A new insight of the distributivity for S-uninorms

Wang, Chun Yong; Duan, Wen Yu; Zhang, Bo

doi:10.22111/ijfs.2025.49247.8686

Document Type : Research Paper

Authors

¹ School of Mathematical Sciences, Shandong Normal University, Jinan, PR China

² School of Mathematics and Statistics, Shandong Normal University, Jinan, PR China

³ Foundational Courses Department, Wuhan Donghu College, Wuhan, PR China

https://doi.org/10.22111/ijfs.2025.49247.8686

Abstract

Although the scholars studied the distributivity for S-uninorms, at least, the underlying
uninorm of one S-uninorm in the distributivity equations was assumed to be in Umin except
the distributivity for S-uninorms over t-(co)norms. In this paper, we further characterize the
distributivity for S-uninorms, where the conjunctive underlying uninorms of the S-uninorms
in the distributivity equations are not fixed in Umin but arbitrary. Firstly, we discuss the
distributivity between S-uninorms. Secondly, we analyze the distributivity for S-uninorms
over T-uninorms. Moreover, we obtain the distributivity for T-uninorms over S-uninorms by
duality. Thirdly, we investigate the distributivity for S-uninorms over disjunctive uninorms.
Because S-uninorms in those distributivity equations are arbitrary, our results are extensions
of the previous results on the distributivity for S-uninorms.

Keywords

Main Subjects

Fuzzy Math

References

[1] J. Acz´el, Lectures on functional equations and their applications, Academic Press, New York, 1966. https://doi.
org/10.1016/s0076-5392(09)x6011-x

[2] M. Baczy´ nski, F. Qin, Some remarks on the distributive equation of fuzzy implication and the contrapositive symmetry for continuous, Archimedean t-norms, International Journal of Approximate Reasoning, 54 (2013), 290-296. http: //dx.doi.org/10.1016/j.ijar.2012.10.001

[3] J. Balasubramaniam, C. J. M. Rao, On the distributivity of implication operators over T and S norms, IEEE
Transactions on Fuzzy Systems, 12 (2004), 194-198. http://dx.doi.org/10.1109/TFUZZ.2004.825075

[4] G. Beliakov, A. Pradera, T. Calvo, Aggregation functions: A guide for practitioners, Springer, Berlin, Heidelberg,
2007. https://doi.org/10.1007/978-3-540-73721-6

[5] J. M. Benitez, J. L. Castro, I. Requena, Are artificial neural networks black boxes?, IEEE Transactions on Neural
Networks, 8 (1997), 1156-1163. http://dx.doi.org/10.1109/72.623216

[6] H. Bustince, M. Pagola, E. Barrenechea, Construction of fuzzy indices from fuzzy DI-subsethood measures: Applica
tion to the global comparison of images, Information Sciences, 177 (2007), 906-929. http://dx.doi.org/10.1016/
j.ins.2006.07.021

[7] T. Calvo, G. Mayor, R. Mesiar, Aggregation operators: New trends and applications, Springer, Berlin, Heidelberg,
2002. http://dx.doi.org/10.1007/978-3-7908-1787-4

[8] Z. Chen, A. Xie, Q. Yang, On the distributivity property for S-uninorms, Fuzzy Sets and Systems, 462 (2023),
108453. http://dx.doi.org/10.1016/j.fss.2022.12.009

[9] B. De Baets, Idempotent uninorms, European Journal of Operational Research, 118 (1999), 631-642. http://dx.
doi.org/10.1016/S0377-2217(98)00325-7

[10] B. De Baets, J. Fodor, Van Melle’s combining function in MYCIN is a representable uninorm: An alternative
proof, Fuzzy Sets and Systems, 104 (1999), 133-136. http://dx.doi.org/10.1016/S0165-0114(98)00265-6

[11] P. Dryga´s, Discussion of the structure of uninorms, Kybernetika, 41 (2005), 213-226. https://dml.cz/handle/
10338.dmlcz/135651

