Interval-valued fuzzy logical connectives with respect to admissible orders

Document Type : Original Manuscript


1 Southwest Jiaotong University

2 School of Economics Information Engineering, Southwestern University of Finance and Economics, Chengdu 611130, Sichuan, China


In this work, we examine several widely-used interval-valued fuzzy logical connectives with respect to admissible orders. We are concerned with interval-valued fuzzy negations, automorphisms, fuzzy implications and aggregation functions with respect to $K_{\alpha,\beta}$ orders and arbitrary intervals on $L([0, 1])$. We also make a discussion of width-preserving interval-valued fuzzy equivalence functions and fuzzy dissimilarity functions with respect to arbitrary admissible orders and the intervals with the same width on $L([0, 1])$. Then we bring some approaches to constructing the proposed interval-valued fuzzy logical connectives with respect to admissible orders.


Main Subjects

[1] M. M. Alsolmia, A. A. Bakery, Multiplication mappings on a new stochastic space of a sequence of fuzzy functions, Journal of Mathematics and Computer Science, 29 (2023), 306-316.
[2] M. J. Asiain, H. Bustince, R. Mesiar, A. Koles´arov´a, Z. Tak´a˘c, Negations with respect to admissible orders in the interval-valued fuzzy set theory, IEEE Transactions on Fuzzy Systems, 26(2) (2018), 556-568.
[3] M. Baczy´nski, B. Jayaram, Fuzzy implications, Springer, Berlin, Heidelberg, 2008.
[4] B. De Baets, Idempotent uninorms, European Journal of Operational Research, 118(3) (1999), 631-642.
[5] B. De Baets, J. Fodor, A single-point characterization of representable uninorms, Fuzzy Sets and Systems, 202 (2012), 89-99.
[6] B. C. Bedregal, On interval fuzzy negations, Fuzzy Sets and Systems, 161(17) (2010), 2290-2313.
[7] B. C. Bedregal, G. P. Dimuro, R. H. N. Santiago, R. H. S. Reiser, On interval fuzzy S-implications, Information Sciences, 180(8) (2010), 1373-1389.
[8] B. C. Bedregal, A. Takahashi, The best interval representation of t-norms and automorphisms, Fuzzy Sets and Systems, 157(24) (2006), 3220-3230.
[9] A. Bobin, P. Thangaraja, E. Prabu, V. Chinnadurai, Interval-valued picture fuzzy hypersoft TOPSIS method based on correlation coefficient, Journal of Mathematics and Computer Science, 27 (2022), 142-163.
[10] H. Bustince, E. Barrenechea, M. Pagola, Restricted equivalence functions, Fuzzy Sets and Systems, 157(17) (2006), 2333-2346.
[11] H. Bustince, P. Burillo, F. Soria, Automorphisms, negations and implication operators, Fuzzy Sets and Systems, 134 (2003), 209-229.
[12] H. Bustince, J. Fernandez, A. Koles´arov´a, R. Mesiar, Generation of linear orders for intervals by means of aggre-gation functions, Fuzzy Sets and Systems, 220 (2013), 69-77.
[13] H. Bustince, M. Galar, B. Bedregal, A. Koles´arov´a, R. Mesiar, A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications, IEEE Transactions on Fuzzy Systems, 21(6) (2013), 1150-1162.
[14] H. Bustince, C. Marco-Detchart, J. Fernandez, C. Wagner, J. M. Garibaldi, Z. Tak´a˘c, Similarity between interval-valued fuzzy sets taking into account the width of the intervals and admissible orders, Fuzzy Sets and Systems, 390 (2020), 23-47.
[15] J. R. Castro, O. Castillo, P. Melin, A. Rodr´ıguez-D´ıaz, A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks, Information Sciences, 179(13) (2009), 2175-2193.
[16] J. Fodor, M. Roubens, Fuzzy preference modelling and multicriteria decision support, Kluwer Academic Publishers, Dordrecht, 1994.
[17] J. Fodor, J. Torrens, An overview of fuzzy logic connectives on the unit interval, Fuzzy Sets and Systems, 281 (2015), 183-187.
