. equations other than $\iota^2=\iota$ and $\kappa^2=\kappa$ and those that can be proved from these two starting points using the monoid axioms) necessarily hold in the monoid $M_X$?
Edit. So far its been stated that $(\kappa \iota)^2 = \kappa \iota$ and $(\iota \kappa)^2= \iota \kappa$ hold. I'd like to know if these equations are complete for the class of monoids that can be expressed in the form $M_X$ (with respect to the inference methods of equational logic).
If not, what further axioms do we need?
Kuratowski's closure-complement theorem may gives an answer of your question. First note that $c\kappa c = \iota$, where $c$ is the complement operator. Kuratowski's theorem gives following equation:
$$(\kappa \iota)^2 = \kappa\iota.$$