Distributivity of ordinal sum implications over overlap and grouping functions

Pan, Deng ; Zhou, Hongjun

Kybernetika, Tome 57 (2021) no. 4, pp. 647-670

Abstract (VO)

In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given.

Détail

DOI : 10.14736/kyb-2021-4-0647

Classification : 03B52, 03E72
Keywords: distributivity; fuzzy implication functions; ordinal sum; overlap functions; grouping functions

Bibliographie

[1] Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York 1966. | Zbl

[2] Baczyński, M., Jayaram, B.: Fuzzy Implications. Springer, Berlin 2008. | Zbl

[3] Baczyński, M., Jayaram, B.: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms. IEEE Trans. Fuzzy Syst. 17 (2009), 590-603. | DOI

[4] Bedregal, B., Dimuro, G. P., Bustince, H., Barrenechea, E.: New results on overlap and grouping functions. Inf. Sci. 249 (2013), 148-170. | DOI

[5] Bedregal, B., Bustince, H., Palmeira, E., Dimuro, G., Fernandez, J.: Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions. Int. J. Approx. Reason. 90 (2017), 1-16. | DOI

[6] Bustince, H., Barrenechea, E., Pagola, M.: Image thresholding using restricted e-quivalent functions and maximizing the measures of similarity. Fuzzy Sets Syst. 158 (2007), 496-516. | DOI

[7] Bustince, H., Fernandez, J., Mesiar, R., Montero, J., Orduna, R.: Overlap index, overlap functions and migrativity. In: Proc. IFSA/EUSFLAT Conference, 2009, pp. 300-305.

[8] Bustince, H., Fernandez, J., Mesiar, R., Montero, J., Orduna, R.: Overlap functions. Nonlinear Anal. 72 (2010), 1488-1499. | DOI

[9] Bustince, H., Pagola, M., Mesiar, R., Hüllermeier, E., Herrera, F.: Grouping, overlaps, and generalized bientropic functions for fuzzy modeling of pairwise comparisions. IEEE Trans. Fuzzy Syst. 20 (2012), 405-415. | DOI

[10] Cao, M., Hu, B. Q., Qiao, J.: On interval $(G,N)$-implications and $(O,G,N)$-implications derived from interval overlap and grouping functions. Int. J. Approx. Reason. 100 (2018), 135-160. | DOI

[11] Combs, W. E., Andrews, J. E.: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 6 (1998), 1-11. | DOI

[12] al., L. De Miguel et: General overlap functions. Fuzzy Sets Syst. 372 (2019), 81-96. | DOI

[13] Dimuro, G. P., Bedregal, B.: On residual implications derived from overlap functions. Inf. Sci. 312 (2015), 78-88. | DOI

[14] Dimuro, G. P., Bedregal, B.: Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties. Fuzzy Sets Syst. 252 (2014), 39-54. | DOI

[15] Dimuro, G. P., Bedregal, B., Bustince, H., Asiáin, M. J., Mesiar, R.: On additive generators of overlap functions. Fuzzy Sets Syst. 287 (2016), 76-96. | DOI

[16] Dimuro, G. P., Bedregal, B., Bustince, H., Jurio, A., Baczyński, M., Miś, K.: $QL$-operations and $QL$-implications constructed from tuples $(O,G,N)$ and the generation of fuzzy subsethood and entropy measures. Int. J. Approx. Reason. 82 (2017), 170-192. | DOI

[17] Dimuro, G. P., Bedregal, B., Santiago, R. H. N.: On $(G,N)$-implications derived from grouping functions. Inf. Sci. 279 (2014), 1-17. | DOI

[18] Dimuro, G. P., Bedregal, B., Santiago, R. H. N., Reiser, R. H. S.: Interval additive generators of interval t-norms and interval t-conorms. Inf. Sci. 181 (2011), 3898-3916. | DOI

[19] Elkano, M., Galar, M., Sanz, J., Bustince, H.: Fuzzy rule based classification systems for multi-class problems using binary decomposition strategies: On the influence of n-dimensional overlap functions in the fuzzy reasoning method. Inf. Sci. 332 (2016), 94-114. | DOI

[20] Elkano, M., Galar, M., Sanz, J., Fernández, A., Barrenechea, E., Herrera, F., Bustince, H.: Enhancing multi-class classification in FARC-HD fuzzy classifier: On the synergy between n-dimensional overlap functions and decomposition strategies. IEEE Trans. Fuzzy Syst. 23 (2015), 1562-1580. | DOI

[21] Elkano, M., Galar, M., Sanz, J., Schiavo, P. F., Pereira, S., Dimuro, G. P., Borges, E. N., Bustince, H.: Consensus via penalty functions for decision making in ensembles in fuzzy rulebased classification systems. Appl. Soft Comput. 67 (2018), 728-740. | DOI

[22] Gómez, D., Rodríguez, J. T., Montero, J., Bustince, H., Barrenechea, E.: $n$-dimensional overlap functions. Fuzzy Sets Syst. 287 (2016), 57-75. | DOI

[23] Jurio, A., Bustince, H., Pagola, M., Pradera, A., Yager, R.: Some properties of overlap and grouping functions and their application to image thresholding. Fuzzy Sets Syst. 229 (2013), 69-90. | DOI

[24] Jayaram, B.: Yager's new class of implications $I_{f}$ and some classical tautologies. Inf. Sci. 177 (2007), 930-946. | DOI

[25] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Acdemic Publisher, Dordrecht, 2000. | Zbl

[26] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Second edition. (A. Gilányi, ed.), Boston 2009.

