Math Transit
Kuratowski’s Closure-Complement Cornucopia
There is a large and scattered literature on Kuratowski’s theorem, most of which focuses on topological spaces; an admirable survey is the paper of Gardner and Jackson.
—Brzozowski, Grant, and Shallit
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2024L. Staiger and K. W. Wagner,
The forty nine Kuratowski lattices in the Cantor space.
Research Report 573, April 2024.
CDMTCS
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2024P. Caron, J.-G. Luque, B. Patrou,
Presentation of monoids generated by a projection and an involution.
26 pp., 2 February 2024.
arXiv
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2024M. Bowron,
Boundary-border extensions of the Kuratowski monoid.
Topology and its Applications,
341(108703):1-35, 1 January 2024.
Elsevier
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arXiv
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2022P. Caron, A. Durand, B. Patrou,
A study of monoids generated by three unary regular operations: reversal, root and star.
Elsevier preprint, 9 November 2022.
SSRN
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2022J. Dassow,
The orbit of closure-involution operations: the case of Boolean functions.
Contributions to Algebra and Geometry,
63(2):321-334, 1 June 2022.
Springer
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2021J. Dassow,
Some remarks on the orbit of closure-involution operations on languages.
Information and Computation,
281(104811), 8 pp., December 2021.
Elsevier
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2021S. Canilang, M. P. Cohen, N. Graese, I. Seong,
The closure-complement-frontier problem in saturated polytopological spaces.
New Zealand Journal of Mathematics,
51:3-27, 6 August 2021.
NZMS
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arXiv
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2020M. P. Cohen, T. Johnson, A. Kral, A. Li, J. Soll,
A Kuratowski closure-complement variant whose solution is independent of ZF.
arXiv preprint, 26 May 2020.
arXiv
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2020E. Przemska,
The lattices of families of regular sets in topological spaces.
Mathematica Slovaca,
70(2):477-488, 2020.
De Gruyter
ResearchGate
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2020I. Cohen, G. Wences, J. R. Contreras, A. R. Linero, R. S. S. Anillo,
A linear version of the Kuratowski problem (in Spanish).
Journal of the Foundations of Science,
5(1):63-71, January 2020.
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English translation by Google Translate
2019J. Jirásek, M. Palmovský, J. Šebej,
Kuratowski Algebras Generated by Prefix-, Suffix-, Factor-, and Subword-Free Languages Under Star and Complementation.
International Journal of Foundations of Computer Science,
30(6-7):1091-1115, September 2019.
World Scientific
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Springer
HAL
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2019J. Dassow,
On the orbit of closure-involution operations - the case of formal languages.
Theoretical Computer Science,
777:192-203, 19 July 2019.
Elsevier
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2019J. H. Santiago,
The group‑theoretic analog of Kuratowski’s closure‑complement theorem.
American Mathematical Monthly,
126(6)519-526, June 2019.
Taylor & Francis
JSTOR
UNAM
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2019B. Nikolić,
Degenerated topological spaces.
Novi Sad Journal of Mathematics,
49(2)95-107, 31 March 2019.
University of Novi Sad
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2019M. V. Volkov,
The identities of the free product of a pair of two-element monoids.
Algebra Universalis,
80(14), 6 pp., 22 February 2019.
Springer
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arXiv
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2018T. Banakh, O. Chervak, T. Martynyuk, M. Pylypovych, A. Ravsky, M. Simkiv,
Kuratowski monoids of n‑topological spaces.
Topological Algebra and its Applications,
6:1-25, 28 March 2018.
De Gruyter
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arXiv
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2018A. P. Farajzadeh and A. Shafie,
On Kuratowski’s problem in vector spaces.
Bulletin of L. N. Gumilyov Eurasian National University
(Series: Mathematics, Computer Science, Mechanics), 125(4):117-119, 2018.
DSpace
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arXiv
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2017P. Gammie and G. Gioiosa,
The Kuratowski Closure-Complement Theorem.
Archive of Formal Proofs,
27 October 2017.
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2017H. Wang, Y. Wang, D. Tao, Y. Peng,
Kuratowski monoids of various orders on lattices.
Journal of Suzhou University of Science and Technology
(Natural Science Edition), 34(3):15-18, September 2017.
