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Pliant Operator System

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Recent Advances in Intelligent Engineering Systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 378))

Abstract

We give a new representation theorem of negation based on the generator function of the strict operator. We study a certain class of strict monotone operators which build the DeMorgan class with infinitely many negations. We show that the necessary and sufficient condition for this operator class is f c (x) f d (x) = 1, where f c (x) and f d (x) are the generator functions of the strict t-norm and strict t-conorm. On the other hand our starting point is study of the relationship for Dombi aggregative operators, uninorms, strict t-norms and t-conorms. We present new representation theorem of strong negations where two explicitly contain the neutral value. Then relationships for aggregative operators and strong negations are verified as well as those for t-norm and t-conorm using the Pan operator concept. We will study a certain class of aggregative operators which build a self-DeMorgan class with infinitely many negation operators. We introduce the multiplicative pliant concept and characterize it by necessary and sufficient conditions.

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References

  1. Aczél, J.: Lectures on Functional Equations and Applications. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Aguil, I., Suner, J., Torrens, J.: A characterization of residual implications derived from left-continuous uninorms. Information Sciences 180(20), 3992–4005 (2010)

    Article  MathSciNet  Google Scholar 

  3. Alsina, C., Schweizer, B., Frank, M.J.: Associative functions: triangular norms and copulas. Word Scientific Publishing, Singapore (2006)

    Google Scholar 

  4. De Baets, B., De Meyer, H., Mesiar, R.: Piecewise linear aggregation functions based on triangulation. Information Sciences (2010) (in Press)

    Google Scholar 

  5. De Baets, B., Fodor, J.: Residual operators of uninorms. Soft Computing 3, 89–100 (1999)

    Google Scholar 

  6. Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuzziness and Soft Computing, vol. 221. Springer, Heidelberg (2007)

    Google Scholar 

  7. Boulkroune, A., M’Saad, M., Chekireb, H.: Design of a fuzzy adaptive controller for MIMO nonlinear time-delay systems with unknown actuator nonlinearities and unknown control direction. Information Sciences 180(24), 5041–5059 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calvo, T., Baets, B.D., Fodor, J.: The functional equations of frank and alsina for uninorms and nullnorms. Fuzzy Sets and Systems 120, 385–394 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Calvo, T., Mayor, G., Mesiar, R.: Aggregation Operators. New Trends and Applications Studies in Fuzziness and Soft Computing 97 (2002)

    Google Scholar 

  10. Calvo, T., Mesiar, R.: Weighted triangular norms-based aggregation operators. Fuzzy Sets and Systems 137(1), 3–10 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cintula, P., Klement, E.P., Mesiar, R., Navara, M.: Fuzzy logics with an additional involutive negation. Fuzzy Sets and Systems 161, 390–411 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dombi, J.: General class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness included by fuzzy operators. Fuzzy Sets and Systems 8, 149–168 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dombi, J.: Basic concepts for a theory of evaluation: The aggregation operator. European Journal of Operation Research 10, 282–293 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dombi, J.: Towards a General Class of Operators for Fuzzy Systems. IEEE Transaction on Fuzzy Systems 16, 477–484 (2008)

    Article  Google Scholar 

  15. Dombi, J.: DeMorgan systems with an infinitely many negations in the strict monotone operator case. Information Sciences (under print, 2011)

    Google Scholar 

  16. Dombi, J.: DeMorgan systems with infinite number of negation. Information Science (2010) (under Review Process)

    Google Scholar 

  17. Dombi, J.: On a certain class of aggregative operators. Information Sciences (2011) (under Refree Process)

    Google Scholar 

  18. Esteva, F.: Some Representable DeMorgan Algebras. Journal of Mathematical Analysis and Applications 100, 463–469 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fodor, J., Yager, R.R., Rybalov, A.: Structure of uninorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5(4), 411–427 (1997)

    Article  MathSciNet  Google Scholar 

  20. Garcia, P., Valverde, L.: Isomorphisms between DeMorgan triplets. Fuzzy Sets and Systems 30, 27–36 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gehrke, M., Walker, C., Walker, E.: DeMorgan Systems on the Unit Interval. International Journal of Intelligent Systems 11, 733–750 (1990)

    Article  Google Scholar 

  22. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and Its Applications, vol. 127. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  23. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation functions: Means. Information Sciences (2010) (in Press)

    Google Scholar 

  24. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes. Information Sciences (2010) (in Press)

    Google Scholar 

  25. Guadarrama, S., Ruiz-Mayor, A.: Approximate robotic mapping from sonar data by modeling perceptions with antonyms. Information Sciences 180(21), 4164–4188 (2010)

    Article  Google Scholar 

  26. Hu, S.-K., Li, Z.-F.: The structure of continuous uni-norms. Fuzzy Sets and Systems 124, 43–52 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Klement, E.P., Mesiar, R., Pap, E.: On the relationship of associative compensatory operators to triangular norms and conorms. Uncertainty, Fuzziness and Knowledge-Based Systems 4, 129–144 (1996)

    Article  MathSciNet  Google Scholar 

  28. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  29. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper I: basic analytical and algebraic properties 143(1), 5–26 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper II: general constructions and parameterized families 145(3), 411–438 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper III: continuous t-norms 145(3), 439–454 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Kolesrov, A., Mesiar, R.: Lipschitzian De Morgan triplets of fuzzy connectives. Information Sciences 180(18), 3488–3496 (2010)