[27] Lu, J., Zhao, B.: Distributivity of a class of ordinal sum implications over t-norms and t-conorms. Fuzzy Sets Syst. 378 (2020), 103-124. | DOI

[28] Mesiar, R., Mesiarová, A.: Residual implications and left-continuous t-norms which are ordinal sums of semigroups. Fuzzy Sets Syst. 143 (2004), 47-57. | DOI

[29] Qin, F.: Distributivity between semi-uninorms and semi-t-operators. Fuzzy Sets Syst. 299 (2016), 66-88. | DOI

[30] Qin, F., Baczyński, M., Xie, A.: Distributive equations of implications based on continuous triangular norms (I). IEEE Trans. Fuzzy Syst. 21 (2012), 153-167. | DOI

[31] Qin, F., Baczyński, M.: Distributive equations of implications based on continuous triangular conorms (II). Fuzzy Sets Syst. 240 (2014), 86-102. | DOI

[32] Qiao, J., Hu, B. Q.: On multiplicative generators of overlap and grouping functions. Fuzzy Sets Syst. 332 (2018), 1-24. | DOI

[33] Qiao, J., Hu, B. Q.: The distributive laws of fuzzy implications over overlap and grouping functions. Inf. Sci. 438 (2018), 107-126. | DOI

[34] Qiao, J., Hu, B. Q.: On generalized migrativity property for overlap functions. Fuzzy Sets Syst. 357 (2019), 91-116. | DOI

[35] Qiao, J., Hu, B. Q.: On the distributive laws of fuzzy implications over additively generated overlap and grouping functions. IEEE Trans. Fuzzy Syst. {\mi26} (2018), 2421-2433. | DOI

[36] Qiao, J., Hu, B. Q.: On interval additive generators of interval overlap functions and interval grouping functions. Fuzzy Sets Syst. 323 (2017), 19-55. | DOI

[37] Su, Y., Xie, A., Liu, H. W.: On ordinal sum implications. Inf. Sci. 293 (2015), 251-262. | DOI

[38] Su, Y., Zong, W. W., Liu, H. W.: On distributivity equations for uninorms over semi-t-operators. Fuzzy Sets Syst. 299 (2016), 41-65. | DOI

[39] Su, Y., Zong, W. W., Liu, H. W.: Distributivity of the ordinal sum implications over t-norms and t-conorms. IEEE Trans. Fuzzy Syst. 24 (2016), 827-840. | DOI

[40] Ti, L., Zhou, H.: On $(O,N)$-coimplications derived from overlap functions and fuzzy negations. J. Intell. Fuzzy Syst. 34 (2018), 3993-4007. | DOI

[41] Trillas, E., Alsina, C.: On the law $[(p\wedge q)\rightarrow r]\equiv [(p\rightarrow r)\vee (q\rightarrow r)]$ in fuzzy logic. IEEE Trans. Fuzzy Syst. 10 (2002), 84-88. | DOI

[42] Wang, Y. M., Liu, H. W.: The modularity conditon for overlap and grouping functions. Fuzzy Sets Syst. 372 (2019), 97-110. | DOI

[43] Xie, A., Li, C., Liu, H.: Distributive equations of fuzzy implications based on continuous triangular conorms given as ordinal sums. IEEE Trans. Fuzzy Syst. 21 (2013), 541-554. | DOI

[44] Xie, A., Liu, H., Zhang, F., Li, C.: On the distributivity of fuzzy implications over continuous Archimedeant-conorms and continuous t-conorms given as ordinal sums. Fuzzy Sets Syst. 205 (2012), 76-100. | DOI

[45] Zhang, T. H., Qin, F., Li, W. H.: On the distributivity equations between uni-nullnorms and overlap (grouping) functions. Fuzzy Sets Syst. 403 (2021), 56-77. | DOI

[46] Zhang, T. H., Qin, F.: On distributive laws between 2-uninorms and overlap (grouping) functions. Int. J. Approx. Reason. 119 (2020), 353-372. | DOI

[47] Zhou, H.: Characterizations of fuzzy implications generated by continuous multiplicative generators of T-norms. IEEE Trans. Fuzzy Syst. | DOI

[48] Zhu, K., Wang, J., Yang, Y.: A note on the modularity condition for overlap and grouping functions. Fuzzy Sets Syst. 408 (2021), 108-117. | DOI

[49] Zhu, K., Wang, J., Yang, Y.: New results on the modularity condition for overlap and grouping functions. Fuzzy Sets Syst. 403 (2021), 139-147 | DOI

[50] Zhu, K., Wang, J., Yang, Y.: A short note on the migrativity properties of overlap functions over uninorms. Fuzzy Sets Syst. 414 (2021), 135-145 | DOI

[51] Zhu, K. Y., Hu, B. Q.: Addendum to "On the migrativity of uninorms and nullnorms over overlap and grouping functions'' [Fuzzy Sets Syst. 346 (2018) 1-54]. Fuzzy Sets Syst. 386 (2020), 48-59. | DOI

[52] Chang, Q., Zhou, H.: Distributivity of $N$-ordinal sum fuzzy implications over t-norms and t-conorms. Int. J. Approx. Reason. 131 (2021), 189-213. | DOI

[53] Zhou, H.: Two general construction ways toward unified framework of ordinal sums of fuzzy implications. IEEE Trans. Fuzzy Syst. 29 (2021), 846-860. | DOI