New constructions and characterizations of semi-t-operators on bounded lattices

[1] G. Birkhoff, Lattice theory, American Mathematical Society Colloquium Publishers, Providence, RI, 1948. https: //doi.org/10.1090/coll/025

[2] T. Calvo, B. De Baets, J. Fodor, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets and Systems, 120(3) (2001), 385-394. https://doi.org/10.1016/S0165-0114(99)00125-6

[3] T. Calvo, A. Fraile, G. Mayor, Algunes consideracions sobre connectius generalitzats, in: Actes del VI Congres Catala de Logica, Barcelona, (1986), 45-46.

[4] Y. F. Cheng, B. Zhao, On the modularity equation for overlap (grouping) functions and semi-t-operators, Iranian Journal of Fuzzy Systems, 20(4) (2023), 121-136. https://doi.org/10.22111/ijfs.2023.43268.7603

[5] P. Drygaś, Distributivity between semi-t-operators and semi-nullnorms, Fuzzy Sets and Systems, 264 (2015), 100 109. https://doi.org/10.1016/j.fss.2014.09.003

[6] P. Drygaś, E. Rak, Distributivity equations in the class of semi-t-operators, Fuzzy Sets and Systems, 291 (2016), 66-81. https://doi.org/10.1016/j.fss.2015.01.015

[7] D. Dubois, H. Prade, Fundamentals of fuzzy sets, Springer New York, NY, 2000. https://doi.org/10.1007/ 978-1-4615-4429-6

[8] B. W. Fang, B. Q. Hu, Semi-t-operators on bounded lattices, Information Sciences, 490 (2019), 191-209. https: //doi.org/10.1016/j.ins.2019.03.077

[9] X. J. Hua, B. W. Fang, Constructions of semi-t-operators on bounded lattices by semi-beam operations, Fuzzy Sets and Systems, 462 (2023), 108439. https://doi.org/10.1016/j.fss.2022.11.010

[10] D. Jočić, P. Drygaś, I. Štajner-Papuga, Left and right distributivity equations in the class of semi-t-operators, Fuzzy Sets and Systems, 372 (2019), 62-80. https://doi.org/10.1016/j.fss.2018.10.016

[11] D. Jočić, I. Štajner -Papuga, Conditional distributivity of continuous semi-t-operators over disjunctive uninorms with continuous underlying t-norms and t-conorms, Fuzzy Sets and Systems, 423 (2021), 89-106. https://doi. org/10.1016/j.fss.2020.10.012

[12] Y. L. Jun, Q. Feng, Migrativity properties of 2-uninorms over semi-t-operators, Kybernetika, 58(3) (2022), 354-375. https://doi.org/10.14736/kyb-2022-3-0354

[13] G. J. Klir, B. Yuan, Fuzzy sets and fuzzy logic: Theory and application, Prentice Hall PTR, Upper Saddle River, New Jersey, 1995. https://doi.org/10.1021/ci950144a

 

[14] X. R. Li, L. Q. Li, C. Z. Jia, Weighted multi-granularity fuzzy probabilistic rough set based on semi-overlapping function and its application in three-way decision, Engineering Applications of Artificial Intelligence, 161(Part A) (2025), 112023. https://doi.org/10.1016/j.engappai.2025.112023

[15] W. H. Li, F. Qin, Conditional distributivity equation for uninorms with continuous underlying operators, IEEE Transactions on Fuzzy Systems, 28(8) (2019), 1664-1678.

[16] J. L. Marichal, On the associativity functional equation, Fuzzy Sets and Systems, 114(3) (2000), 381-389. https: //doi.org/10.1016/S0165-0114(98)00332-7

[17] M. Mas, G. Mayor, J. Torrens, The distributivity condition for uninorms and t-operators, Fuzzy Sets and Systems, 128(2) (2002), 209-225. https://doi.org/10.1016/S0165-0114(01)00123-3

[18] P. Orlovs, S. Asmuss, General aggregation operators based on a fuzzy equivalence relation in the context of approximate systems, Fuzzy Sets and Systems, 291 (2016), 114-131. https://doi.org/10.1016/j.fss.2015.08.015

[19] F. Qin, Distributivity between semi-uninorms and semi-t-operators, Fuzzy Sets and Systems, 299 (2016), 66-88. https://doi.org/10.1016/j.fss.2015.10.012

[20] Z. Q. Shi, L. Q. Li, S. R. Xie, J. L. Xie, The variable precision fuzzy rough set based on overlap and grouping functions with double weight method to MADM, Applied Intelligence, 54(17) (2024), 7696-7715. https://doi.org/ 10.1007/s10489-024-05554-3