[12] P. Dryga´s, On the structure of continuous uninorms, Kybernetika, 43 (2007), 183-196. https://dml.cz/handle/
10338.dmlcz/135765

[13] P. Dryga´ s, On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums, Fuzzy Sets and
Systems, 161 (2010), 149-157. http://dx.doi.org/10.1016/j.fss.2009.09.017

[14] D. Dubois, J. Fodor, H. Prade, M. Roubens, Aggregation of decomposable measures with applications to utility
theory, Theory and Decision, 41 (1996), 59-95. http://dx.doi.org/10.1007/BF00134116

[15] D. Dubois, E. Pap, H. Prade, Hybrid probabilistic-possibilistic mixtures and utility functions, Preferences and
Decisions Under Incomplete Knowledge, Studies in Fuzziness and Soft Computing, 51, Springer-Verlag, (2000),
51-73. http://dx.doi.org/10.1007/978-3-7908-1848-2_4

[16] B. W. Fang, B. Q. Hu, Distributivity and conditional distributivity for S-uninorms, Fuzzy Sets and Systems, 372
(2019), 1-33. http://dx.doi.org/10.1016/j.fss.2018.11.003

[17] J. C. Fodor, R. R. Yager, A. Rybalov, Structure of uninorms, International Journal of Uncertainty, Fuzziness and
Knowledge-Based Systems, 5 (1997), 411-427. http://dx.doi.org/10.1142/S0218488597000312

[18] M. Grabisch, J. Marichal, R. Mesiar, E. Pap, Aggregations functions, Cambridge University Press, 2009. http:
//dx.doi.org/10.1017/CBO9781139644150

[19] S. K. Hu, Z. F. Li, The structure of continuous uni-norms, Fuzzy Sets and Systems, 124 (2001), 43-52. http:
//dx.doi.org/10.1016/S0165-0114(00)00044-0

[20] D. Joˇ ci´ c, I. ˇ Stajner-Papuga, Some implications of the restricted distributivity of aggregation operators with absorbing elements for utility theory, Fuzzy Sets and Systems, 291 (2016), 54-65. https://doi.org/10.1016/j.fss.2015.
06.003

[21] D. Joˇci´c, I. ˇ Stajner-Papuga, Distributivity and conditional distributivity for T-uninorms, Information Sciences, 424
(2018), 91-103. http://dx.doi.org/10.1016/j.ins.2017.09.065

[22] D. Joˇci´c, I. ˇ Stajner-Papuga, On distributivity and conditional distributivity of S-uninorms over uninorms, Acta
Polytechnica Hungarica, 16 (2019), 209-226. http://dx.doi.org/10.12700/APH.16.1.2019.1.11

[23] D. Joˇci´c, I. ˇ Stajner-Papuga, Corrigendum to “Distributivity and conditional distributivity for T-uninorms” [Infor
mation Sciences, (424) (2018) 91-103], Information Sciences, 568 (2021), 384-385. http://dx.doi.org/10.1016/
j.ins.2021.04.049

[24] M. Kalina, B. Mihailovi´c, M. ˘ Strboja, Bipolar ordered weighted qauasi-averages and induced bipolar ordered weighted averages: BIGOWA and IBIOWA operators, Iranian Journal of Fuzzy Systems, 22 (2025), 39-54. http://dx.doi.
org/10.22111/ijfs.2025.50487.8917

[25] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000. http://dx.
doi.org/10.1007/978-94-015-9540-7

[26] G. Li, H. W. Liu, Distributivity and conditional distributivity of a uninorm with continuous underlying operators
over a continuous t-conorm, Fuzzy Sets and Systems, 287 (2016), 154-171. http://dx.doi.org/10.1016/j.fss.
2015.01.019

[27] G. Li, H. W. Liu, On properties of uninorms locally internal on the boundary, Fuzzy Sets and Systems, 332 (2018),
116-128. http://dx.doi.org/10.1016/j.fss.2017.07.014