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2017K. P. Shum,
A note on Kuratowski’s theorem and its related topics.
Advances in Pure Mathematics
7(8):383-406, August 2017.
SCIRP
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2017R. Berghammer,
Tool-based relational investigation of closure-interior relatives for finite topological spaces.
In P. Höfner, D. Pous, and G. Struth, editors,
Relational and Algebraic Methods in Computer Science, RAMICS 2017, Lyon, France,
volume 10226 of
Lecture Notes in Computer Science (LNICS),
pp. 60-76, 2017.
Springer
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2017A. Gupta and R. D. Sarma,
A note on some generalized closure and interior operators in a topological space.
Mathematics for Applications,
6:11-20, 2017.
Brno University of Technology
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2017R. C. Schwiebert,
The radical-annihilator monoid of a ring.
Communications in Algebra,
45(4):1601-1617, 2017.
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arXiv
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2016C. Câmpeanu, N. Moreira, R. Reis,
Distinguishability operations and closures.
Fundamenta Informaticae,
148(3-4):243-266, 24 December 2016.
IOS Press
University of Porto
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arXiv
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2016J. Jirásek and J. Šebej,
Kuratowski algebras generated by prefix‑free languages.
In Y. S. Han and K. Salomaa, editors, Implementation and Application of Automata, 21st International Conference, CIAA 2016, Seoul, South Korea, July 19-22, 2016, Proceedings,
volume 9705 of Lecture Notes in Computer Science (LNICS),
pp. 150-162, 2016.
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2015J. Jirásek and G. Jirásková,
On the boundary of regular languages.
Theoretical Computer Science,
578:42-57, 3 May 2015.
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St. Mary’s University
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2014R. D. Sarma and A. Gupta,
On the k‑numbers of some generalized Kuratowski operators in topology.
In 3rd International Conference On Innovative Approach in Applied Physical, Mathematical/Statistical, Chemical Sciences and Emerging Energy Technology for Sustainable Development, APMSCSET 2014, New Delhi, India, September 27-28, 2014, Proceedings,
pp. 50-54, 2014.
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2014E. A. Ok and G. Riella,
Topological closure of translation invariant preorders.
Mathematics of Operations Research,
39(3):737-745, August 2014.
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2014A. Grabowski,
Topological interpretation of rough sets.
Formalized Mathematics,
22(1):89-97, March 2014.
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2014J. Brzozowski, G. Jirásková, C. Zou,
Quotient complexity of closed languages.
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54:277-292, February 2014.
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2013É. Charlier, M. Domaratzki, T. Harju, J. Shallit,
Composition and orbits of language operations: finiteness and upper bounds.
International Journal of Computer Mathematics,
90(6):1171-1196, 2013.
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2013S. Plewik and M. Walczyńska,
The monoid consisting of Kuratowski operations.
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2013(289854), 9 pp., 6 March 2013.
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2013M. Bowron (proposer), B. S. Burdick (solver),
Problem 1898: More on Kuratowski 14-sets.
Mathematics Magazine,
85(3):228, March 2012 (proposal), 86(3):231, March 2013 (solution).
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2013A. Aguado (proposer), M. Bowron (solver),
Problem 1888: Closures and complements.
Mathematics Magazine,
85(1):62 January 2012 (proposal), 86(1):67–68, February 2013 (solution).
Taylor & Francis
JSTOR
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2012G. Jirásková and J. Shallit,
The state complexity of star-complement-star.
In H. C. Yen, O. H. Ibarra (editors), Developments in Language Theory (DLT 2012), Taipei, Taiwan, August 14-17, 2012, volume 7410 of Lecture Notes in Computer Science (LNICS), Springer, Berlin, Heidelberg,
pp. 380-391, 2012.
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arXiv
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2012O. Al-Hassani, Q. Mahesar, C. Sacerdoti Coen, V. Sorge,
A term rewriting system for Kuratowski’s closure-complement problem.
In 23rd International Conference on Rewriting Techniques and Applications (RTA ’12), Nagoya, Japan, May 28 to June 2, 2012, Leibniz International Proceedings in Informatics (LIPIcs),
15:38-52, 2012.
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ARW 2012
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UBIRA
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2012S. Cato,
Complements and consistent closures.
