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Roughness in Residuated Lattices

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Advances on Computational Intelligence (IPMU 2012)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 297))

Abstract

Commutative bounded integral residuated lattices (= residuated lattices) form a large class of algebras containing among others several classes of algebras of fuzzy logics which are related to reasoning under uncertainty. The paper investigates approximation spaces in residuated lattices based on their filters.

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References

  1. Biswas, R., Nanda, S.: Rough groups and rough subgroups. Bulletin of the Polish Academy of Sciences Mathematics 42, 251–254 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Cattaneo, G., Ciucci, D.: Algebraic Structures for Rough Sets. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 208–252. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. Ciucci, D.: On the axioms of residuated structures independence dependencies and rough approximations. Fundam. Inform. 69, 359–387 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Ciucci, D.: A Unifying Abstract Approach for Rough Models. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 371–378. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  5. Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundation of Many-valued Reasoning. Kluwer Acad. Publ., Dordrecht (2000)

    Google Scholar 

  6. Ciungu, L.C.: Bosbach and Riečan states on residuated lattices. J. Appl. Funct. Anal. 3, 175–188 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Davvaz, B.: Roughness in rings. Inf. Sci. 164, 147–163 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Davvaz, B.: Roughness based on fuzzy ideals. Inf. Sci. 176, 2417–2437 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dvurečenskij, A., Rachůnek, J.: Probabilistic averaging in bounded commutative residuated ℓ-monoids. Discret. Math. 306, 1317–1326 (2006)

    Article  MATH  Google Scholar 

  10. Estaji, A.A., Khodaii, S., Bahrami, S.: On rough set and fuzzy sublattice. Inf. Sci., doi:10.1016/j.ins.2011.04.043

    Google Scholar 

  11. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124, 271–288 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices: An Algebraic Glimpse at Substructural Logics. Elsevier Sudies in Logic and Foundations. Elsevier, New York (2007)

    MATH  Google Scholar 

  13. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Amsterdam (1998)

    Book  MATH  Google Scholar 

  14. Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer Acad. Publ., Dordrecht (2002)

    Google Scholar 

  15. Kondo, M.: Algebraic Approach to Generalized Rough Sets. In: Ślęzak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y., et al. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 132–140. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  16. Kuroki, N.: Rough ideals in semigroups. Inf. Sci. 100, 139–163 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leoreanu-Fotea, V., Davvaz, B.: Roughness in n-ary hypergroups. Inf. Sci. 178, 4114–4124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, T.J., Leung, Y., Zhang, W.X.: Generalized fuzzy rough approximation operators based on fuzzy coverings. Int. J. Approx. Reasoning 48, 836–856 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, X., Liu, S.: Matroidal approaches to rough sets via closure operators. Int. J. Approx. Reasoning 53, 513–527 (2012)

    Article  Google Scholar 

  20. Liu, G.L., Zhu, W.: The algebraic structures of generalized rough set theory. Inf. Sci. 178, 4105–4113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mundici, D.: Interpretation of AF C *-algebras in sentential calculus. J. Funct. Analys. 65, 15–63 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pawlak, Z.: Rough sets. Int. J. Inf. Comput. Sci. 11, 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. System Theory, Knowledge Engineering and Problem Solving, vol. 9. Kluwer Academic Publishers, Dordrecht (1991)

    Google Scholar 

  24. Pawlak, Z., Skowron, A.: Rudiments of rough sets. Inf. Sci. 177, 3–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pawlak, Z., Skowron, A.: Rough sets: some extensions. Inf. Sci. 177, 28–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pawlak, Z., Skowron, A.: Rough sets and Boolean reasoning. Inf. Sci. 177, 41–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pei, D.: On definable concepts of rough set models. Inf. Sci. 177, 4230–4239 (2007)

    Article  MATH  Google Scholar 

  28. Radzikowska, A.M., Kerre, E.E.: Fuzzy Rough Sets Based on Residuated Lattices. In: Peters, J.F., Skowron, A., Dubois, D., Grzymała-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 278–296. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  29. Rasouli, S., Davvaz, B.: Roughness in MV-algebras. Inf. Sci. 180, 737–747 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. She, Y.H., Wang, G.J.: An approximatic approach of fuzzy rough sets based on residuated lattices. Comput. Math. Appl. 58, 189–201 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xiao, Q.M., Zhang, Z.L.: Rough prime ideals and rough fuzzy prime ideals in semigroups. Inf. Sci. 176, 725–733 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, L., Xu, L.: Algebraic aspects of generalized approximation spaces. Int. J. Approx. Reasoning 51, 151–161 (2009)

    Article  MATH  Google Scholar 

  33. Zhu, P.: Covering rough sets based on neighborhoods: An approach without using neighborhoods. Int. J. Approx. Reasoning 52, 461–472 (2011)

    Article  MATH  Google Scholar 

  34. Zhu, W.: Relationship between generalized rough set based on binary relation and covering. Inf. Sci. 179, 210–225 (2009)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Sciences, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Jiří Rachůnek

  2. Department of Mathematical Methods in Economy, Faculty of Economics, VŠB–Technical University Ostrava, Sokolská 33, 701 21, Ostrava, Czech Republic

    Dana Šalounová

Authors
  1. Jiří Rachůnek
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  2. Dana Šalounová
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Editor information

Editors and Affiliations

  1. Faculty of Economics, University of Catania, Corso Italia, 55, 95129, Catania, Italy

    Salvatore Greco  & Benedetto Matarazzo  & 

  2. Université Pierre et Marie Curie - Paris6, CNRS UMR 7606, DAPA, LIP6 8, rue du Capitaine Scott, F-75015, Paris, France

    Bernadette Bouchon-Meunier

  3. Dip. Matematica e Informatica, Università di Perugia, 06123, Perugia, Italy

    Giulianella Coletti

  4. Department of Computer and Management Science, University of Trento, Via Inama 5, 38122, Trento, Italy

    Mario Fedrizzi

  5. Machine Intelligence Institute, IONA College, 10801, New Rochelle, NY, USA

    Ronald R. Yager

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Rachůnek, J., Šalounová, D. (2012). Roughness in Residuated Lattices. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances on Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 297. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31709-5_60

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  • DOI: https://doi.org/10.1007/978-3-642-31709-5_60

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31708-8

  • Online ISBN: 978-3-642-31709-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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