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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.

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The radical or annihilator of an arbitrary subset of a ring is an ideal, so you'd gain exactly one set by changing "ideal" to "arbitrary subset." –  Qiaochu Yuan Feb 25 '10 at 4:29
    
Well, unless you're working in the zero ring, every subset of which is an ideal! –  Qiaochu Yuan Feb 25 '10 at 4:31
    
Atiyah-Macdonald, p.9 says that the radical of an arbitrary subset of a ring is not necessarily an ideal - an example would be, in $\mathbb{Z}$, $rad({2})={2}$. –  Zev Chonoles Feb 25 '10 at 5:05
    
Ah. I was thinking of defining the radical of a subset as the intersection of the prime ideals containing it. This is the analogue of defining the closure of a subset as the intersection of the closed sets containing it. –  Qiaochu Yuan Feb 25 '10 at 5:05
    
Good point - that might make more sense. Well, let's just see what people come up with for the case of ideals, first. –  Zev Chonoles Feb 25 '10 at 5:29
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up vote 5 down vote accepted

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).

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Thanks for the reference and intriguing subquestion! –  Zev Chonoles Feb 27 '10 at 1:55
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