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Fuzzy Information Relations and Operators: An Algebraic Approach Based on Residuated Lattices

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Theory and Applications of Relational Structures as Knowledge Instruments II

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4342))

Abstract

We discuss fuzzy generalisations of information relations taking two classes of residuated lattices as basic algebraic structures. More precisely, we consider commutative and integral residuated lattices and extended residuated lattices defined by enriching the signature of residuated lattices by an antitone involution corresponding to the De Morgan negation. We show that some inadequacies in representation occur when residuated lattices are taken as a basis. These inadequacies, in turn, are avoided when an extended residuated lattice constitutes the basic structure. We also define several fuzzy information operators and show characterizations of some binary fuzzy relations using these operators.

This work was carried out in the framework of COST Action 274/TARSKI on Theory and Applications of Relational Structures as Knowledge Instruments (www.tarski.org).

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References

  1. Allwein, G., Dunn, M.: Kripke models for linear logic. Journal of Symbolic Logic 58(2), 514–545 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bahls, P., Cole, J., Galatos, N., Jipsen, P., Tsinakis, C.: Cancellative residuated lattices. Algebra Universalis 50(1), 83–106 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  4. Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. of Algebra Comput. 13(4), 437–461 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. De Baets, B., Mesiar, R.: Pseudo-metrics and \(\mathcal{T}\)–equivalences. Journal of Fuzzy Mathematics 5(2), 471–481 (1997)

    MATH  MathSciNet  Google Scholar 

  6. Demri, S., Orłowska, E., Vakarelov, D.: Indiscernibility and complementarity relations in information systems, In: Gerbrandy, J., Marx, M., de Rijke, M., Venema, Y. (eds.) JFAK, Essays Dedicated to Johan van Benthem on the Occasion of his 50th Birthday. Amsterdam University Press (1999)

    Google Scholar 

  7. Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. In: EATCS Monographs in Theoretical Computer Science. Springer, Heidelberg (2002)

    Google Scholar 

  8. Dilworth, R.P., Ward, N.: Residuated lattices. Transactions of the American Mathematical Society 45, 335–354 (1939)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dilworth, R.P.: Non–commutative residuated lattices. Transactions of the American Mathematical Society 46, 426–444 (1939)

    Article  MATH  MathSciNet  Google Scholar 

  10. Düntsch, I., Orłowska, E.: Logics of complementarity in information systems. Mathematical Logic Quarterly 46, 267–288 (2000)

    Article  MATH  Google Scholar 

  11. Düntsch, I., Orłowska, E.: Beyond modalities: sufficiency and mixed algebras. In: Orłowska, E., Szałas, A. (eds.) Relational Methods for Computer Science Applications, pp. 263–285. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  12. Düntsch, I.: Private communication (2003)

    Google Scholar 

  13. Fitting, M.: Many–valued modal logics. Fundamenta Informaticae 15, 235–254 (1991)

    MATH  MathSciNet  Google Scholar 

  14. Fitting, M.: Many–valued modal logics (II). Fundamenta Informaticae 17, 55–73 (1992)

    MATH  MathSciNet  Google Scholar 

  15. Flondor, P., Sularia, M.: On a class of residuated semilattice monoids. Fuzzy Sets and Systems 138(1), 149–176 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Georgescu, G., Iorgulescu, A.: Pseudo–MV algebras. Multiple Valued Logic 6, 95–135 (2001)

    MATH  MathSciNet  Google Scholar 

  17. Godo, L., Rodriguez, R.O.: A Fuzzy Modal Logic for Similarity Reasoning. In: Chen, G., Ying, M., Cai, K.-Y. (eds.) Fuzzy Logic and Soft Computing, pp. 33–48. Kluwer, Dordrecht (1999)

    Google Scholar 

  18. Gougen, J.A.: L–fuzzy sets. Journal of Mathematical Analysis and Applications 18, 145–174 (1967)

    Article  MathSciNet  Google Scholar 

  19. Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    MATH  Google Scholar 

  20. Hajek, P.: Fuzzy Logics with Noncommutative Conjunctions. Journal of Logic and Computation 13(4), 469–479 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hart, J.B., Rafter, L., Tsinakis, C.: The structure of commutative residuated lattices. Int. J. Algebra Comput. 12(4), 509–524 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jipsen, P., Tsinaksis, C.: A Survey of Residuated Lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  23. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice–Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  24. Nachtegael, M., Kerre, E.E. (eds.): Fuzzy Techniques in Image Processing. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  25. Orłowska, E.: Kripke models with relative accessibility and their application to inferences from incomplete information. In: Mirkowska, G., Rasiowa, H. (eds.) Mathematical Problems in Computation Theory, vol. 21, pp. 329–339. Banach Center Publications (1988)

