The original [Kuratowski closure-complement problem](http://en.wikipedia.org/wiki/Kuratowski%27s_closure-complement_problem) asks:



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


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



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> 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 $R$? 
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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. 



Kuratowski closure-complement problem for other mathematical objects?
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).


