The original Kuratowski closure-complement problem asks:

How many distinct sets can be obtained by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space?

My question is: **what is known about analogous questions in other settings?**

Here's an example of what I'm thinking of, for rings:

How many distinct ideals can be obtained by repeatedly applying the operations of radical and annihilator to a given starting ideal $I$ of a commutative ring $R$?

Note that $r(r(I))=r(I)$ and $I\subseteq Ann(Ann(I))=\{x\in R: x\cdot Ann(I)=(0)\}$, which are the best analogs I could think of to $\overline{\overline{S}}=\overline{S}$ and $(S^C)^C=S$.

Also: **what is the structure necessary to formulate this kind of question called, and where does it occur naturally?**

It seems like we need at least a poset, but with distinguished idempotent and involution operations to generalize the closure and complement, respectively.