[28] G. Li, H. W. Liu, A note on “Distributivity and conditional distributivity of a uninorm with continuous underlying
operators over a continuous t-conorm”, Fuzzy Sets and Systems, 334 (2018), 126-131. http://dx.doi.org/10.
1016/j.fss.2017.04.016

[29] H. W. Liu, Distributivity and conditional distributivity of semi-uninorms over continuous t-conorms and t-norms,
Fuzzy Sets and Systems, 268 (2015), 27-43. http://dx.doi.org/10.1016/j.fss.2014.07.025

[30] J. Martin, G. Mayor, J. Torrens, On locally internal monotonic operations, Fuzzy Sets and Systems, 137 (2003),
27-42. http://dx.doi.org/10.1016/S0165-0114(02)00430-X

[31] M. Mas, G. Mayor, J. Torrens, The distributivity condition for uninorms and t-operators, Fuzzy Sets and Systems,
128 (2002), 209-225. http://dx.doi.org/10.1016/S0165-0114(01)00123-3

[32] M. Mas, R. Mesiar, M. Monserrat, J. Torrens, Aggregation operators with annihilator, International Journal of
General Systems, 34 (2005), 17-38. http://dx.doi.org/10.1080/03081070512331318347

[33] A. Mesiarov´a-Zem´ankov´a, Commutative, associative and non-decreasing functions continuous around diagonal,
Iranian Journal of Fuzzy Systems, 19 (2022), 31-48. https://doi.org/10.22111/IJFS.2022.6786

[34] M. Petr´ık, R. Mesiar, On the structure of special classes of uninorms, Fuzzy Sets and Systems, 240 (2014), 22-38.
https://doi.org/10.1016/j.fss.2013.09.013

[35] F. Qin, M. Baczy´nski, A. Xie, Distributivity equations of implications based on continuous triangular norms (I),
IEEE Transactions on Fuzzy Systems, 20 (2012), 153-167. http://dx.doi.org/10.1109/TFUZZ.2011.2171188

[36] F. Qin, L. Fu, A characterization of uninorms not internal on the boundary, Fuzzy Sets and Systems, 469 (2023),
108641. http://dx.doi.org/10.1016/j.fss.2023.108641

[37] F. Qin, L. Yang, Distributive equations of implications based on nilpotent triangular norms, International Journal
of Approximate Reasoning, 51 (2010), 984-992. http://dx.doi.org/10.1016/j.ijar.2010.07.005

[38] E. Rak, P. Dryga´s, Distributivity between uninorms, Journal of Electrical Engineering, 57 (2006), 35-38. https:
//www.researchgate.net/publication/265706173

[39] D. Ruiz-Aguilera, J. Torrens, Distributive idempotent uninorms, International Journal of Uncertainty, Fuzziness
and Knowledge-Based Systems, 11 (2003), 413-428. http://dx.doi.org/10.1142/S0218488503002168

[40] D. Ruiz-Aguilera, J. Torrens, Distributivity and conditional distributivity of uninorm and a continuous t-conorm,
IEEE Transactions on Fuzzy Systems, 14 (2006), 180-190. http://dx.doi.org/10.1109/TFUZZ.2005.864087

[41] D. Ruiz-Aguilera, J. Torrens, S- and R-implications from uninorms continuous in ]0,1[2 and their distributivity
over uninorms, Fuzzy Sets and Systems, 160 (2009), 832-852. https://doi.org/10.1016/j.fss.2008.05.015

[42] D. Ruiz-Aguilera, J. Torrens, B. De Baets, J. Fodor, Some remarks on the characterization of idempotent uni
norms, in: E. H¨ullermeier, R. Kruse, F. Hoffmann (Eds.), IPMU 2010, in: LNAI, vol.6178, Springer-Verlag, Berlin,
Heidelberg, (2010), 425-434. http://dx.doi.org/10.1007/978-3-642-14049-5_44

