Methods for obtaining uninorms on some special classes of bounded lattices

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey

2 STU Bratislava

Abstract

In this article, we go on to discuss the structure of uninorms on bounded lattices. We suggest two techniques to yield  uninorms with some constraints on the identity element by applying that the t-norms and t-conorms are always present on the considered bounded lattices. These techniques ensure new approaches for getting idempotent uninorms on bounded lattices when regarding infimum t-norm and supremum t-conorm. Furthermore, we display the distinctness between our new construction techniques and the published ones. 

Keywords

References

[1] C. Alsina, M. J. Frank, B. Schweizer, Associative functions: Triangular norms and copulas, World Scientific, Singapore, 2006.
[2] E. Aşıcı, R. Mesiar, On the construction of uninorms on bounded lattices, Fuzzy Sets and Systems, 408 (2021), 65-85.
[3] E. Aşıcı, R. Mesiar, On generating uninorms on some special classes of bounded lattices, Fuzzy Sets and Systems, 439 (2022), 102-125.
[4] G. Birkhoff, Lattice theory, American Mathematical Society Colloquium Publishers, Providence, RI, 1967.
[5] S. Bodjanova, M. Kalina, Construction of uninorms on bounded lattices, in: IEEE 12th International Symposium on Intelligent Systems and Informatics, SISY 2014, Subotica, Serbia, 11-13 September, 2014.
[6] S. Bodjanova, M. Kalina, Uninorms on bounded lattices-recent development, in: J. Kacprzyk (Ed.), Advances in Fuzzy Logic and Technology 2017, IWIFSGN 2017, EUSFLAT 2017, in: Advances in Intelligent Systems and Computing, 641, Springer, Cham, (2018), 224-234.
[7] D. Butnariu, E. P. Klement, Triangular norm-based measures and games with fuzzy coalitions, Kluwer Academic Publishers, Dordrecht, 1993.
[8] G. D. Çaylı, Alternative approaches for generating uninorms on bounded lattice, Information Sciences, 488 (2019), 111-139.
[9] G. D. Çaylı, New methods to construct uninorms on bounded lattices, International Journal of Approximate Reasoning, 115 (2019), 254-264.
[10] G. D. Çaylı, Uninorms on bounded lattices with the underlying t-norms and t-conorms, Fuzzy Sets and Systems, 395 (2020), 107-129.
[11] G. D. Çaylı, On generating of t-norms and t-conorms on bounded lattices, International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, 28 (2020), 807-835.
[12] G. D. Çaylı, F. Kara¸cal, R. Mesiar, On internal and locally internal uninorms on bounded lattices, International Journal of General Systems, 48 (2019), 235-259.
[13] Y. Dan, B. Q. Hu, A new structure for uninorms on bounded lattices, Fuzzy Sets and Systems, 386 (2020), 77-94.
[14] Y. Dan, B. Q. Hu, J. Qiao, New constructions of uninorms on bounded lattices, International Journal of Approximate Reasoning, 110 (2019), 185-209.
[15] B. De Baets, Idempotent uninorms, European Journal of Operational Research, 118 (1999), 631-642.
[16] B. De Baets, J. Fodor, Van Melle’s combining function in MYCIN is a representable uninorm: An alternative proof, Fuzzy Sets and Systems, 104(1) (1999), 133-136.
[17] B. De Baets, J. Fodor, D. Ruiz-Aguilera, J. Torrens, Idempotent uninorms on finite ordinal scales, International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, 17(1) (2009), 1-14.
[18] J. Drewniak, P. Drygaś, On a class of uninorms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10 (2002), 5-10.
[19] P. Drygaś, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and Systems, 161 (2010), 149-157.
[20] P. Drygaś, E. Rak, Distributivity equation in the class of 2-uninorms, Fuzzy Sets and Systems, 291 (2016), 82-97.
[21] D. Dubois, H. Prade, A review of fuzzy set aggregation connective, Information Sciences, 36 (1985), 85-121.
[22] D. Dubois, H. Prade, Fundamentals of fuzzy sets, Kluwer Academic Publisher, Boston, 2000.
[23] A. Dvořák, M. Holčapek, Ordinal sums of t-norms and t-conorms on bounded lattices, In: R. Halaś et al. (eds) New Trends in Aggregation Theory. AGOP 2019. Advances in Intelligent Systems and Computing, Springer, Cham, Vol. 981 (2019), 289-301.
[24] A. Dvořák, M. Holčapek, New construction of an ordinal sum of t-norms and t-conorms on bounded lattices, Information Sciences, 515 (2020), 116-131.
