| Title:
|
Cauchy-like functional equation based on a class of uninorms (English) |
| Author:
|
Qin, Feng |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
51 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
678-698 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting $n$-ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that the unary function is non-decreasing and distributive over special aggregation operators, for examples, continuous t-norms, continuous t-conorms and two classes of uninorms. Along this way of thinking, in this paper, we will investigate and fully characterize the following unary distributive functional equation $f(U(x,y))=U(f(x),f(y))$, where $f\colon[0,1]\rightarrow[0,1]$ is an unknown function but unnecessarily non-decreasing, a uninorm $U\in{\mathcal U}_{\min}$ has a continuously underlying t-norm $T_U$ and a continuously underlying t-conorm $S_U$. Our investigation shows that the key point is a transformation from this functional equation to the several known ones. Moreover, this equation has also non-monotone solutions completely different with already obtained ones. Finally, our results extend the previous ones about the Cauchy-like equation $f(A(x,y))=B(f(x),f(y))$, where $A$ and $B$ are some continuous t-norm or t-conorm. (English) |
| Keyword:
|
fuzzy connectives |
| Keyword:
|
distributivity functional equations |
| Keyword:
|
T-norms |
| Keyword:
|
T-conorms |
| Keyword:
|
uninorms |
| MSC:
|
03B52 |
| MSC:
|
03E72 |
| idZBL:
|
Zbl 06530338 |
| idMR:
|
MR3423194 |
| DOI:
|
10.14736/kyb-2015-4-0678 |
| . |
| Date available:
|
2015-11-20T12:22:40Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144475 |
| . |
| Reference:
|
[1] Aczél, J.: Lectures on Functional Equations and Their Applications..Acad. Press, New York 1966. Zbl 0139.09301, MR 0208210, 10.1002/zamm.19670470321 |
| Reference:
|
[2] Aczél, J., Maksa, G., Taylor, M.: Equations of generalized bisymmetry and of consistent aggregation: Weakly surjective solutions which may be discontinuous at places..J. Math. Anal. Appl. 214 (1997), 1, 22-35. Zbl 0936.39009, MR 1645499, 10.1006/jmaa.1997.5580 |
| Reference:
|
[3] Baczyński, M., Jayaram, B.: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms..IEEE Trans. Fuzzy Systems 17 (2009), 3, 590-603. 10.1109/TFUZZ.2008.924201 |
| Reference:
|
[4] Beliakov, G., Calvo, T., Pradera, A.: Aggregation Functions: A Guide for Practitioners..Springer-Verlag, Berlin - Heidelberg 2007. |
| Reference:
|
[5] Benvenuti, P., Vivona, D.: General theory of the fuzzy integral..Mathware Soft Comput. 3 (1996), 199-209. Zbl 0857.28014, MR 1414267 |
| Reference:
|
[6] Combs, W. E., Andrews, J. E.: Combinatorial rule explosion eliminated by a fuzzy rule configuration..IEEE Trans. Fuzzy Systems 6 (1) (1998), 1, 1-11. 10.1109/91.660804 |
| Reference:
|
[7] Baets, B. De, Meyer, H. De, Mesiar, R.: Binary survival aggregation functions..Fuzzy Sets and Systems 191 (2012), 82-102. Zbl 1237.62063, MR 2874825, 10.1016/j.fss.2011.09.013 |
| Reference:
|
[8] Díaz, S., Montes, S., Baets, B. De: Transitivity bounds in additive fuzzy preference structures..IEEE Trans. Fuzzy Systems 15 (2007), 2, 275-286. 10.1109/tfuzz.2006.880004 |
| Reference:
|
