Three Lessons on the Topological and Algebraic Hidden Core of Rough Set Theory

  • Piero Pagliani
Part of the Trends in Mathematics book series (TM)


In what follows the reader will find an exposition of the basic, albeit not elementary, connections between Rough Set Theory and relation algebra, topology and algebraic logic.

Many algebraic aspects of Rough Set Theory, are known nowadays. Other are less known, although they are important, for instance because they unveil the “epistemological meaning” of some “unexplained” mathematical features of well-known algebraic structures.

We shall wrap everything in a simple exposition, illustrated by many examples, where just a few basic notions are required. Some new results will help the connection of the topics taken into account.

Important features in Rough Set Theory will be explained by means of notions connecting relation algebra, pre-topological and topological spaces, formal (pre) topological systems, algebraic logic and logic.

Relation algebra provides basic tools for the definition of approximations in general (that is, not confined to particular kind of relations). Indeed, these tools lead to pairs of operators fulfilling Galois adjointness, whose combinations, in turn, provide pre-topological and topological operators, which, in some cases, turn into approximation operators.

Once one has approximation operators, rough sets can be defined. In turn, rough set systems can be made into different logico-algebraic systems, such as Nelson algebras, three-valued Łukasiewicz algebras, Post algebras of order three, Heyting and co-Heyting algebras.

In addition, in the process of approximation, one has to deal with both exact and inexact pieces of information (definable and non-definable sets). Therefore, the concept of local validity comes into picture. It will be extensively discussed because it links the construction of Nelson algebras from Heyting ones with the notions of a Grothendieck topology and a Lawvere-Tierney operator.

As a side effect, we obtain an information-oriented explanation of the above logico-algebraic constructions which usually are given on the basis of pure formal motivations.

The exposition will move from abstract levels (pointless) to concrete levels of analysis.


