| Title:
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Migrativity of continuous t-conorms with respect to ordinal sum implications (English) |
| Author:
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Yan, Xinxin |
| Author:
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Zhou, Hongjun |
| Language:
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English |
| Journal:
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Kybernetika |
| ISSN:
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0023-5954 (print) |
| ISSN:
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1805-949X (online) |
| Volume:
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62 |
| Issue:
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2 |
| Year:
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2026 |
| Pages:
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163-189 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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The topic of migrativity among aggregation functions is of significant interest from both theoretical and practical perspectives within the field of fuzzy set theory. Nonetheless, there is a scarcity of characterizations in the existing literature concerning the migrativity of ordinal sum implications, especially when the ordinal summands are positioned along the major diagonal line of $[0,1]^{2}$, and this area has not been thoroughly investigated. The present paper aims to fill this gap by conducting a detailed study on the migrativity of t-conorms with respect to ordinal sum implications. We provide the structural solutions to the migrative functional equation for t-conorms with respect to ordinal sum implications, which depend on the position of parameter $\alpha$ within the range of natural negation $N$. The characterizations under which t-conorms are $\alpha$-migrative with respect to ordinal sum implications are obtained by presenting ordinal sum representations of the underlying functions. (English) |
| Keyword:
|
Migrativity |
| Keyword:
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T-conorm |
| Keyword:
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Ordinal sum implication |
| MSC:
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03B52 |
| MSC:
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03E72 |
| DOI:
|
10.14736/kyb-2026-2-0163 |
| . |
| Date available:
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2026-05-21T10:58:27Z |
| Last updated:
|
2026-05-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153630 |
| . |
| Reference:
|
[1] Aczél, J.: Lectures on Functional Equations and Their Applications..Academic Press, New York 1966. Zbl 0139.09301 |
| Reference:
|
[2] Aczél, J., Belousov, V. D., Hosszú, M.: Generalized associativity and bisymmetry on quasigroups..Acta Math. Acad. Sci. Hung. 11 (1963), 127-136. |
| Reference:
|
[3] Alsina, C., Frank, M. J., Schweizer, B.: Associative Functions: Triangular Norms and Copulas..World Scientific, New Jersey 2006. Zbl 1100.39023 |
| Reference:
|
[4] Baczyński, M., Jayaram, B.: Fuzzy Implications..Springer, Berlin 2008. Zbl 1293.03012 |
| Reference:
|
[5] Baczyński, M., Drygás, P., Król, A., Mesiar, R.: New types of ordinal sum of fuzzy implications..In: 2017 IEEE International Conference on Fuzzy Systems, Naples, Italy. (2017) 1-6. |
| Reference:
|
[6] Baczyński, M., Drygás, P., Król, A., Pusz, P.: Developing idea of ordinal sum of fuzzy implications..In: 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Glasgow, UK (2020) 1-7. |
| Reference:
|
[7] Baczyński, M., Jayaram, B., Mesiar, R.: Fuzzy implication: alpha migrativity and generalised laws of importation..Inf. Sci. 531 (2020), 87-96. |
| Reference:
|
[8] Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles..Springer New York, NY 2002. |
| Reference:
|
[9] Bustince, H., Baets, B. De, Fernandez, J., Mesiar, R., Montero, J.: A generalization of the migrativity property of aggregation functions..Inf. Sci. 191 (2012), 76-85. |
| Reference:
|
[10] Bustince, H., Montero, J., Mesiar, R.: Migrativity of aggregation functions..Fuzzy Sets Syst. 160 (2009), 766-777. Zbl 1186.68459, |
| Reference:
|
