A famous theorem/puzzle/problem from topology is to show that given any subset of a topological space, at most 14 distinct sets can be obtained from it by iteratively using the operations closure and complement. (See Wikipedia's article on the Kuratowski's closure-complement problem.)
In the realm of formal languages, it appears to me that one can argue that the Kleene star operator can be seen as a type of closure. And complementation is complementation.
So I was curious about the following question: Given an alphabet $\Sigma$ and language $L \subseteq \Sigma^*$ what is the maximum number of different languages that can be obtained from $L$ by iteratively applying the Kleene star and complementation operators? An example of a language for which this maximum is achieved would also be instructive.
Since Kleene star is a closure operator, combining it with complementation can produce at most 14 distinct languages (P. C. Hammer. Kuratowski’s closure theorem, Nieuw Arch. Wisk., 8(2):74–80, 1960).
In Section 4.1 of the following paper, the author states without proof that $L=\{a,aab,bbb\}$ generates the maximum where $\Sigma=\{a,b\}$.
D. Peleg. A generalized closure and complement phenomenon. Discrete Math., 50:285–293, 1984. http://dx.doi.org/10.1016/0012-365X(84)90055-4
The following papers address similar topics.
J. Brzozowski, E. Grant, and J. Shallit. Closures in formal languages: Concatenation, separation, and algorithms. Comp. Research Rep. – CORR, abs/0901.3. arXiv:1109.1227 [math.GN], arXiv.org, 2009. http://arxiv.org/abs/0901.3763
J. Brzozowski, E. Grant, and J. Shallit. Closures in formal languages and Kuratowski’s theorem. Intl. J. Found. Comp. Sci., 22:301–321, 2011. http://dx.doi.org/10.1142/S0129054111008052
J. Brzozowski, G. Jirásková, and C. Zou. Quotient complexity of closed languages. Theory Comput. Syst., 54(2):277–292, 2014. ISSN 1432-4350. http://dx.doi.org/10.1007/s00224-013-9515-7
É. Charlier, M. Domaratzki, T. Harju, and J. Shallit. Finite orbits of language operations. Language and Automata Theory and Applications, Proceedings of the 5th International Conference, pages 204–215, 2011. http://dx.doi.org/10.1007/978-3-642-21254-3_15
É. Charlier, M. Domaratzki, T. Harju, and J. Shallit. Composition and orbits of language operations: Finiteness and upper bounds. Intl. J. Comp. Math., pages 1–26, 2012. http://dx.doi.org/10.1080/00207160.2012.681305
Š. Holub and J. Kortelainen. On partitions separating words. Intl. J. Algebra Comp., 21(8):1305–1316, 2011. http://dx.doi.org/10.1142/S0218196711006650
J. Jirásek, M. Palmovský, and J. Šebej. Kuratowski algebras generated by factor-, subword-, and suffix-free languages. International Conference on Descriptional Complexity of Formal Systems, pages 189–201, 2017. http://dx.doi.org/10.1007/978-3-319-60252-3_15
J. Jirásek and Juraj Šebej. Kuratowski algebras generated by prefix-free languages. Implementation and Application of Automata, CIAA 2016, pages 150–162, 2016. http://dx.doi.org/10.1007/978-3-319-40946-7_13