  1. 1.
    Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1975)zbMATHGoogle Scholar
  2. 2.
    Banerjee, M.: Algebras from Rough Sets. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough Neural Computing, pp.157–184. Springer, Berlin (2004)CrossRefGoogle Scholar
  3. 3.
    Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fund. Inform. 28(3–4), 211–221 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Banerjee, M., Chakraborty, M.K.: Rough sets and three-valued Łukasiewicz logic. Fund. Inform. 32, 213–220 (1997)zbMATHGoogle Scholar
  5. 5.
    Bartol, W., Miró, J., Pióro, K., Rosselló, F.: On the coverings by tolerance classes. Inf. Sci. 166, 193–211 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cattaneo, G.: Abstract approximation spaces for rough theories. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 2. Applications, Case Studies and Software Systems, pp. 59–106. Springer, Berlin (1998)Google Scholar
  7. 7.
    Cattaneo, G., Ciucci, D.: Heyting Wajsberg algebras as an abstract environment linking fuzzy and rough sets. In: Alpigini, J., Peters, J., Skowron, A., Zhong, N. (eds.) Rough Sets and Current Trends in Computing: Third International Conference, RSCTC 2002. Lecture Notes in Artificial Intelligence, vol. 2475, pp. 77–84. Springer, Berlin (2002)CrossRefGoogle Scholar
  8. 8.
    Comer, S.: An algebraic approach to the approximation of information. Fund. Inform. 14, 492–502 (1991)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Düntsch, I.: Rough sets and algebras of relations. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 95–108. Physica-Verlag, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Düntsch, I., Gegida, G.: Modal-style operators in qualitative data analysis. In: Proceedings of the 2002 IEEE International Conference on Data Mining, pp. 155–162 (2002)Google Scholar
  11. 11.
    Düntsch, I., Orłowska, E.: Mixing modal and sufficiency operators. Bull. Sect. Logic 28, 99–106 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Epstein, G.: The underlying ground for hypothetical propositions. In: Logic in the Twentieth Century, Scientia, pp. 101–124 (1983)Google Scholar
  13. 13.
    Epstein, G., Horn, A.: Chain based lattices. Pac. J. Math 55(1), 65–85 (1974)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Epstein, G., Horn, A.: Logics characterized by subresiduated lattices. Math. Log. Q. 22(1), 199–210 (1976)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fourman, M.P., Scott, D.S.: Sheaves and logic. In: Fourman, M.P., Mulvey, C.J., Scott, D.S. (eds.) Applications of Sheaves, Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Lecture Notes in Computer Science, vol. 753, pp. 302–401. Springer, Berlin (1977)Google Scholar
  16. 16.
    Greco, S., Matarazzo, B., Słowiński, R.: Dominance-based rough set approach and bipolar abstract rough approximation spaces. In: Chan, C., Grzymala-Busse, J.W., Ziarko, W.P. (eds.) Rough Sets and Current Trends in Computing. RSCTC 2008. Lecture Notes in Computer Science, vol. 5306, pp. 31–40. Springer, Berlin (2008)Google Scholar
  17. 17.
    Järvinen, J.: Knowledge representation and rough sets. Ph.D. Dissertation, University of Turku (1999)Google Scholar
  18. 18.
    Järvinen, J., Radeleczki, S.: Tolerances induced by irredundant coverings. Fund. Inform. 137, 341–353 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Järvinen, J., Pagliani, P., Radeleczki, S.: Information completeness in Nelson algebras of rough sets induced by quasiorders. Stud. Logica 101(5), 1073–1092 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Johnston, P.T.: Conditions related to De Morgan’s law. In: Fourman, M.P., Mulvey, C.J., Scott, D.S. (eds.) Applications of Sheaves, Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis. Lecture Notes in Computer Science, vol. 753, pp. 579–491. Springer, Berlin (1977)Google Scholar
  21. 21.
    Lawvere, F.W.: Introduction to part I. In: Lawvere, F.W., Maurer, C., Wraith, C.G. (eds.) Model Theory and Topoi, pp. 3–14. Springer, Berlin (1975)CrossRefGoogle Scholar
  22. 22.
    Lawvere, F.W.: Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes. In: Carboni, A., Pedicchio, M.C., Rosolini, G. (eds.) Category Theory, Lecture Notes in Computer Science, vol. 1488, pp. 279–281. Springer, Berlin (1991)Google Scholar
  23. 23.
    MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Springer, Berlin (1992)Google Scholar
  24. 24.
    Miglioli, P., Moscato, U., Ornaghi, M., Usberti, G.: A constructivism based on classical truth. Notre Dame J. Formal Logic 30(1), 67–90 (1988)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Obtułowicz, A.: Rough sets and Heyting algebra valued sets. Bull. Polish Acad. Sci. Math. 35(9–10), 667–671 (1987)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ore, O.: Galois connexions. Trans. Am. Math. Soc. 55, 493–531 (1944)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pagliani, P.: Remarks on special lattices and related constructive logics with strong negation. Notre Dame J. Formal Logic 41(4), 515–528 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pagliani, P.: A pure logic-algebraic analysis of rough top and rough bottom equalities. In: Ziarko, W.P. (ed.) Rough Sets, Fuzzy Sets and Knowledge Discovery. Workshops in Computing, pp. 227–236. Springer, Berlin (1994)CrossRefGoogle Scholar