[43] W. Sander, J. Siedekum, Multiplication, distributivity and fuzzy-integral I, II, III, Kybernetika, 41 (2005), 397-422,
469-496, 497-518. https://dml.cz/handle/10338.dmlcz/135664,135670,135671

[44] Y. Su, H. W. Liu, J. V. Riera, D. Ruiz-Aguilera, J. Torrens, The distributivity equation for uninorms revisited,
Fuzzy Sets and Systems, 334 (2018), 1-23. http://dx.doi.org/10.1016/j.fss.2016.11.015

[45] Y. Su, H. W. Liu, D. Ruiz-Aguilera, J. V. Riera, J. Torrens, On the distributivity property for uninorms, Fuzzy
Sets and Systems, 287 (2016), 184-202. http://dx.doi.org/10.1016/j.fss.2015.06.023

[46] Y. Su, W. W. Zong, J. Wu, Distributivity and conditional distributivity for uninorms with continuous underlying
operators over a given continuous t-norm, IEEE Transactions on Fuzzy Systems, 29 (2021), 2239-2245. http:
//dx.doi.org/10.1109/TFUZZ.2020.2996378

[47] C. Y. Wang, L. Wan, B. Zhang, A note on distributivity and conditional distributivity for S-uninorms, Fuzzy Sets
and Systems, 408 (2021), 86-107. http://dx.doi.org/10.1016/j.fss.2020.04.007

[48] C. Y. Wang, P. Wang, B. Zhang, Distributivity for uninorms with noncontinuous underlying operators, Fuzzy Sets
and Systems, 462 (2023), 108403. http://dx.doi.org/10.1016/j.fss.2022.09.009

[49] A. Xie, Z. Chen, Q. Yang, A note on the distributivity for uninorms with noncontinuous underlying operators,
Fuzzy Sets and Systems, 467 (2023), 108508. http://dx.doi.org/10.1016/j.fss.2023.03.011

[50] R. R. Yager, Uninorms in fuzzy systems modeling, Fuzzy Sets and Systems, 122 (2001), 167-175. http://dx.doi.
org/10.1016/S0165-0114(00)00027-0

[51] R. R. Yager, V. Kreinovich, On the relation between two approaches to combining evidence: Ordered Abelian groups
and uninorms, Journal of Intelligent and Fuzzy Systems: Applications in Engineering and Technology, 14 (2003),
7-12. http://dx.doi.org/10.3233/IFS-2003-00186

[52] R. R. Yager, V. Kreinovich, Universal approximation theorem for uninorm-based fuzzy systems modeling, Fuzzy
Sets and Systems, 140 (2003), 331-339. http://dx.doi.org/10.1016/S0165-0114(02)00521-3

[53] R. R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80 (1996), 111-120. http:
//dx.doi.org/10.1016/0165-0114(95)00133-6

[54] P. Yan, G. Chen, Discovering a cover set of ARsi with hierarchy from quantitative databases, Information Sciences,
173 (2005), 319-336. http://dx.doi.org/10.1016/j.ins.2005.03.003

[55] F. X. Zhang, H. W. Liu, On a new class of implications: (g,u)-implications and the distributive equations, Interna
tional Journal of Approximate Reasoning, 54 (2013), 1049-1065. http://dx.doi.org/10.1016/j.ijar.2013.04.
008

[56] B. Zhang, L. Wan, C. Y. Wang, On the distributivity property of uninorms locally internal on the boundary over
noncontinuous t-(co)norms, Iranian Journal of Fuzzy Systems, 20 (2023), 137-152. http://dx.doi.org/10.22111/
IJFS.2023.43521.7653

A new insight of the distributivity for S-uninorms

References

References

Volume 22, Issue 4July and August 2025Pages 17-35

Volume 22, Issue 4
July and August 2025
Pages 17-35