| Title:
|
On the distributivity equation for uni-nullnorms (English) |
| Author:
|
Wang, Ya-Ming |
| Author:
|
Liu, Hua-Wen |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
55 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
24-43 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A uni-nullnorm is a special case of 2-uninorms obtained by letting a uninorm and a nullnorm share the same underlying t-conorm. This paper is mainly devoted to solving the distributivity equation between uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms and some binary operators, such as, continuous t-norms, continuous t-conorms, uninorms, and nullnorms. The new results differ from the previous ones about the distributivity in the class of 2-uninorms, which have not yet been fully characterized. (English) |
| Keyword:
|
fuzzy connectives |
| Keyword:
|
uni-nullnorms |
| Keyword:
|
T-norms |
| Keyword:
|
T-conorms |
| Keyword:
|
nullnorms |
| Keyword:
|
uninorms |
| Keyword:
|
distributivity equation |
| MSC:
|
46F10 |
| MSC:
|
62E86 |
| idZBL:
|
Zbl 07088877 |
| idMR:
|
MR3935413 |
| DOI:
|
10.14736/kyb-2019-1-0024 |
| . |
| Date available:
|
2019-05-07T11:03:40Z |
| Last updated:
|
2020-02-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147704 |
| . |
| Reference:
|
[1] Aczél, J.: Lectures on Functional Equations and Their Applications..Academia, New York 1966. Zbl 0139.09301, MR 0208210, 10.1002/zamm.19670470321 |
| Reference:
|
[2] Akella, P.: Structure of n-uninorms..Fuzzy Sets Syst. 158 (2007), 1631-1651. MR 2341328, 10.1016/j.fss.2007.02.015 |
| Reference:
|
[3] Calvo, T., Baets, B. De, Fodor, J. C.: The functional equations of Frank and Alsina for uninorms and nullnorms..Fuzzy Sets Syst. 120 (2001), 385-394. Zbl 0977.03026, MR 1829256, 10.1016/s0165-0114(99)00125-6 |
| Reference:
|
[4] Baets, B. De: Idempotent uninorms..Eur. J. Oper. Res. 118 (1999), 631-642. Zbl 1178.03070, 10.1016/s0377-2217(98)00325-7 |
| Reference:
|
[5] Drygaś, P., Rak, E.: Distributivity equation in the class of 2-uninorms..Fuzzy Sets Syst. 291 (2015), 82-97. MR 3463655, 10.1016/j.fss.2015.02.014 |
| Reference:
|
[6] Fechner, W., Rak, E., Zedam, L.: The modularity law in some classes of aggregation operators..Fuzzy Sets Syst. 332 (2018), 56-73. MR 3732249, 10.1016/j.fss.2017.03.010 |
| Reference:
|
[7] Fodor, J. C., Yager, R. R., Rybalov, A.: Structure of uninorms..Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 5 (1997), 411-427. Zbl 1232.03015, MR 1471619, 10.1142/s0218488597000312 |
| Reference:
|
[8] Fodor, J. C., Baets, B. De: A single-point characterization of representable uninorms..Fuzzy Sets Syst. 202 (2012), 89-99. Zbl 1268.03027, MR 2934788, 10.1016/j.fss.2011.12.001 |
| Reference:
|
[9] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms..Kluwer, Dordrecht 2000. Zbl 1087.20041, MR 1790096, 10.1007/978-94-015-9540-7 |
| Reference:
|
[10] Li, G., Liu, H. W., Fodor, J.: Single-point characterization of uninorms with nilpotent underlying t-norm and t-conorm..Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 22 (2014), 591-604. MR 3252143, 10.1142/s0218488514500299 |
| Reference:
|
[11] Li, G., Liu, H. W.: Distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t-conorm..Fuzzy Sets Syst. 287 (2016), 154-171. MR 3447024, 10.1016/j.fss.2015.01.019 |
| Reference:
|
[12] Mas, M., Mayor, G., Torrens, J.: T-operators..Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 7 (1999), 31-50. MR 1691482, 10.1142/s0218488599000039 |