    Google Scholar 

  26. Orłowska, E. (ed.): Incomplete Information – Rough Set Analysis. Studies in Fuzziness and Soft Computing. Springer, Heidelberg (1998)

    Google Scholar 

  27. Orłowska, E.: Multi–valuedness and uncertainty. Multiple–Valued Logics 4, 207–227 (1999)

    MATH  Google Scholar 

  28. Orłowska, E., Radzikowska, A.M.: Information relations and operators based on double residuated lattices. In: Proceedings of 6th International Workshop on Relational Methods in Computer Science RelMiCS–2001, pp. 185–199. Oisterwijk, The Netherlands (2001)

    Google Scholar 

  29. Orłowska, E., Radzikowska, A.M.: Double residuated lattices and their applications. In: de Swart, H. (ed.) RelMiCS 2001. LNCS, vol. 2561, pp. 171–189. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  30. Radzikowska, A.M., Kerre, E.E.: On Some Classes of Fuzzy Information Relations. In: Proceedings of 31st IEEE International Symposium of Multiple–Valued Logics ISMVL–2001, Warsaw, pp. 75–80 (2001)

    Google Scholar 

  31. Radzikowska, A.M., Kerre, E.E.: Towards studying of fuzzy information relations. In: Proceedings of International Conference in Fuzzy Logics and Technology EUSFLAT–2001, Leicester, UK, pp. 365–369 (2001)

    Google Scholar 

  32. Radzikowska, A.M., Kerre, E.E.: A Comparative Study of Fuzy Rough Sets. Fuzzy Sets and Systems 126, 137–155 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Radzikowska, A.M., Kerre, E.E.: A fuzzy generalisation of information relations. In: Orłowska, E., Fitting, M. (eds.) Beyond Two: Theory and Applications of Multiple–Valued Logics, pp. 264–290. Springer, Heidelberg (2002)

    Google Scholar 

  34. Radzikowska, A.M., Kerre, E.E.: On L–valued fuzzy rough sets. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS, vol. 3070, pp. 526–531. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  35. Radzikowska, A.M., Kerre, E.E.: An algebraic characterisation of fuzzy rough sets. In: Proceedings of IEEE International Conference on Fuzzy Systems FUZZ–IEEE 2004, Budapest, Hungary, vol. 1, pp. 115–120 (2004)

    Google Scholar 

  36. Radzikowska, A.M., Kerre, E.E.: Lattice–based fuzzy information relations and operators. In: De Baets, B., De Caluve, R., Kacprzyk, J., De Tré, G., Zadrożny, S. (eds.) Current Issues in Data and Knowledge Engineering, EXIT, Warsaw, pp. 433–443 (2004)

    Google Scholar 

  37. 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 

  38. Radzikowska, A.M., Kerre, E.E.: Characterisations of main classes of fuzzy relations using fuzzy modal operators. Fuzzy Sets and Systems 152(2), 223–247 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics, Warsaw (1970)

    Google Scholar 

  40. Rauszer, C.: Semi-Boolean algebras and their applications to intuitionistic logic with dual operations. Fundamenta Mathematicae 83, 219–249 (1974)

    MATH  MathSciNet  Google Scholar 

  41. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North Holland, Amsterdam (1983)

    MATH  Google Scholar 

  42. Turunen, E.: Mathematics Behind Fuzzy Logic. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  43. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–358 (1965)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00–661, Warsaw, Poland

    Anna Maria Radzikowska

  2. Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000, Gent, Belgium

    Etienne E. Kerre

Authors
  1. Anna Maria Radzikowska
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  2. Etienne E. Kerre
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Editor information

Editors and Affiliations

  1. Faculty of Philosophy, Tilburg University, P.O. Box 90153, 5000, Tilburg, LE, The Netherlands

    Harrie de Swart

  2. National Institute of Telecommunications, Warsaw,  

    Ewa Orłowska

  3. Institute for Software Technology, Department of Computing Science, Universität der Bundeswehr München, 85577, Neubiberg, Germany

    Gunther Schmidt

  4. Faculté Polytechnique de Mons, Rue de Houdain 9, B-7000, Mons, Belgium

    Marc Roubens

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Radzikowska, A.M., Kerre, E.E. (2006). Fuzzy Information Relations and Operators: An Algebraic Approach Based on Residuated Lattices. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds) Theory and Applications of Relational Structures as Knowledge Instruments II. Lecture Notes in Computer Science(), vol 4342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11964810_8

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  • DOI: https://doi.org/10.1007/11964810_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69223-2

  • Online ISBN: 978-3-540-69224-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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