[11] Chang, Q., Zhou, H., Baczyński, M.: Characterizations for the migrativity of uninorms over N-ordinal sum implications..Comput. Appl. Math. 42 (172) (2023), 1-38. |
| Reference:
|
[12] Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic..College Publications, London 2011. |
| Reference:
|
[13] Clifford, A. H.: Naturally totally ordered commutative semigroups..Amer. J. Math. 76 (1954), 631-646. |
| Reference:
|
[14] Cox, E.: The Fuzzy Systems Handbook: A Practitioner's Guide to Building, Using, and Maintaining Fuzzy Systems..Academic Press Professional, Inc. 1994. |
| Reference:
|
[15] Cutello, V., Montero, J.: Recursive connective rules..Int. J. Intell. Syst. 14 (1999), 3-20. 10.1002/(SICI)1098-111X(199901)14:1<3::AID-INT2>3.0.CO;2-K |
| Reference:
|
[16] Baets, B. De, Kerre, E., Gupta, M.: The fundamentals of fuzzy mathematical morphology part 2: idempotence, convexity and decomposition..Int. J. Gen. Syst. 23 (1995), 307-322. |
| Reference:
|
[17] Nola, A. Di, Sessa, S., Pedrycz, W., Sánchez, E.: Fuzzy Relation Equations and Their Applications to Knowledge Engineering..Kluwer Academic Publishers, Kluwer 1989. |
| Reference:
|
[18] Drygás, P., Król, A.: Two constructions of ordinal sums of fuzzy implications..In: Uncertainty and Imprecision in Decision Making and Decision Support: Cross-Fertilization, New Models and Applications, IWIFSGN 2016. Advances in Intelligent Systems and Computing, (K.T. Atanassov, ed.) Springer, Berlin, Germany, (2018) 102-111. |
| Reference:
|
[19] Dubois, D., Prade, H.: Fuzzy sets in approximate reasoning, Part I: inference with possibility distributions..Fuzzy Sets Syst. 40 (1991), 143-202. |
| Reference:
|
[20] Dubois, D., Lang, J., Prade, H.: Fuzzy sets in approximate reasoning, Part II: logical approaches..Fuzzy Sets Syst. 40 (1991), 203-244. |
| Reference:
|
[21] Durante, F., Fernández-Sánchez., J., Quesada-Molina, J. J.: On the $\alpha$-migrativity of multivariate semi-copulas..Inf. Sci. 187 (2012), 216-223. |
| Reference:
|
[22] Fang, X., Zhu, K.: Characterizations on the cross-migrativity between uni-nullnorms (null-uninorms) and overlap (grouping) functions..Comput. Appl. Math. 44 (2025), 147. |
| Reference:
|
[23] Fang, X., Zhu, K.: A note on the cross-migrativity between uninorms and overlap (grouping) functions..Fuzzy Sets Syst. 499 (2025), 109190. |
| Reference:
|
[24] Durante, F., Sarkoci, P.: A note on the convex combinations of triangular norms..Fuzzy Sets Syst. 159 (2008), 77-80. |
| Reference:
|
[25] Fodor, J., Klement, E. P., Mesiar, R.: Cross-migrative triangular norms..Int. J. Intell. Syst. 27 (2012), 411-428. |
| Reference:
|
[26] Gottwald, S.: A Treatise on Many-Valued Logic..Research studies press LTD 2001. |
| Reference:
|
[27] Jenei, S.: On the convex combination of left-continuous t-norms..Aequ. Math. 72 (2006), 47-59. |
| Reference:
|
[28] Kerre, E., Huang, C., Ruan, D.: Fuzzy Set Theory and Approximate Reasoning..Wu Han University Press, Wu Chang 2004. |
| Reference:
|
[29] Kerre, E., Nachtegael, M.: Fuzzy Techniques in Image Processing..Springer-Verlag, New York 2000. |
| Reference:
|
[30] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms..Kluwer Academic Publisher, Dordrecht 2000. Zbl 1087.20041 |
| Reference:
|
[31] Li, G., Liu, H.-W.: Some results on the convex combination of uninorms..Fuzzy Sets Syst. 372 (2019) 50-61. |
| Reference:
|
[32] Liang, S., Wang, X.-P.: On the migrativity of 2-uninorms..Fuzzy Sets Syst. 472 (2023), 108703. |
| Reference:
|
[33] Lopez-Molina, C., Bates, B. De, Bustince, H., Induráin, E., Stup\u{n}anová, A., Mesiar, R.: Bimigrativity of binary aggregation functions..Inf. Sci. 274 (2014), 225-235. |
| Reference:
|
[34] Luo, Y., Zhu, K.: Characterizations for the cross-migrativity between overlap functions and commutative aggregation functions..Inf. Sci. 622 (2023), 303-318. |
| Reference:
|
[35] Mas, M., Monserrat, M., Ruiz-Aguilera, D., Torrens, J.: An extension of the migrative property for uninorms..Inf. Sci. 246 (2013), 191-198. |
| Reference:
|
[36] Massanet, S., Riera, J. V., Torrens, J.: A new look on the ordinal sum of fuzzy implication functions..In: Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, (J. Carvalho, M. J. Lesot, U. Kaymak, S. Vieira, B. Bouchon-Meunier, R. Yager, ed.), Springer, Cham. (2016) 399-410. |
