A generalized closure and complement phenomenon. (English) Zbl 0542.54001

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 C. Kuratowski [Fundam. Math. 3, 182-199 (1922)] for topological closures (with 14 as a bound), and later by R. L. Graham, D. E. Knuth and 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.


54A05 Topological spaces and generalizations (closure spaces, etc.)
03E20 Other classical set theory (including functions, relations, and set algebra)
68Q45 Formal languages and automata


Zbl 0309.04002
Full Text: DOI


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[2] Graham, R. L.; Knuth, D. E.; Motzkin, T. S., Complements and transitive closures, Discrete Math., 2, 17-29 (1972) · Zbl 0309.04002
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