| Reference:
|
[13] Mas, M., Mayor, G., Torrens, J.: The distributivity condition for uninorms and t-operators..Fuzzy Sets Syst. 128 (2002), 209-225. Zbl 1005.03047, MR 1908427, 10.1016/s0165-0114(01)00123-3 |
| Reference:
|
[14] Mas, M., Mesiar, R., Monserat, M., Torrens, J.: Aggregation operations with annihilator..Internat, J. Gen. Syst. 34 (2015), 17-38. MR 2179736, 10.1080/03081070512331318347 |
| Reference:
|
[15] Min, Y. M., Qin, F.: The distributivity for semi-nullnorms over 2-uninorms. (In Chinese.).J. Jiangxi Normal Univ. (Nature Science) 40 (2016), 3, 263267. |
| Reference:
|
[16] Qin, F., Zhao, B.: The distributive equations for idempotent uninorms and nullnorms..Fuzzy Sets Syst. 155 (2005), 446-458. Zbl 1077.03514, MR 2181001, 10.1016/j.fss.2005.04.010 |
| Reference:
|
[17] Qin, F.: Distributivity between semi-uninorms and semi-t-operators..Fuzzy Sets Syst. 299 (2015), 66-88. MR 3510877, 10.1016/j.fss.2015.10.012 |
| Reference:
|
[18] Rak, E.: Distributivity equation for nullnorms..J. Electr. Eng. 56 (2005), 53-55. 10.1109/t-aiee.1936.5057143 |
| Reference:
|
[19] Rak, E., Drygaś, P.: Distributivity equation between uninorms..J. Electr. Engrg. 57 (2006), 35-38. |
| Reference:
|
[20] Rak, E.: Some remarks about distributivity equation between uninorms..J. Electr. Engrg. 58 (2007), 41-42. |
| Reference:
|
[21] Ruiz-Aguilera, D., Torrens, J.: Distributivity of strong implications over conjunctive and disjunctive uninorms..Kybernetika 42 (2006), 319-336. Zbl 1249.03030, MR 2253392 |
| Reference:
|
[22] Ruiz, D., Torrens, J.: Distributivity and conditional distributivity of a uninorm and a continuous t-conorm..IEEE Trans. Fuzzy Syst. 14 (2006), 180-190. 10.1109/tfuzz.2005.864087 |
| Reference:
|
[23] Su, Y., Zong, W., Liu, H. W.: On distributivity equations for uninorms over semi-t-operators..Fuzzy Sets Syst. 287 (2015), 41-65. MR 3510876, 10.1016/j.fss.2015.08.001 |
| Reference:
|
[24] Su, Y., Zong, W., Liu, H. W., Xue, P.: On distributivity equations for semi-t-operators over uninorms..Fuzzy Sets Syst. 287 (2016), 172-183. MR 3447025, 10.1016/j.fss.2015.03.009 |
| Reference:
|
[25] Su, Y., Liu, H. W., Ruiz-Aguilera, D., Riera, J. Vicente, Torrens, J.: On the distributivity property for uninorms..Fuzzy Sets Syst. 287 (2016), 184-202. MR 3447026, 10.1016/j.fss.2015.06.023 |
| Reference:
|
[26] Sun, F., Wang, X. P., Qu, X. B.: Uni-nullnorms and null-uninorms..J. Intell. Fuzzy Syst. 32 (2017), 1969-1981. 10.3233/jifs-161495 |
| Reference:
|
[27] Sun, F., Wang, X. P., Qu, X. B.: Characterizations of uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms..Fuzzy Sets Syst. 334 (2018), 24-35. MR 3742230, 10.1016/j.fss.2017.03.001 |
| Reference:
|
[28] Wang, Y. M., Qin, F.: Distributivity for 2-uninorms over semi-uninorms..Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 25 (2017), 317-345. MR 3631939, 10.1142/s0218488517500131 |
| Reference:
|
[29] Xie, A. F., Liu, H. W.: On the distributivity of uninorms over nullnorms..Fuzzy Sets Syst. 211 (2013), 62-72. Zbl 1279.03047, MR 2991797, 10.1016/j.fss.2012.05.008 |
| Reference:
|
[30] Yager, R. R., Rybalov, A.: Uninorm aggregation operators..Fuzzy Sets Syst. 80 (1996), 111-120. Zbl 0871.04007, MR 1389951, 10.1016/0165-0114(95)00133-6 |
| Reference:
|
[31] Yager, R. R.: Uninorms in fuzzy system modeling..Fuzzy Sets Syst. 122 (2001), 167-175. MR 1839955, 10.1016/s0165-0114(00)00027-0 |
| . |
