Kuratowski closure-complement problem for other mathematical objects? 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. 
 A: Here's a paper that might be of interest:
D. Peleg, A generalized closure and complement phenomenon, Discrete Math., v.50 (1984) pp.285-293.
Other than what's found in the above paper I do not know of any general theory or framework specifically aimed at organizing results similar to the Kuratowski closure-complement problem, i.e., those which involve starting with a seed object (or objects) and repeatedly applying operations to generate further objects of the same type in a given space.
Here's a general sub-question I thought of recently, that might be interesting to study:
"What's the minimum possible cardinality of a seed set that generates the maximum number of sets via the given operations?"
A few years ago I proposed a challenging Monthly problem (11059) that essentially asks this question for the operations of closure, complement, and union in a topological space.  It does turn out there's a space containing a singleton that generates infinitely many sets under the three operations, but it's a bit tricky to find.  I haven't looked into the question yet for other operations.  As far as I know it hasn't been discussed yet in the literature (apart from the specific case addressed by my problem proposal).