[21] Y. Su, H. W. Liu, J. V. Riera, et al., The migrativity equation for uninorms revisited, Fuzzy Sets and Systems, 323 (2017), 56-78. https://doi.org/10.1016/j.fss.2017.03.003

[22] Y. Su, W. Zong, H. W. Liu, On distributivity equations for uninorms over semi-t-operators, Fuzzy Sets and Systems, 299 (2016), 41-65. https://doi.org/10.1016/j.fss.2015.08.001

[23] Y. Su, W. Zong, H. W. Liu, P. Xue, Migrativity property for uninorms and semi-t-operators, Information Sciences, 325 (2015), 455-465. https://doi.org/10.1016/j.ins.2015.07.030

[24] X. R. Sun, H. W. Liu, The left (right) distributivity of semi-t-operators over 2-uninorms, Iranian Journal of Fuzzy Systems, 17(3) (2020), 103-116. https://doi.org/10.22111/ijfs.2020.5352

[25] C. Torres-Blanc, S. Cubillo, P. Hernández, Aggregation operators on type-2 fuzzy sets, Fuzzy Sets and Systems, 324 (2017), 74-90. https://doi.org/10.1016/j.fss.2017.03.015

[26] Y. M. Wang, Z. B. Li, H. W. Liu, On the structure of semi-t-operators on bounded lattices, Fuzzy Sets and Systems, 395 (2020), 130-148. https://doi.org/10.1016/j.fss.2019.06.018

[27] Y. M. Wang, Z. B. Li, H. W. Liu, Semi-t-operators and idempotent semi-t-operators on bounded lattices, Iranian Journal of Fuzzy Systems, 18(2) (2021), 109-125. https://doi.org/10.22111/ijfs.2021.5918

[28] C. Y. Wang, L. Wan, B. Zhang, Distributivity and conditional distributivity of semi-t-operators over S-uninorms, Fuzzy Sets and Systems, 441 (2022), 224-240. https://doi.org/10.1016/j.fss.2021.09.020

[29] X. P. Wang, C. L. Zhu, Construction of semi-t-operators on bounded lattices by order-preserving mappings, Fuzzy Sets and Systems, 462 (2023), 108424. https://doi.org/10.1016/j.fss.2022.10.016

[30] K. Y. Xiao, Y. Su, W. Zong, The modularity condition for semi-t-operators and Bi-uninorms, Fuzzy Sets and Systems, 528 (2026), 109729. https://doi.org/10.1016/j.fss.2025.109729

[31] H. Zhan, Y. M. Wang, H. W. Liu, The modularity condition for semi-t-operators and semi-uninorms, Fuzzy Sets and Systems, 334 (2018), 36-59. https://doi.org/10.1016/j.fss.2017.05.025

[32] Y. Q. Zhang, H. W. Liu, Y. Y. Zhao, New classes of semi-t-operators on bounded lattices, Information Sciences, 718 (2025), 122380. https://doi.org/10.1016/j.ins.2025.122380

[33] Y. Q. Zhang, Y. M. Wang, H. W. Liu, New constructions of semi-t-operators on bounded lattices by order-preserving functions, Fuzzy Sets and Systems, 473 (2023), 108714. https://doi.org/10.1016/j.fss.2023.108714

[34] X. H. Zhang, F. X. Zhang, M. P. Zhang, M. E. Qin, Several findings about the weak dominance between t-norms and semi-t-operators, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 33(8) (2025), 985-1002. https://doi.org/10.1142/S0218488525500291

[35] Y. Y. Zhao, H. W. Liu, The modularity equation for semi-t-operators and T-uninorms, International Journal of Approximate Reasoning, 146 (2022), 106-118. https://doi.org/10.1016/j.ijar.2022.04.005

[36] B. Zhao, J. Q. Shi, Distributivity of uni-nullnorms over semi-t-operators, Aequationes Mathematicae, 99 (2025), 2047-2071. https://doi.org/10.1007/s00010-025-01229-7

[37] C. L. Zhu, X. P. Wang, Alternative approaches for generating semi-t-operators on bounded lattices, Fuzzy Sets and Systems, 439 (2022), 75-88. https://doi.org/10.1016/j.fss.2021.09.003

[38] H. Zong, F. X. Zhang, X. X. Liu, Y. Song, Distributivity and conditional distributivity equations for S-uninorms over T-uninorms, Aequationes Mathematicae, 100(2) (2026), 1-15. https://doi.org/10.1007/s00010-025-01241-x