    Article  MathSciNet  Google Scholar 

  33. Ling, C.M.: Representation of associative functions. Publ.Math. Debrecen 12, 189–212 (1965)

    MathSciNet  Google Scholar 

  34. Li, Y.-M., Shi, Z.-K.: Weak uninorm aggregation operators. Information Sciences 124, 317–323 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, Y.M., Shi, Z.K.: Remarks on uninorms aggregation operators. Fuzzy Sets and Systems 114, 377–380 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, J., Mesiar, R., Struk, P.: Pseudo-optimal measures. Information Sciences 180(20), 4015–4021 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mas, M., Mayor, G., Torrens, J.: The distributivity condition for uninorms and t-operators. Fuzzy Sets and Systems 128, 209–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mas, M., Monserrat, M., Torrens, J.: The law of importation for discrete implications. Information Sciences 179(24), 4208–4218 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mesiar, R., Rybárik, J.: Pan-operations structure. Fuzzy Sets and Systems 74, 365–369 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mesiar, R., Kolesrov, A., Calvo, T., Komornkov, M.: A Review of Aggregation Functions. In: Fuzzy Sets and Their Extension: Representation, Aggregation and Models. Studies in Fuzziness and Soft Computing, vol. 220 (2008)

    Google Scholar 

  41. Monserrat, M., Torrens, J.: On the reversibility of uninorms and t-operators. Fuzzy Sets and Systems 131, 303–314 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mostert, P.S., Shields, A.L.: On the structure of semi-groups on a compact manifold with boundary. Ann. Math. II. Ser. 65, 117–143 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  43. Nguyen, H.T., Walker, E.: A First Course in Fuzzy Logic. CRC Press, Boca Raton (1997); 2nd. edn. (1999), 3rd edn. (2006)

    MATH  Google Scholar 

  44. Ouyang, Y., Fang, J.: Some results of weighted quasi-arithmetic mean of continuous triangular norms. Information Sciences 178(22), 4396–4402 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Roychowdhury, S.: New triangular operator generators for fuzzy systems. IEEE Trans. of Fuzzy Systems 5, 189–198 (1997)

    Article  Google Scholar 

  46. Ruiz, D., Torrens, J.: Distributivity and Conditional Distributivity of a Uninorm and a Continous t-Conorm. IEEE Transactions on Fuzzy Systems 14(2), 180–190 (2006)

    Article  Google Scholar 

  47. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983)

    MATH  Google Scholar 

  48. Silvert, W.: Symmetric summation: a class of operations on fuzzy sets. IEEE Transactions on Systems, Man and Cybernetics 9, 659–667 (1979)

    MathSciNet  Google Scholar 

  49. Tong, X., Chen, M., Li, H.: Pan-operations structure with non-idempotent pan-addition. Fuzzy Sets and Systems 145, 463–470 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. Trillas, E.: Sobre funciones de negación en la teoría de conjuntos difusos. Stochastica III, 47–60 (1979)

    MathSciNet  Google Scholar 

  51. Wang, Z., Klir, G.J.: Fuzzy Measure Theory. Plenum Press, New York (1992)

    MATH  Google Scholar 

  52. Waegeman, W., De Baets, B.: A transitivity analysis of bipartite rankings in pairwise multi-class classification. Information Sciences 180(21), 4099–4117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80(1), 111–120 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  54. Yager, R.R.: Uninorms in fuzzy systems modeling. Fuzzy Sets and Systems 122, 167–175 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  55. Yager, R.R.: Weighted triangular norms using generating functions. International Journal of Intelligent Systems 19(3), 217–231 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, K.-L., Li, D.-H., Song, L.-X.: Solution of an open problem on pseudo-Archimedean t-norms. Information Sciences 178(23), 4542–4549 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zimmermann, H.J., Zysno, P.: Latent connectives in human decision making. Fuzzy Sets and Systems 4, 37–51 (1980)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Szeged, 6720, Szeged, Árpád tér 2., Hungary

    József Dombi

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  1. József Dombi
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Editors and Affiliations

  1. Institute of Intelligent Engineering Systems, Óbuda University, Bécsi út 96/b, H-1034, Budapest, Hungary

    János Fodor

  2. Institute of Computer Engineering, Control and Robotics, Wrocław University of Technology, 27 Wybrzeze Wyspianskiego st., 50-372, Wrocław, Poland

    Ryszard Klempous

  3. Departamento de Informática y Sistemas, Universidad de Las Palmas de Gran Canaria, Instituto Universitario de Ciencias y Tecnologías Cibernéticas, Campus Universitario de Tafira s/n, 35017, Las Palmas de Gran, Canaria, Spain

    Carmen Paz Suárez Araujo

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Dombi, J. (2012). Pliant Operator System. In: Fodor, J., Klempous, R., Suárez Araujo, C.P. (eds) Recent Advances in Intelligent Engineering Systems. Studies in Computational Intelligence, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23229-9_2

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  • DOI: https://doi.org/10.1007/978-3-642-23229-9_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23228-2

  • Online ISBN: 978-3-642-23229-9

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