Discrete Mathematics,
312(6):1218-1221, 28 March 2012.
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2011Š. Holub and J. Kortelainen,
On partitions separating words.
International Journal of Algebra and Computation,
21(8):1305-1316, December 2011.
World Scientific
Charles University
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2011J. Shallit and R. Willard,
Kuratowski’s theorem for two closure operators.
6 pp., 7 September 2011.
arXiv
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2011J. Brzozowski, E. Grant, J. Shallit,
Closures in formal languages and Kuratowski’s theorem.
International Journal of Foundations of Computer Science,
22(2):301-321, 2011.
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arXiv
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2011M. Kudlek,
A relation between modal logic and language closure operators.
Fundamenta Informaticae,
109(3):297-304, 2011.
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2010D. Sherman,
Variations on Kuratowski’s 14‑set theorem.
American Mathematical Monthly,
117(2):113-123, February 2010.
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arXiv
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2008B. J. Gardner and M. Jackson,
The Kuratowski closure-complement theorem.
New Zealand Journal of Mathematics,
38:9-44, 2008.
University of Tasmania
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2007A. E. McCluskey, D. W. McIntyre, W. S. Watson,
The propositional theory of closure.
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16(4):501-512, 2007.
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2006A. Grabowski and M. Jastrzębska,
Enhancing MML via Isomichi's classification of domains.
Mechanized Mathematics and Its Applications,
5(2):11-18, December 2006.
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2006M. Bowron (proposer), B. S. Burdick (solver),
Problem 11059: Closure, complement, and union.
American Mathematical Monthly,
111(1):64-65, January 2004 (proposal), 113(1):83, January 2006 (solution).
Taylor & Francis
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ResearchGate
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2006A. Grabowski,
Solving two problems in general topology via types.
In J. C. Filliâtre, C. Paulin-Mohring, and B. Werner, editors,
Types for Proofs and Programs International Workshop
(TYPES 2004), Jouy-en-Josas, France, December 15-18, 2004,
Revised Selected Papers, volume 3839 of
Lecture Notes in Computer Science (LNICS),
pp. 138-153, 2006.
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2005P. Muhm,
Kuratowski’s 14‑set theorem — a modal logic view.
PhD thesis, University of Houston, 58 pp., December 2005.
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2005A. Grabowski,
On the computer-checked solution of the Kuratowski closure-complement problem.
Mechanized Mathematics and Its Applications,
4(1):25-34, March 2005.
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2004M. Jackson,
Semilattices with closure.
Algebra Universalis,
52(1):1-37, November 2004.
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2004F. D. Farmer,
Containment for c, s, t operators on a binary relation.
Algebra Universalis,
51:153-175, August 2004.
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2004M. Jackson and T. Stokes,
Semilattice pseudo-complements on semigroups.
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32(8):2895-2918, 2004.
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2003L. K. Bagińska and A. Grabowski,
On the Kuratowski closure-complement problem.
Journal of Formalized Mathematics,
11, 6 pp., 2003.
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2002J. A. Pomykala,
Similarity in relational databases and in information systems theory.
Demonstratio Mathematica,
35(4):927-953, 1 October 2002.
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University of Amsterdam
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2002G. Voutsadakis,
On some operations on classes of algebras and coalgebras from a bialgebraic viewpoint.
17 pp., 2002.
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2001B. Tasić,
On the partially ordered monoid generated by the operators H, S, P, Ps on classes of algebras.
Journal of Algebra,
245(1):1-19, 1 November 2001.
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2001D. Mašulović and B. Tasić,
Operators on classes of coalgebras.
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2000W. He and Y. Zhang,
Interior and boundary in a locale.
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1999B. J. Gardner and T. Stokes,
Closure rings.
Commentationes Mathematicae Universitatis Carolinae,
40(3):413-427, 1999.
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1998M. Bowron and S. Rabinowitz (proposers), J. Rickard (solver),
Problem 10577: Closure, complement, and arbitrary union.
American Mathematical Monthly,
104(2):169, February 1997 (proposal), 105(3):282-283, March 1998 (solution).
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1997E. Q. F. Rezende, R. A. Bacci, R. C. Bassanezi,
Semi-group generated by K-limits.
