Kybernetika 56 no. 4, 662-694, 2020

The study on semicopula based implications

Zuming PengDOI: 10.14736/kyb-2020-4-0662

Abstract:

Recently, Baczyński et al. (2017) proposed a new family of implication operators called semicopula based implications, which combines a given a priori fuzzy implication and a semicopula. In this paper, firstly, the relationship between the basic properties of the priori fuzzy implication and the semicopula based implication are analyzed. Secondly, the conditions such that the semicopula based implication is a fuzzy implication are studied, the study is carried out mainly in the case that the semicopula is a special family semicopula and the priori fuzzy implication is a ($U,N$)-implication. Moreover, the case that the semicopula based implication is 2-increasing (directionally decreasing, respectively) is also considered.

Keywords:

semicopula, fuzzy implications, semicopula based implications, ($U, N$)-implications, 2-increasing

Classification:

03E72, 03B52

paper.pdf

References:

  1. M. Baczyński and B. Jayaram: On the characterization of (S, N)-implications. Fuzzy Sets Systems. 158 (2007), 1713-1727.   DOI:10.1016/j.fss.2007.02.010
  2. M. Baczyński and B. Jayaram: Fuzzy Implications. Studies in Fuzziness and Soft Computing 231, Springer, Berlin Heidelberg 2008.   CrossRef
  3. M. Baczyński and B. Jayaram: (S, N)-and R-implications: a state-of-the-artsurvey. Fuzzy Sets Systems 159 (2008), 1836-1859.   DOI:10.1016/j.fss.2007.11.015
  4. M. Baczyński and B. Jayaram: QL-implications: some properties and intersections. Fuzzy Sets Systems 161 (2010), 158-188.   DOI:10.1016/j.fss.2008.09.021
  5. M. Baczyński, P. Grzegorzewski, R. Mesiar, P. Helbin and W. Niemyska: Fuzzy implications based on semicopulas. Fuzzy Sets Systems 323 (2017), 138-151.   DOI:10.1016/j.fss.2016.09.009
  6. M. Baczyński, G. Beliakov, H. B. Sola and A. Pradera: Advances in Fuzzy Implication Functions. Springer-Verlag, Berlin 2013.   DOI:10.1007/978-3-642-35677-3
  7. H. Bustince, J. Fernandez, A. Kolesárová and R. Mesiar: Directional monotonicity of fusion functions. Europ. J. Oper. Res. 244 (2015), 300-308.   DOI:10.1016/j.ejor.2015.01.018
  8. F. Durante and C. Sempi: Semicopul{\ae}. Kybernetika 41 (2005), 315-328.   CrossRef
  9. J. C. Fodor: Contrapositive symmetry of fuzzy implications. Fuzzy Sets Systems 69 (1995), 141-156.   DOI:10.1016/0165-0114(94)00210-x
  10. P. Grzegorzewski: Probabilistic implications. Fuzzy Sets Systems 226 (2013), 53-66.   DOI:10.1016/j.fss.2013.01.003
  11. S. Jenei: A new approach for interpolation and extrapolation of compact fuzzy quantities. In: Proc. 21th Linz Seminar on Fuzzy Set Theory 2000, pp. 13-18.   CrossRef
  12. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  13. H. Liu: On a new class of implications: (g, min)-implications and several classical tautologies. Int. J. Uncert. Fuzziness Knowl. Based Systems 20 (2012), 1-20.   DOI:10.1142/s0218488512500018
  14. H. Liu: Fuzzy implications derived from generalized additive generators of representable uninorms. IEEE Trans. Fuzzy Systems 21 (2013), 555-566.   DOI:10.1109/tfuzz.2012.2222892
  15. M. Mas, M. Monserrat, J. Torrens and E. Trillas: A survey on fuzzy implication functions. IEEE Trans. Fuzzy Systems 15 (2007), 1107-1121.   DOI:10.1109/tfuzz.2007.896304
  16. S. Massanet and J. Torrens: On a new class of fuzzy implication: $h$-implication and generalization. Inform. Sci. 181 (2011), 2111-2127.   DOI:10.1016/j.ins.2011.01.030
  17. S. Massanet and J. Torrens: Threshold generation method of construction of a new implication from two given ones. Fuzzy Sets Systems 205 (2012), 50-75.   DOI:10.1016/j.fss.2012.01.013
  18. S. Massanet and J. Torrens: On some properties of threshold generated implications. Fuzzy Sets Systems 205 (2012), 30-49.   DOI:10.1016/j.fss.2012.01.018
  19. S. Massanet and J. Torrens: On the vertical threshold generation method of fuzzy implication and its properties. Fuzzy Sets Systems 226 (2013), 32-52.   DOI:10.1016/j.fss.2013.03.003
  20. R. B. Nelsen: An Introduction to Copulas (Second edition). Springer, New York 2006.   DOI:10.1007/0-387-28678-0
  21. Y. Ouyang: On fuzzy implications determined by aggregation operators. Inform. Sci. 193 (2012), 153-162.   DOI:10.1016/j.ins.2012.01.001
  22. Z. Peng: A new family of (A, N)-implications: construction and properties. Iran. J. Fuzzy Systems 17 (2020), 129-145.   DOI:10.1016/0165-0114(94)90082-5
  23. A. Pradera: A review of the relationships between aggregation, implication and negation functions. Adv. Intell. Systems Comput. 228 (2013), 31-36.   DOI:10.1007/978-3-642-39165-1\_6
  24. A. Pradera, G. Beliakov, H. Bustince and B. De Baets: A review of the relationships between implication, negation and aggregation functions from the point of view of material implication. Inform. Sci. 329 (2016), 357-380.   DOI:10.1016/j.ins.2015.09.033
  25. Y. Su, A. Xie and H. Liu: On ordinal sum implications. Inform. Sci. 293 (2015), 251-262.   DOI:10.1016/j.ins.2014.09.021
  26. R. R. Yager: On some new classes of implication operators and their role in approximate reasoning. Inform. Sci. 167 (2004), 193-216.   DOI:10.1016/j.ins.2003.04.001