The distributivity equation for uninorms revisited. (English) Zbl 1380.03056
Summary: The distributivity equation has been widely studied involving different classes of aggregation functions from t-norms and t-conorms to uninorms, nullnorms and generalizations of them. It is important in the framework of logical connectives because of its applications in fuzzy logic and approximate reasoning as well as in image processing. Since uninorms have been used in these topics, the study of the distributivity between two uninorms becomes specially interesting. In a recent paper by the same authors, the already known solutions were compiled and completed when the first uninorm is in any of the most studied classes of uninorms and the second uninorm is anyone. In this paper, we want to achieve this study by focusing on the reverse direction, that is, for the cases when the second uninorm lies in any of the most studied classes of uninorms and the first one is any uninorm. We show along the paper that this new point of view leads to many new solutions.
Keywords:
fuzzy connectives; aggregation functions; functional equations; distributivity equation; uninorm; t-norm; t-conormReferences:
| [1] | Aczél, J., Lectures on Functional Equations and Their Applications (1966), Acad. Press: Acad. Press New York · Zbl 0139.09301 |
| [2] | Baczyński, M.; Jayaram, B., Fuzzy Implications, Studies in Fuzziness and Soft Computing, vol. 231 (2008), Springer: Springer Berlin, Heidelberg · Zbl 1147.03012 |
| [3] | Beliakov, G.; Pradera, A.; Calvo, T., Aggregation Functions: A Guide for Practitioners, Stud. Fuzziness Soft Comput., vol. 221 (2007), Springer: Springer Berlin, Heidelberg · Zbl 1123.68124 |
| [4] | Calvo, T., On some solutions of the distributivity equation, Fuzzy Sets Syst., 104, 1, 85-96 (1999) · Zbl 0928.03023 |
| [5] | Calvo, T.; De Baets, B.; Fodor, J., The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets Syst., 120, 385-394 (2001) · Zbl 0977.03026 |
| [6] | Calvo, T.; Mayor, G.; Mesiar, R., Aggregation Operators: New Trends and Applications, Stud. Fuzziness Soft Comput., vol. 97 (2002), Springer: Springer Berlin, Heidelberg · Zbl 0983.00020 |
| [7] | De Baets, B., Idempotent uninorms, Eur. J. Oper. Res., 118, 631-642 (1999) · Zbl 0933.03071 |
| [8] | Drewniak, J.; Drygaś, P.; Rak, E., Distributivity between uninorms and nullnorms, Fuzzy Sets Syst., 159, 13, 1646-1657 (2008) · Zbl 1216.03058 |
| [9] | Drygaś, P., Discussion of the structure of uninorms, Kybernetika, 41, 213-226 (2005) · Zbl 1249.03093 |
| [10] | Drygaś, P., On monotonic operations which are locally internal on some subset of their domain, (Stepnicka, M.; Novák, V.; Bodenhofer, U., New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference. New Dimensions in Fuzzy Logic and Related Technologies. Proceedings of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, vol. 2: Regular Sessions (2007)), 185-191 |
| [11] | Drygaś, P., On the structure of continuous uninorms, Kybernetika, 43, 183-196 (2007) · Zbl 1132.03349 |
| [12] | Drygaś, P., On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets Syst., 161, 149-157 (2010) · Zbl 1191.03039 |
| [13] | Drygaś, P., Distributivity between semi-t-operators and semi-nullnorms, Fuzzy Sets Syst., 264, 100-109 (2015) · Zbl 1361.03017 |
| [14] | Drygaś, P.; Rak, E., Distributivity equation in the class of semi-t-operators, Fuzzy Sets Syst., 291, 66-81 (2016) · Zbl 1368.03051 |
| [15] | Drygaś, P.; Rak, E., Distributivity equation in the class of 2-uninorms, Fuzzy Sets Syst., 291, 82-97 (2016) · Zbl 1368.03052 |
| [16] | Drygaś, P.; Ruiz-Aguilera, D.; Torrens, J., A characterization of uninorms locally internal in A(e) with continuous underlying operators, Fuzzy Sets Syst., 287, 137-153 (2016) · Zbl 1392.03050 |
| [17] | Fodor, J.; De Baets, B., A single-point characterization of representable uninorms, Fuzzy Sets Syst., 202, 89-99 (2012) · Zbl 1268.03027 |
| [18] | Fodor, J.; Roubens, M., Fuzzy Preference Modeling and Multicriteria Decision Support (1994), Kluwer Acad. Publ.: Kluwer Acad. Publ. New York · Zbl 0827.90002 |
| [19] | Fodor, J.; Yager, R. R.; Rybalov, A., Structure of uninorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5, 4, 411-427 (1997) · Zbl 1232.03015 |
| [20] | Grabisch, M.; Marichal, J. L.; Mesiar, R.; Pap, E., Aggregation Functions, Encyclopedia of Mathematics and Its Applications, vol. 127 (2009), Cambridge University Press: Cambridge University Press New York · Zbl 1196.00002 |
| [21] | Hu, S. K.; Li, Z. F., The structure of continuous uninorms, Fuzzy Sets Syst., 124, 1, 43-52 (2001) · Zbl 0989.03058 |