| Reference:
|
[37] Mesiar, R., Novák, V.: Open problems from the 2nd International Conference on Fuzzy Sets Theory and Its Applications..Fuzzy Sets Syst. 81 (1996), 185-190. |
| Reference:
|
[38] Mesiar, R., Bustince, H., Fernandez, J.: On the $\alpha$-migrativity of semicopulas, quasi-copulas, and copulas..Inf. Sci. 246 (2010), 1967-1976. |
| Reference:
|
[39] Montero, J., Gómez, D., Muñoz, S.: Fuzzy information representation for decision aiding..In: Proceedings of IPMU 08, (L. Magdalena, M. Ojeda-Aciego, J. L. VerdegayMálaga ed.), Spain, Torremolinos (Malaga) (2008) 1425-1430. |
| Reference:
|
[40] Nguyen, H. T., Sugeno, M.: Fuzzy Systems: Modeling and Control..Springer, New York 1998. |
| Reference:
|
[41] Ouyang, Y.: Generalizing the migrativity of continuous t-norms..Fuzzy Sets Syst. 211 (2013), 73-83. |
| Reference:
|
[42] Ouyang, Y., Fang, J., Li, G.: On the convex combination of $T_{D}$ and continuous triangular norms..Inf. Sci. 177 (2007), 2945-2953. |
| Reference:
|
[43] Pan, D., Zhou, H., Yan, X.: Characterizations for the migrativity of continuous t-conorms over fuzzy implications..Fuzzy Sets Syst. 456 (2023), 173-196. |
| Reference:
|
[44] Qiao, J., Hu, B. Q.: On generalized migrativity property for overlap functions..Fuzzy Sets Syst. 357 (2019), 91-116. |
| Reference:
|
[45] Su, Y., Xie, A., Liu, H. -W.: On ordinal sum implications..Inf. Sci. 293 (2015), 251-262. |
| Reference:
|
[46] Su, Y., Zong, W., Liu, H.-W., Xue, P.: Migrativity property for uninorms and semi t-operators..Inf. Sci. 325 (2015), 455-465. |
| Reference:
|
[47] Wang, C., Wan, L., Zhang, B.: Notes on alpha-cross-migrativity of t-conorms over fuzzy implications..Fuzzy Sets Syst. 473 (2023), 108741. |
| Reference:
|
[48] Wang, W., Zhu, K.: The necessary and sufficient conditions for bimigrativity of uninorms over overlap functions..Fuzzy Sets Syst. 507 (2025), 109319. |
| Reference:
|
[49] Wu, L., Ouyang, Y.: On the migrativity of triangular subnorms..Fuzzy Sets Syst. 226 (2013), 89-98. |
| Reference:
|
[50] Yan, X., Zhou, H.: Migrativity properties of general grouping (overlap) functions with respect to null-norms..Comput. Appl. Math. 43 (251) (2024), 1-23. |
| Reference:
|
[51] Zeng, X., Zhu, K.: On the migrativity properties between uni-nullnorms and overlap (grouping) functions..Soft Comput. 28 (2024), 7671-7685. |
| Reference:
|
[52] Zhan, H., Liu, H.-W.: The cross-migrative property for uninorms..Aequ. Math. 90 (2016), 1219-1239. |
| Reference:
|
[53] Zhang, T., Zhu, K., Wang, J., Pan, D.: Characterizations of some classes of generated implication solutiond to the cross-migrativity..Fuzzy Sets Syst. 511 (2025), 109375. |
| Reference:
|
[54] Zhou, H., Yan, X.: Migrativity properties of overlap functions over uninorms..Fuzzy Sets Syst. 403 (2021), 10-37. |
| Reference:
|
[55] Zhou, H.: Two general construction ways toward unified framework of ordinal sums of fuzzy implications..IEEE Trans. Fuzzy Syst. 29 (4) (2021), 846-860. |
| Reference:
|
[56] Zhou, H.: Probabilistially Quantitative Logic and its Applications..Science Press, Beijing 2015. |
| Reference:
|
[57] Zhou, H., Chang, Q., Baczyński, M.: Characterizations on migrativity of continuous triangular conorms with respect to $N$-ordinal sum implications..Inf. Sci. 637 (2023), 118926. |
| Reference:
|
[58] Zhu, K., Wang, J., Yang, Y.: Migrative uninorms and nullnorms over t-norms and t-conorms revisited..Fuzzy Sets Syst. 423 (2021), 74-88. |
| Reference:
|
[59] Zhu, K., Wang, J., Yang, Y.: Some new results on the migrativity of uninorms over overlap and grouping functions..Fuzzy Sets Syst. 427 (2022), 55-70. |
| Reference:
|
[60] Zhu, K., 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. |
| Reference:
|
[61] 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. |
| Reference:
|
[62] Zhu, K., Zeng, X., Qiao, J.: On the cross-migrativity between uninorms and overlap (grouping) functions..Fuzzy Sets Syst. 451 (2022), 113-129. |
| Reference:
|
[63] Zong, W., Su, Y., Liu, H.-W.: Migrative property for nullnorms..Int. J. Unc. Fuzz. Knowl. Based Syst. 22 (5) (2014), 749-759. |
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