Towards Multi-adjoint Property-Oriented Concept Lattices

  • Jesús Medina
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6401)


In this paper we present some properties related to adjoint triples when we consider dual supports. These results are used in order to generalize the classical property oriented concept lattices, which itself embeds rough set theory. Specifically, we define a fuzzy environment based on the philosophy of the multi-adjoint paradigm, which is related to the multi-adjoint concept lattice. As a consequence, we can move the properties from one to another.


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  1. 1.
    Abdel-Hamid, A., Morsi, N.: Associatively tied implicacions. Fuzzy Sets and Systems 136(3), 291–311 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bělohlávek, R.: Fuzzy concepts and conceptual structures: induced similarities. In: Joint Conference on Information Sciences, pp. 179–182 (1998)Google Scholar
  3. 3.
    Bělohlávek, R.: Concept lattices and order in fuzzy logic. Annals of Pure and Applied Logic 128, 277–298 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Burusco, A., Fuentes-González, R.: The study of L-fuzzy concept lattice. Mathware & Soft Computing 3, 209–218 (1994)Google Scholar
  5. 5.
    Chen, X., Li, Q.: Construction of rough approximations in fuzzy setting. Fuzzy Sets and Systems 158(23), 2641–2653 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, Y., Yao, Y.: A multiview approach for intelligent data analysis based on data operators. Information Sciences 178(1), 1–20 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Davey, B., Priestley, H.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  8. 8.
    Dubois, D., de Saint-Cyr, F.D., Prade, H.: A possibility-theoretic view of formal concept analysis. Fundamenta Informaticae 75(1-4), 195–213 (2007)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dubois, D., Prade, H.: Putting fuzzy sets and rough sets together. In: Slowiński, R. (ed.) Intelligent Decision Support, pp. 203–232. Kluwer Academic, Dordrecht (2004)Google Scholar
  10. 10.
    Düntsch, I., Gediga, G.: Approximation operators in qualitative data analysis. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) Theory and Applications of Relational Structures as Knowledge Instruments. LNCS, vol. 2929, pp. 214–230. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Gediga, G., Düntsch, I.: Modal-style operators in qualitative data analysis. In: Proc. IEEE Int. Conf. on Data Mining, pp. 155–162 (2002)Google Scholar
  12. 12.
    Georgescu, G., Popescu, A.: Non-dual fuzzy connections. Arch. Math. Log. 43(8), 1009–1039 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Julian, P., Moreno, G., Penabad, J.: On fuzzy unfolding: A multi-adjoint approach. Fuzzy Sets and Systems 154(1), 16–33 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Krajci, S.: A generalized concept lattice. Logic Journal of IGPL 13(5), 543–550 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lai, H., Zhang, D.: Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory. International Journal of Approximate Reasoning 50(5), 695–707 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lei, Y., Luo, M.: Rough concept lattices and domains. Annals of Pure and Applied Logic 159(3), 333–340 (2009),
  17. 17.
    Liu, G.L.: Construction of rough approximations in fuzzy setting. Information Sciences 178(6), 1651–1662 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Liu, M., Shao, M., Zhang, W., Wu, C.: Reduction method for concept lattices based on rough set theory and its application. Computers & Mathematics with Applications 53(9), 1390–1410 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Medina, J., Ojeda-Aciego, M., Ruiz-Calviño, J.: Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets and Systems 160(2), 130–144 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Medina, J., Ojeda-Aciego, M., Vojtáš, P.: Multi-adjoint logic programming with continuous semantics. In: Eiter, T., Faber, W., Truszczyński, M. (eds.) LPNMR 2001. LNCS (LNAI), vol. 2173, pp. 351–364. Springer, Heidelberg (2001)Google Scholar
  21. 21.
    Medina, J., Ojeda-Aciego, M., Vojtáš, P.: Similarity-based unification: a multi-adjoint approach. Fuzzy Sets and Systems 146, 43–62 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pawlak, Z.: Rough sets. International Journal of Computer and Information Science 11, 341–356 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pollandt, S.: Fuzzy Begriffe. Springer, Berlin (1997)zbMATHGoogle Scholar
  24. 24.
    Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets and Systems 126(2), 137–155 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Wang, L., Liu, X.: Concept analysis via rough set and afs algebra. Information Sciences 178(21), 4125–4137 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel (1982)Google Scholar
  27. 27.
    Yao, Y.: A comparative study of formal concept analysis and rough set theory in data analysis. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 59–68. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  28. 28.
    Yao, Y.Y.: Concept lattices in rough set theory. In: Proceedings of Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS 2004), pp. 796–801 (2004)Google Scholar
  29. 29.
    Yao, Y.Y., Chen, Y.: Rough set approximations in formal concept analysis. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 285–305. Springer, Heidelberg (2006)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jesús Medina
    • 1
  1. 1.Department of MathematicsUniversity of Cádiz 

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