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Revising the link between L-Chu correspondences and completely lattice L-ordered sets

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Abstract

Continuing our categorical study of L-fuzzy extensions of formal concept analysis, we provide a representation theorem for the category of L-Chu correspondences between L-formal contexts and prove that it is equivalent to the category of completely lattice L-ordered sets.

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References

  1. Alcalde, C., Burusco, A., Fuentes-González, R.: Interval-valued linguistic variables: an application to the L-fuzzy contexts with absent values. Int. J. Gen. Syst. 39(3), 255–270 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alcalde, C., Burusco, A., Fuentes-González, R.: Analysis of certain L-fuzzy relational equations and the study of its solutions by means of the L-fuzzy concept theory. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 20(1), 21–40 (2012)

    Article  MATH  Google Scholar 

  3. Alcalde, C., Burusco, A., Fuentes-González, R., Zubia, I.: The use of linguistic variables and fuzzy propositions in the L-fuzzy concept theory. Comput. Math. Appl. 62(8), 3111–3122 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Awodey, S.: Category Theory. Oxford Logic Guides, vol. 49 (2006)

  5. Barr, M. *-Autonomous categories, volume 752. Lecture Notes in Mathematics. Springer (1979)

  6. Bělohlávek, R.: Fuzzy concepts and conceptual structures: induced similarities. In: Joint Conference on Information Sciences, pp. 179–182 (1998)

  7. Bělohlávek, R.: Lattices of fixed points of fuzzy Galois connections. Math. Log. Q. 47(1), 111–116 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Norwell (2002)

    Book  Google Scholar 

  9. Bělohlávek, R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Log. 128, 277–298 (2004)

    Article  MATH  Google Scholar 

  10. Butka, P., Pócs, J., Pócsová, J., Sarnovský, M.: Multiple data tables processing via one-sided concept lattices. Adv. Intell. Syst. Comput. 183, 89–98 (2013)

    Article  Google Scholar 

  11. Butka, P., Pócsová, J., Pócs, J.: On generation of one-sided concept lattices from restricted context. In: IEEE 10th Jubilee Intl Symp on Intelligent Systems and Informatics (SISY), pp. 111–115 (2012)

  12. Chen, R.-C., Bau, C.-T., Yeh, C.-J.: Merging domain ontologies based on the wordnet system and fuzzy formal concept analysis techniques. Appl. Soft Comput. 11(2), 1908–1923 (2011)

    Article  Google Scholar 

  13. Chen, X., Li, Q., Deng, Z.: Chu space and approximable concept lattice in fuzzy setting. In: Proceeding of FUZZ-IEEE, pp. 1–6 (2007)

  14. Denniston, J. T., Melton, A., Rodabaugh, S. E.: Formal concept analysis and lattice-valued Chu systems. Fuzzy Sets Syst. 216, 52–90 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. du Boucher-Ryana, P., Bridge, D.: Collaborative recommending using formal concept analysis. Knowl.-Based Syst. 19(5), 309–315 (2006)

    Article  Google Scholar 

  16. Dubois, D., Prade, H.: Possibility theory and formal concept analysis: Characterizing independent sub-contexts. Fuzzy Sets Syst. 196, 4–16 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Formica, A.: Concept similarity in formal concept analysis: an information content approach. Knowl.-Based Syst. 21(1), 80–87 (2008)

    Article  MathSciNet  Google Scholar 

  18. Formica, A.: Semantic web search based on rough sets and fuzzy formal concept analysis. Knowl.-Based Syst. 26, 40–47 (2012)

    Article  Google Scholar 

  19. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated lattices. An Algebraic Glimpse at Substructural Logics. Elsevier (2007)

  20. García, J. G., Mardones-Pérez, I., de Prada-Vicente, M. A., Zhang, D.: Fuzzy Galois connections categorically. Math. Log. Q 56(2), 131–147 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goguen, J.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goguen, J.: What is a concept? Lect. Notes Comput. Sci. 3596, 52–77 (2005)

    Article  Google Scholar 

  23. Hitzler, P., Krötzsch, M., Zhang, G.-Q.: A categorical view on algebraic lattices in formal concept analysis. Fundam. Informatica 74(2,3), 301–328 (2006)

    MATH  Google Scholar 

  24. Hitzler, P., Zhang, G.-Q.: A cartesian closed category of approximable concept structures. Lect. Notes Comput. Sci. 3127, 170–185 (2004)

