A generalized closure and complement phenomenon (Q795354)

The number of different sets that can be generated from a given set by applications of complement and closure operators is finite and small. This fact, stated originally by \textit{C. Kuratowski} [Fundam. Math. 3, 182-199 (1922)] for topological closures (with 14 as a bound), and later by \textit{R. L. Graham}, \textit{D. E. Knuth} and \textit{T. S. Motzkin} [Discrete Math. 2, 17-29 (1972; Zbl 0309.04002)] for transitive closures of binary relations (with 10 as a bound), is generalized to other closure operators, with different bounds. Several examples are given, including Kleene closures of languages, unions and intersections with a fixed set, transitive closures of non-binary relations and difunctional closures of binary relations.