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16(1):67-82, 1997.
Catholic University of the North
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1996K. P. Shum,
Closure functions on the set of positive integers.
Science in China Series A: Mathematics,
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1995E. Garel and J. P. Olivier,
On closures verifying that the interior of a closed element is closed.
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1995E. Garel and J. P. Olivier,
The opoid generated by transitive closure and interior and symmetric closure and interior: a characterization using generators and relations.
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1995M. S. Moslehian and N. Tavallaii,
A generalization of the Kuratowski closure-complement problem.
Punjab University Journal of Mathematics,
28:1-9, 1995.
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1993E. Garel and J. P. Olivier,
About the finiteness of monoids generated by closures and interiors.
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1993K. P. Shum,
The amalgamation of closure and boundary functions on semigroups and partially ordered sets.
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Conference on Ordered Structures and Algebra of Computer Languages: Hong Kong,
26-29 June 1991, pp. 232–243, 1993.
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1992S. M. Kam and K. P. Shum,
On a problem of P. C. Hammer.
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1991J. H. Fife,
The Kuratowski closure-complement problem.
Mathematics Magazine,
64(3):180-182, June 1991.
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1988E. Buchman (proposer), J. Ferrer (solver),
Problem E3144: Generating sets in a topological space.
American Mathematical Monthly,
93(4):299, April 1986 (proposal), 95(4):353, April 1988 (solution).
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1988C. E. Aull,
Classification of topological spaces II.
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1986S. Wroński,
On the operations definable in terms of the complementation and the closure.
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1984D. Peleg,
A generalized closure and complement phenomenon.
Discrete Mathematics,
50:285-293, 1984.
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1984K. P. Shum,
On the boundary of Kuratowski 14‑sets in connected spaces.
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1982A. Capelo and L. Sanchez,
On certain operators expressible in a topological space (in Portuguese).
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1982A. V. Chagrov,
Kuratowski numbers (in Russian).
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1982V. P. Soltan,
Problems of Kuratowski type (in Russian).
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1980J. Anderson,
Closure operators on languages and binary relations.
Semigroup Forum,
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1980E. Langford (proposer), W. Myers (solver),
Problem 6260: Sets formed by iterated closure, interior, and union.
American Mathematical Monthly,
86(3):226-226, March 1979 (proposal), 87(8):680-681, October 1980 (solution).
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1980V. P. Soltan,
On Kuratowski’s problem (in Russian).
Bulletin of the Polish Academy of Sciences, Mathematics,
28:369-375, 1980.
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1980M. J. Fischer and M. S. Paterson,
The fast skew-closure algorithm.
L'Enseignement Mathématique,
26(3):345-360, 1980.
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1979V. P. Soltan,
The Kuratowski number of a closure space (in Russian).
IV Tiraspol Symposium on General Topology and its Applications,
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1978A. Smith (proposer), C. Y. Yu (solver),
Problem 5996: Kuratowski sets.
American Mathematical Monthly,
81(9):1034, November 1974 (proposal), 85(4):283-284, April 1978 (solution).
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1978P. C. Fishburn,
Operations on binary relations.
Discrete Mathematics,
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1977W. J. Blok,
The free closure algebra on finitely many generators.
Indagationes Mathematicae (Proceedings),
80:362-379, 1977.
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1977I. Z. Kleiner,
Closure and boundary operators in topological spaces.
Ukrainian Mathematical Journal (English edition),
29(3):295-296, 1977.
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1977L. E. Moser,
Closure, interior, and union in finite topological spaces.
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1976J. C. Oxtoby,
The kernel operation on subsets of a T1-space.
Fundamenta Mathematicae,
90(3):275-284, 1976.
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1975J. Berman and S. L. Jordan,
The Kuratowski closure-complement problem.
American Mathematical Monthly,
82(8):841-842, October 1975.
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1975E. Langford,
Problems of Kuratowski type involving unions and intersections.
Notices of the American Mathematical Society,
22(4):A-485, June 1975.
AMS
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1975K. W. Yip and K. P. Shum,
On the structure of Kuratowski sets.
Journal of the Chinese University of Hong Kong,
3:429-439, 1975.
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1974Y. R. Gaida and A. É. Eremenko,
On the frontier operator in Boolean algebras with a closure.