    Article  Google Scholar 

  25. Kent, R. E.: The IFF foundation for ontological knowledge organization. Cat. Classif. Q 37(1–2), 187–203 (2003)

    Google Scholar 

  26. Krajči, S.: A categorical view at generalized concept lattices. Kybernetika 43(2), 255–264 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Krídlo, O., Krajči, S., Ojeda-Aciego, M.: The category of L-Chu correspondences and the structure of L-bonds. Fundam. Informaticae 115(4), 297–325 (2012)

    MATH  Google Scholar 

  28. Krídlo, O., Ojeda-Aciego, M.: On L-fuzzy Chu correspondences. Intl. J. Comput. Math. 88(9), 1808–1818 (2011)

    Article  MATH  Google Scholar 

  29. Krötzsch, M., Hitzler, P., Zhang, G.-Q.: Morphisms in context. Lect. Notes Comput. Sci. 3596, 223–237 (2005)

    Article  Google Scholar 

  30. Lai, H., Zhang, D.: Formal concept analysis vs. rough set theory. Int. J. Approx. Reason. 50(5), 695–707 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lei, Y., Luo, M.: Rough concept lattices and domains. Ann. Pure Appl. Log. 159(3), 333–340 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, S.-T., Tsai, F.: Noise control in document classification based on fuzzy formal concept analysis. In: Proc. of FUZZ-IEEE’11, pp. 2583–2588 (2011)

  33. Medina, J.: Multi-adjoint property-oriented and object-oriented concept lattices. Inf. Sci. 190, 95–106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Medina, J., Ojeda-Aciego, M.: Multi-adjoint t-concept lattices. Inf. Sci. 180(5), 712–725 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Medina, J., Ojeda-Aciego, M.: On multi-adjoint concept lattices based on heterogeneous conjunctors. Fuzzy Sets Syst. 208, 95–110 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Medina, J., Ojeda-Aciego, M., Ruiz-Calviño, J.: Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets Syst. 160(2), 130–144 (2009)

    Article  MATH  Google Scholar 

  37. Mora, A., Cordero, P., Enciso, M., Fortes, I., Aguilera, G.: Closure via functional dependence simplification. Int. J. Comput. Math. 89(4), 510–526 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mori, H.: Chu correspondences. Hokkaido Math. J. 37, 147–214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Seely, R.A.G.: Linear logic, *-autonomous categories and cofree coalgebras. In: Gray, J., Scedrov, A. (eds.) Categories in Computer Science and Logic, Contemporary Mathematics. Am. Math. Soc. 92, 371–382 (1989)

  40. Shen, L., Zhang, D.: The concept lattice functors. Int. J. Approx. Reason. 54(1), 166–183 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Solovyov, S.: Lattice-valued topological systems as a framework for lattice-valued formal concept analysis. J. Math. 2013, 506275: 33 (2013)

  42. Wu, Q., Liu, Z.: Real formal concept analysis based on grey-rough set theory. Knowl.-Based Syst. 22(1), 38–45 (2009)

    Article  Google Scholar 

  43. Zhang, D.: Galois connections between categories of L-topological spaces. Fuzzy Sets Syst. 152(2), 385–394 (2005)

    Article  MATH  Google Scholar 

  44. Zhang, G.-Q.: Chu spaces, concept lattices, and domains. Electron. Notes Theor. Comput. Sci. 83, 287–302 (2003)

  45. Zhang, G.-Q., Shen, G.: Approximable concepts, Chu spaces, and information systems. Theor. Appl. Categories 17(5), 80–102 (2006)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Pavol Jozef Šafárik, Košice, Slovakia

    Ondrej Krídlo

  2. Department Matemática Aplicada, Universidad de Málaga, Málaga, Spain

    Manuel Ojeda-Aciego

Authors
  1. Ondrej Krídlo
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  2. Manuel Ojeda-Aciego
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Correspondence to Ondrej Krídlo.

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Krídlo, O., Ojeda-Aciego, M. Revising the link between L-Chu correspondences and completely lattice L-ordered sets. Ann Math Artif Intell 72, 91–113 (2014). https://doi.org/10.1007/s10472-014-9416-8

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  • DOI: https://doi.org/10.1007/s10472-014-9416-8

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