[9] Dubois, D., Fodor, J., Prade, H., Roubens, M.: Aggregation of decomposable measures with application to utility theory..Theory Decision 41 (1996), 59-95. Zbl 0863.90020, MR 1401341, 10.1007/bf00134116 |
| Reference:
|
[10] Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support..Kluwer, Dordrecht 1994. Zbl 0827.90002, 10.1007/978-94-017-1648-2 |
| Reference:
|
[11] Fodor, J., Yager, R. R., Rybalov, A.: Structure of uninorms..Int. J. Uncertainly and Knowledge-Based Systems 5 (1997), 411-427. Zbl 1232.03015, MR 1471619, 10.1142/s0218488597000312 |
| Reference:
|
[12] Gottwald, S.: A Treatise on Many-Valued Logics..Research Studies Press, Hertfordshire 2001. Zbl 1048.03002, MR 1856623 |
| Reference:
|
[13] Grabisch, M., Marichal, J. L., Mesiar, R., Pap, E.: Aggregation Functions..Cambridge University Press, 2009. Zbl 1206.68299, MR 2538324, 10.1017/cbo9781139644150 |
| Reference:
|
[14] 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:
|
[15] McConway, K. J.: Marginalization and linear opinion pools..J. Amer. Stat. Assoc. 76 (1981), 410-414. Zbl 0455.90004, MR 0624342, 10.1080/01621459.1981.10477661 |
| Reference:
|
[16] Qin, F.: Cauchy like functional equation based on continuous t-conorms and representable uninorms..IEEE Trans. Fuzzy Systems 20 (2013), 126-132. 10.1109/fuzz-ieee.2014.6891537 |
| Reference:
|
[17] Qin, F., Baczyński, M.: Distributivity equations of implications based on continuous triangular norms (I)..IEEE Trans. Fuzzy Syst. 21 (2012), 1, 153-167. 10.1109/TFUZZ.2011.2171188 |
| Reference:
|
[18] Qin, F., Baczyński, M.: Distributivity equations of implications based on continuous triangular conorms (II)..Fuzzy Sets and Systems 240 (2014), 86-102. Zbl 1315.03040, MR 3167514, 10.1016/j.fss.2013.07.020 |
| Reference:
|
[19] Qin, F., Yang, P. C.: On the distributive equation of implication based on a continuous t-norm and a continuous Archimedean t-conorm..In: BMEI, 4th International Conference 4, 2011, pp. 2290-2294. |
| Reference:
|
[20] Qin, F., Yang, L.: Distributivity equations of implications based on nilpotent triangular norms..Int. J. Approx. Reason. 51 (2010), 984-992. MR 2719614, 10.1016/j.ijar.2010.07.005 |
| Reference:
|
[21] Rudas, I. J., Pap, E., Fodor, J.: Information aggregation in intelligent systems: An application oriented approach..Knowledge-Based Systems 38 (2013), 3-13. 10.1016/j.knosys.2012.07.025 |
| Reference:
|
[22] Ruiz, D., Torrens, J.: Residual implications and co-implications from idempotent uninorms..Kybernetika 40 (2004), 21-38. Zbl 1249.94095, MR 2068596 |
| Reference:
|
[23] Saminger, S., Mesiar, R., Bodenhofer, U.: Domination of aggregation operators and preservation of transitivity..Int. J. Uncertainty Fuzziness Knowledge-Based Systems 10/s (2002), 11-35. Zbl 1053.03514, MR 1962666, 10.1142/S0218488502001806 |
| Reference:
|
[24] Saminger-Platz, S., Mesiar, R., Dubois, D.: Aggregation Operators and Commuting..IEEE Trans. Fuzzy Syst. 15 (2007), 6, 1032-1045. 10.1109/TFUZZ.2006.890687 |
| Reference:
|
[25] Su, Y., Wang, Z., Tang, K.: Left and right semi-uninorms on a complete lattice..Kybernetika 49 (2013), 6, 948-961. Zbl 1286.03098, MR 3182650 |
| Reference:
|
[26] Yager, R. R., Rybalov, A.: Uninorm aggregation operators..Fuzzy Sets and Systems 80 (1996), 111-120. Zbl 0871.04007, MR 1389951, 10.1016/0165-0114(95)00133-6 |
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