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1974A. A. Gol'dberg,
On Boolean algebras with a derivation.
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1972R. L. Graham, D. E. Knuth, T. S. Motzkin,
Complements and transitive closures.
Discrete Mathematics,
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1972S. Comer and J. Johnson,
The standard semigroup of operators of a variety.
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1972D. Pigozzi,
On some operations on classes of algebras.
Algebra Universalis,
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1971K. P. Shum (proposer and solver),
Problem 153: Bounds on the Kuratowski family.
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1971S. Baron (proposer), E. Langford (solver),
Problem 5569: Kuratowski’s 14‑sets.
American Mathematical Monthly,
75(2):199, February 1968 (proposal), 78(4):411, April 1971 (solution).
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1971E. Langford,
Characterization of Kuratowski 14‑sets.
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1971J. Anusiak and K. P. Shum,
Remarks on finite topological spaces.
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1971Y. Isomichi,
New concepts in the theory of topological space — supercondensed set, subcondensed set, and condensed set.
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38(3):657-668, 1971.
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1968W. Koenen (proposer), L. F. Meyers (solver),
Problem E1941: On two closure problems.
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73(10):1123, December 1966 (proposal), 75(4):408-409, April 1968 (solution).
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1968D. H. Staley,
On the commutativity of the boundary and interior operators in a topological space.
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1967C. E. Aull,
Classification of topological spaces.
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15(11):773-778, 1967 (errata ibid., 16(6), 1968).
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1967P. J. Coleman,
On the composition semigroup generated by complement and an expansive function.
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1967E. Nelson,
Finiteness of semigroups of operators in universal algebra.
Canadian Journal of Mathematics,
19:764-768, 1967.
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1966S. P. Lloyd (proposer), J. C. Morgan II (solver),
Problem 5349: A Kuratowski closure and complement problem.
American Mathematical Monthly,
72(10):1136, December 1965 (proposal), 73(10):1132-1134, December 1966 (solution).
Taylor & Francis
JSTOR
MR
1966W. Koenen,
The Kuratowski closure problem in the topology of convexity.
American Mathematical Monthly,
73(7):704-708, August 1966.
Taylor & Francis
JSTOR
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Semantic
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1966H. H. Herda and R. C. Metzler,
Closure and interior in finite topological spaces.
Colloquium Mathematicum,
15(2):211-216, 1966.
Library of Science
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1965J. Hintikka,
A closure and complement result for nested topologies.
Fundamenta Mathematicae,
57(1):97-106, 1965.
Library of Science
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1964T. Tamura,
Operations on binary relations and their applications.
Bulletin of the American Mathematical Society,
70:113-120, 1964.
AMS
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1962T. A. Chapman,
A further note on closure and interior operators.
American Mathematical Monthly,
69(6):524-529, June 1962.
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1962T. A. Chapman,
An extension of the Kuratowski closure and complementation problem.
Mathematics Magazine,
35(1):31-35, January 1962.
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1961N. Levine,
On the commutativity of the closure and interior operators in topological spaces.
American Mathematical Monthly,
68(5):474-477, May 1961.
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1961H. Choda and K. Matoba,
On a theorem of Levine.
Proceedings of the Japan Academy, Series A (Mathematical Sciences),
37(8):462-463, 1961.
J-Stage
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1960P. C. Hammer,
Kuratowski’s closure theorem.
Nieuw Archief voor Wiskunde,
8(2):74-80, 1960.
Worldcat
MR
ZB
1957L. Rieger,
A note on the so-called free closure algebras (in Russian).
Czechoslovak Mathematical Journal,
7(1):16-20, 1957.
DML-CZ
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MR
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1955M. Zarycki,
Boolean algebras with a closure and Boolean algebras with a derivation (in Ukrainian).
Dopovidi Akademiї Nauk Ukrains'koї RSR (Reports of the Academy of Sciences of the Ukrainian SSR),
1:3-6, 1955.
Worldcat
1954E. Halfar,
The isolated points of a set.
Portugaliae Mathematica,
13(3):125-128, 1954.
National Library of Portugal
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1951E. Halfar,
A note on point-set operators.
Portugaliae Mathematica,
10(2):103-104, 1951.
