order theory - 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).