[25] J. Fodor, R. R. Yager, A. Rybalov, Structure of uninorms, International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, 5 (1997), 411-427.
[26] M. González-Hidalgo, A. Mir-Torres, D. Ruiz-Aguilera, J. Torrens, Edge-images using a uninorm-based fuzzy mathematical morphology: Opening and closing, in: J. Tavares, N. Jorge (Ed.), Advances in Computational Vision and Medical Image Processing, number 13 in Computational Methods in Applied Sciences, Springer, Netherlands, (2009), 137-157.
[27] M. González-Hidalgo, A. Mir-Torres, D. Ruiz-Aguilera, J. Torrens, Image analysis applications of morphological operators based on uninorms, in: Proceedings of the IFSA-EUSFLAT Conference, Lisbon, Portugal, (2009), 630- 635.
[28] M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap, Aggregation functions, Cambridge University Press, Cambridge, 2009.
[29] M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap, Aggregation functions: Construction methods, conjunctive, disjunctive and mixed classes, Information Sciences, 181 (2011), 23-43.
[30] P. Hájek, T. Havránek, R. Jiroušek, Uncertain information processing in expert systems, CRC Press, Boca Raton, 1992.
[31] W. Homenda, A. Jastrzebska, W. Pedrycz, Multicriteria decision making inspired by human cognitive processes, Applied Mathematics and Computation, 290 (2016), 392-411.
[32] F. Kara¸cal, R. Mesiar, Uninorms on bounded lattices, Fuzzy Sets and Systems, 261 (2015), 33-43.
[33] F. Kara¸cal, Ü. Ertuğrul, R. Mesiar, Characterization of uninorms on bounded lattices, Fuzzy Sets and Systems, 308 (2017), 54-71.
[34] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Academic Publications, Dordrecht, 2000.
[35] G. Li, H. W. Liu, J. Fodor, On weakly smooth uninorms on finite chain, International Journal of Intelligent Systems, 30 (2015), 421-440.
[36] J. Medina, Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices, Fuzzy Sets and Systems, 202 (2012), 75-88.
[37] K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences USA, 28 (1942), 535-537.
[38] R. Mesiar, A. Kolesárová, A. Stupňanová, Quo vadis aggregation, International Journal of General Systems, 47(2) (2018), 97-117.
[39] A. Mesiarová-Zemánková, Ordinal sums of representable uninorms, Fuzzy Sets and Systems, 308 (2017), 42-53.
[40] A. Mesiarová-Zemánková, Characterization of idempotent n-uninorms, Fuzzy Sets and Systems, 427 (2022), 1-22.
[41] E. S. Palmeira, B. C. Bedregal, Extension of fuzzy logic operators defined on bounded lattices via retractions, Computers and Mathematics with Applications, 63 (2012), 1026-1038.
[42] W. Pedrycz, K. Hirota, Uninorm-based logic neurons as adaptive and inter-pretable processing constructs, Soft Computing, 11(1) (2016), 41-52.
[43] S. Saminger, On ordinal sums of triangular norms on bounded lattices, Fuzzy Sets and Systems, 157 (2006), 1403-1416.
[44] B. Schweizer, A. Sklar, Associative functions and statistical triangular inequalities, Publicationes Mathematicae Debrecen, 8 (1961), 169-186.
[45] B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland, New York, 1983.
[46] X. R. Sun, H. W. Liu, Further characterization of uninorms on bounded lattices, Fuzzy Sets and Systems, 427 (2022), 96-108.
[47] X. Wu, G. Chen, Answering an open problem on t-norms for type-2 fuzzy sets, Information Sciences, 522 (2020), 124-133.
[48] A. Xie, S. Li, On constructing the largest and smallest uninorms on bounded lattices, Fuzzy Sets and Systems, 386 (2020), 95-104.
[49] R. R. Yager, Misrepresentations and challenges: A response to Elkan, IEEE Expert, (1994), 41-42.
[50] R. R. Yager, Uninorms in fuzzy systems modeling, Fuzzy Sets and Systems, 122(1) (2001), 167-175.
[51] R. R. Yager, A. Rybalov, Uninorms aggregation operators, Fuzzy Sets and Systems, 80 (1996), 111-120.
[52] B. Zhao, T. Wu, Some further results about uninorms on bounded lattices, International Journal of Approximate Reasoning, 130 (2021), 22-49.
[53] H. P. Zhang, M. Wu, Z. Wang, Y. Ouyang, B. De Baets, A characterization of the classes Umin and Umax of uninorms on a bounded lattice, Fuzzy Sets and Systems, 423 (2021), 107-121.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 7
Invited Paper
November and December 2023
Pages 111-126

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