National Library of Portugal
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1951T. Ogasawara and J. Funakoshi,
On topological operations determined by local characters.
Journal of Science of the Hiroshima University Series A-1 Mathematics,
15:103-112, 1951.
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1949M. Zarycki,
On an operation in the theory of point sets (in Ukrainian).
Naukovi Zapysky: Seriia Fizyko-Matematychna (Scientific Notes: Series Physics and Mathematics),
12(3):35-43, 1949.
Worldcat
1948S. Matsushita,
The algebra of topological operations, I.
Mathematica Japonicae,
1(1):28-35, 1948.
Worldcat
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1947M. Zarycki,
Some properties of the derived set operation in abstract spaces (in Ukrainian).
Naukovi Zapysky: Seriia Fizyko-Matematychna (Scientific Notes: Series Physics and Mathematics),
5(1):22-33, 1947.
Scholar
English translation by M. Bowron
1944J. C. C. McKinsey, A. Tarski,
The algebra of topology.
Annals of Mathematics,
45(1):141-191, January 1944.
Cambridge
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1943E. Hewitt,
A problem of set-theoretic topology.
Duke Mathematical Journal,
10(2):309-333, 1943.
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1941J. C. C. McKinsey,
A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology.
Journal of Symbolic Logic,
6(4):117-134, 1941.
Cambridge
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1941E. W. Chittenden,
On the reduction of topological functions.
In R. L. Wilder and W. L. Ayres, editors, Lectures in Topology:
The University of Michigan Conference of 1940,
pp. 267–285, 1941.
Worldcat
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1941E. W. Chittenden,
A new proof of a formula of Kuratowski.
Proceedings of the Iowa Academy of Sciences (now Journal of the Iowa Academy of Science),
48(1):299-300, 1941.
UNI ScholarWorks
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1941F. Haight,
On the independence of operators on a lattice.
Proceedings of the Iowa Academy of Sciences (now Journal of the Iowa Academy of Science),
48(1):301-304, 1941.
UNI ScholarWorks
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1940H. Terasaka,
The theory of topological lattices (in German).
Fundamenta Mathematicae,
33, 1939.
Reprinted in Collected Papers From the Faculty of Science Osaka University Series A Mathematics,
8(1):1-33, 1940.
Worldcat
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1939E. C. Stopher Jr.,
Point set operators and their interrelations.
Bulletin of the American Mathematical Society,
45(10):758-762, October 1939.
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1937E. C. Stopher Jr.,
Cyclic relations in point set theory.
Bulletin of the American Mathematical Society,
43(10):686-694, October 1937.
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1937H. Terasaka,
The theory of topological lattices: an attempt to formalize general topology and the theory of real functions (in German).
Proceedings of the Imperial Academy, Series A, Mathematical Sciences,
13(10):401-405, 1937.
Project Euclid
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1936S. T. Sanders Jr.,
Derived sets and their complements.
Bulletin of the American Mathematical Society,
42(8):577-584, August 1936.
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1930M. Zarycki,
On the kernel of a set (in German).
Jahresbericht der Deutschen Mathematiker-Vereinigung (Annual Report of the German Mathematical Society),
39:154-158, 1930.
SUB
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1928M. Zarycki,
General properties of Cantorian coherences (in German).
Transactions of the American Mathematical Society,
30(3):498-506, July 1928.
JSTOR
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1928M. Zarycki,
Derivation and coherence in abstract sets (in Ukrainian).
Zbirnik Matematichno-Prirodopisno-Likarskoi Sektsii Naukovogo Tovaristva (Proceedings of the Mathematical-Natural History-Medical Section of the Scientific Society),
27:247-259, 1928.
TNTU
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1927M. Zarycki,
Quelques notions fondamentales de l'analysis situs au point du vue de l'algèbre de la logique (Some fundamental concepts of topology in terms of the algebra of logic).
Fundamenta Mathematicae,
9:3-15, 1927.
Library of Science
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1922C. Kuratowski,
Sur l'opération Ā de l'analysis situs (On the topological closure operation).
Fundamenta Mathematicae,
3:182-199, 1922.
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English translation by M. Bowron
< 1920?Did anyone alternately add limit points
and take complements before Kuratowski?
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