| [22] | Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht · Zbl 0972.03002 |
| [23] | Klir, G. J.; Yuan, B., Fuzzy Sets and Fuzzy Logic, Theory and Application (1995), Prentice Hall PTR: Prentice Hall PTR Upper Saddle River, New Jersey · Zbl 0915.03001 |
| [24] | Li, G.; Liu, H., Distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t-conorm, Fuzzy Sets Syst., 287, 154-171 (2016) · Zbl 1392.03051 |
| [25] | Li, G.; Liu, H.; Fodor, J., Single-point characterization of uninorms with nilpotent underlying t-norm and t-conorm, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 22, 4, 591-604 (2014) · Zbl 1323.03079 |
| [26] | Liu, H., Distributivity and conditional distributivity of semi-uninorms over continuous t-conorms and t-norms, Fuzzy Sets Syst., 268, 27-43 (2015) · Zbl 1361.03021 |
| [27] | Martin, J.; Mayor, G.; Torrens, J., On locally internal monotonic operations, Fuzzy Sets Syst., 137, 27-42 (2003) · Zbl 1022.03038 |
| [28] | Mas, M.; Massanet, S.; Ruiz-Aguilera, D.; Torrens, J., A survey on the existing classes of uninorms, J. Intell. Fuzzy Syst., 29, 1021-1037 (2015) · Zbl 1361.03022 |
| [29] | Mas, M.; Mayor, G.; Torrens, J., The distributivity condition for uninorms and t-operators, Fuzzy Sets Syst., 128, 2, 209-225 (2002) · Zbl 1005.03047 |
| [30] | Mas, M.; Mayor, G.; Torrens, J., Corrigendum to “the distributivity condition for uninorms and t-operators”, Fuzzy Sets Syst.. Fuzzy Sets Syst., Fuzzy Sets Syst., 153, 297-299 (2005) |
| [31] | Mas, M.; Mayor, G.; Torrens, J., The modularity condition for uninorms and t-operators, Fuzzy Sets Syst., 126, 2, 207-218 (2002) · Zbl 0996.03038 |
| [32] | Mas, M.; Monserrat, M.; Ruiz-Aguilera, D.; Torrens, J., An extension of the migrative property for uninorms, Inf. Sci., 246, 191-198 (2013) · Zbl 1320.68193 |
| [33] | Mas, M.; Monserrat, M.; Ruiz-Aguilera, D.; Torrens, J., Migrative uninorms and nullnorms over t-norms and t-conorms, Fuzzy Sets Syst., 261, 20-32 (2015) · Zbl 1360.68844 |
| [34] | Petrík, M.; Mesiar, R., On the structure of special classes of uninorms, Fuzzy Sets Syst., 240, 22-38 (2014) · Zbl 1315.03099 |
| [35] | Qin, F.; Zhao, B., The distributive equations for idempotent uninorms and nullnorms, Fuzzy Sets Syst., 155, 3, 446-458 (2005) · Zbl 1077.03514 |
| [36] | Ruiz, D.; Torrens, J., Distributive idempotent uninorms, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 11, 4, 413-428 (2003) · Zbl 1074.03026 |
| [37] | Ruiz, D.; Torrens, J., Distributivity and conditional distributivity of a uninorm and a continuous t-conorm, IEEE Trans. Fuzzy Syst., 14, 2, 180-190 (2006) · Zbl 1355.03016 |
| [38] | Ruiz-Aguilera, D.; Torrens, J., Distributivity of strong implications over conjunctive and disjunctive uninorms, Kybernetika, 42, 319-336 (2005) · Zbl 1249.03030 |
| [39] | Ruiz-Aguilera, D.; Torrens, J., S- and R-implications from uninorms continuous in \(] 0, 1 [{}^2\) and their distributivity over uninorms, Fuzzy Sets Syst., 160, 832-852 (2009) · Zbl 1184.03014 |
| [40] | Ruiz-Aguilera, D.; Torrens, J.; De Baets, B.; Fodor, J., Some remarks on the characterization of idempotent uninorms, (Hüllermeier, E.; Kruse, R.; Hoffmann, F., IPMU 2010. IPMU 2010, LNAI, vol. 6178 (2010), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 425-434 |
| [41] | Su, Y.; Zong, W.; Liu, H., A note on “an extension of the migrative property for uninorms”, Inf. Sci., 281, 334-337 (2014) · Zbl 1355.68258 |
| [42] | Su, Y.; Zong, W.; Liu, H.; Zhang, X., On migrativity property for uninorms, Inf. Sci., 300, 114-123 (2015) · Zbl 1360.03069 |
| [43] | Su, Y.; Zong, W.; Liu, H., Migrativity property for uninorms, Fuzzy Sets Syst., 287, 213-226 (2016) · Zbl 1392.03053 |
| [44] | Su, Y.; Zong, W.; Liu, H.; Xue, P., The distributivity equations for semi-t-operators over uninorms, Fuzzy Sets Syst., 287, 172-183 (2016) · Zbl 1392.03054 |
| [45] | Su, Y.; Zong, W.; Liu, H., On distributivity equations for uninorms over semi-t-operators, Fuzzy Sets Syst., 299, 41-65 (2016) · Zbl 1374.03056 |
| [46] | Su, Y.; Liu, H.; Ruiz-Aguilera, D.; Riera, J. V.; Torrens, J., On the distributivity property for uninorms, Fuzzy Sets Syst., 287, 184-202 (2016) · Zbl 1392.03052 |
| [47] | Xie, A.; Liu, H., On the distributivity of uninorms over nullnorms, Fuzzy Sets Syst., 211, 62-72 (2013) · Zbl 1279.03047 |
| [48] | Yager, R. R.; Rybalov, A., Uninorm aggregation operators, Fuzzy Sets Syst., 80, 1, 111-120 (1996) · Zbl 0871.04007 |
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