[18] I. Grattan-Guiness, Fuzzy membership mapped onto interval and many-valued quantities, Zeitschrift f¨ur Mathematische Logik und Grundlagen der Mathematik, 22(1) (1976), 149-160.
[19] K. U. Jahn, Intervall-wertige mengen, Mathematische Nachrichten, 68(1) (1975), 115-132.
[20] D. Joˇci´c, I. ˇStajner-Papuga, Distributivity between 2-uninorms and Mayor's aggregation operators, Iranian Journal of Fuzzy Systems, 19(6) (2022), 13-27.
[21] A. Jurio, M. Pagola, D. Paternain, C. Lopez-Molina, P. Melo-Pinto, Interval-valued restricted equivalence func-tions applied on Clustering Techniques, in: 13th International Fuzzy Systems Association World Congress and 6th European Society for Fuzzy Logic and Technology Conference, Portugal, (2009), 831-836.
[22] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer, Dordrecht, 2000.
[23] E. P. Klement, M. Navara, A survey on di erent triangular norm-based fuzzy logics, Fuzzy Sets and Systems, 101(2) (1999), 241-251.
[24] F. Kouchakinejad, A. Stupˇnanov´a, A. ˇSeliga, A note on divisible discrete triangular norms, Iranian Journal of Fuzzy Systems, 19(1) (2022), 73-81.
[25] D. C. Li, Y. M. Li, Y. J. Xie, Interval-valued fuzzy inference, Information Sciences, 181(20) (2011), 4754-4764.
[26] Y. F. Li, K. Y. Qin, X. X. He, Robustness of fuzzy connectives and fuzzy reasoning, Fuzzy Sets and Systems, 225 (2013), 93-105.
[27] Y. F. Li, K. Y. Qin, X. X. He, Dissimilarity functions and divergence measures between fuzzy sets, Information Sciences, 288 (2014), 15-26.
[28] Y. F. Li, K. Y. Qin, X. X. He, D. Meng, Three constructive methods for the definition of interval-valued fuzzy equivalencies, Fuzzy Sets and Systems, 322 (2017), 70-85.
[29] O. Linda, M. Manic, Interval type-2 fuzzy voter design for fault tolerant systems, Information Sciences, 181(14)(2011), 2933-2950.
[30] R. Mesiar, M. Komorn´ıkov´a, Aggregation functions on bounded posets, in: C. Cornelis, et al. (Eds.), 35 Years of Fuzzy Sets Theory, in: Studies in Fuzziness and Soft Computing, 261, Springer, Berlin, (2010), 3-17.
[31] R. Prasertpong, A. Iampan, Approximation approaches for rough hypersoft sets based on hesitant bipolar-valued fuzzy hypersoft relations on semigroups, Journal of Mathematics and Computer Science, 28 (2023), 85-122.
[32] R. Sambuc, ϕ- floues. Application l'aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Universit´e Marseille, Marseille, 1975.
[33] F. Santana, B. Bedregal, P. Viana, H. Bustince, On admissible orders over closed subintervals of [0, 1], Fuzzy Sets and Systems, 399 (2020), 44-54.
[34] L. Verma, R. Meher, Z. Avazzadeh, O. Nikan, Solution for generalized fuzzy fractional Kortewege-de Varies equa-tion using a robust fuzzy double parametric approach, Journal of Ocean Engineering and Science, (2022). DOI: 10.1016/j.joes.2022.04.026.
[35] H. Wang, Z. S. Xu, Admissible orders of typical hesitant fuzzy elements and their application in ordered information fusion in multi-criteria decision making, Information Fusion, 29 (2016), 98-104.
[36] H. Wang, Z. S. Xu, Total orders of extended hesitant fuzzy linguistic term sets: Definitions, generations and applications, Knowledge-Based Systems, 107 (2016), 142-154.
[37] Z. S. Xu, R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General Systems, 35(4) (2006), 417-433.
[38] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, 8(3) (1975), 199-249.
[39] H. Zapata, H. Bustince, S. Montes, et al., Interval-valued implications and interval-valued strong equality index with admissible orders, International Journal of Approximate Reasoning, 88 (2017), 91-109.