A new insight of the distributivity for S-uninorms

[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