  29. 29.
    Pagliani, P.: A modal relation algebra for generalized approximation spaces. In: Tsumoto, S., Kobayashi, S., Yokomori, T., Tanaka, H., Nakamura, A. (eds.) Proceedings of the 4th International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery, pp. 89–96. The University of Tokyo, Tokyo (1996)Google Scholar
  30. 30.
    Pagliani, P.: Algebraic models and proof analysis: a simple case study. In: Orłowska, E. (ed.) Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 651–668. Physica-Verlag, Heidelberg (1998)Google Scholar
  31. 31.
    Pagliani, P.: Intrinsic co-Heyting boundaries and information incompleteness in Rough Set Analysis. In: Polkowski, L., Skowron, A. (eds.) Rough Sets and Current Trends in Computing. RSCTC 1998. Lecture Notes in Computer Science 13, vol. 1424, pp. 123–130. Springer, Berlin (1998)Google Scholar
  32. 32.
    Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 109–190. Physica-Verlag, Heidelberg (1998)CrossRefGoogle Scholar
  33. 33.
    Pagliani, P.: Concrete neighbourhood systems and formal pretopological spaces. Calcutta Logical Circle Conference on Logic and Artificial Intelligence, Calcutta, India, 13–16 October 2002. Now Ch. 15.14 of [39] 41(4), 515–528 (2002)Google Scholar
  34. 34.
    Pagliani, P.: Pretopology and dynamic spaces. Fund. Inform. 59(2–3), 221–239 (2004)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Pagliani, P.: The relational construction of conceptual patterns. Tools, implementation and theory. In: Kryszkiewicz, M., Cornelis, C., Ciucci, D., Medina-Moreno, J., Motoda, H., Ras, Z. (eds.) Rough Sets and Intelligent Systems Paradigms. Lecture Notes in Computer Science, vol. 8537, pp. 14–27. Springer, Berlin (2014)CrossRefGoogle Scholar
  36. 36.
    Pagliani, P.: Covering rough sets and formal topology. A uniform approach through intensional and extensional constructors. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets. Lecture Notes in Computer Science, vol. 10020, pp. 109–145. Springer (2016)Google Scholar
  37. 37.
    Pagliani, P., Chakraborty, M.K.: Information quanta and approximation spaces. I: non-classical approximation operators. In: Peters, J., Skowron, A. (eds.) Proceedings of the IEEE International Conference on Granular Computing, Beijing, P. R. China, 25–27 July 2005, vol. 2, pp. 605–610. IEEE, Los Alamitos (2005)CrossRefGoogle Scholar
  38. 38.
    Pagliani, P., Chakraborty, M.K.: Formal topology and information systems. In: Skowron, A., Peters, J.F., Düntsch, I., Grzymala-Busse, J., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets. Lecture Notes in Computer Science VI, vol. 4374, pp. 253–297. Springer, Berlin (2007)CrossRefGoogle Scholar
  39. 39.
    Pagliani, P., Chakraborty, M.K.: A Geometry of Approximation. Springer, Berlin (2008)CrossRefGoogle Scholar
  40. 40.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic, Dordrecht (1991)CrossRefGoogle Scholar
  41. 41.
    Pomikała, J., Pomikała, J.A.: The Stone algebra of rough sets. Bull. Polish Acad. Sci. Math. 36, 495–508 (1988)MathSciNetGoogle Scholar
  42. 42.
    Rasiowa, H.: An Algebraic Approach to Non-classical Logics. Kluwer Academic, Dordrecht (1974)zbMATHGoogle Scholar
  43. 43.
    Reyes, G.E., Zolfaghari, H.: Bi-Heyting algebras, toposes and modalities. J. Philos. Log. 25(1), 25–43 (1996)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Sambin, G.: Intuitionistic formal spaces - a first communication. In: Skordev, D. (ed.) Mathematical Logic and Its Applications, pp. 187–204. Plenum Press, New York (1987)CrossRefGoogle Scholar
  45. 45.
    Sambin, G., Gebellato, S.: A preview of the basic picture: a new perspective on formal topology. In: Altenkirch, T., Reus, B., Naraschewski, W. (eds.) Proceedings of International Workshop on Types for Proofs and Programs, TYPES ’98. Lecture Notes in Computer Science, vol. 1657, pp. 194–207. Springer, Berlin (1999)zbMATHGoogle Scholar
  46. 46.
    Sen, J., Chakraborty, M.K.: Algebraic structures in the vicinity of pre-rough algebra and their logics. Inf. Sci. 282(20), 296–320 (2014)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Sendlewski, A.: Nelson algebras through Heyting ones: I. Stud. Logica 49(1), 105–126 (1990)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Thomason, R.H.: A semantical study of constructible falsity. Z. Math. Logik Grundlagen Math. 15, 247–257 (1969)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wille, R.: Restructuring lattice theory. In: Rival, I. (ed.) Ordered Sets. NATO ASI Series, vol. 83, pp. 445–470. Reidel, Dordrecht (1982)Google Scholar

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Authors and Affiliations

  • Piero Pagliani
    • 1
  1. 1.International Rough Set SocietyRomeItaly

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