| Title:
|
On the structure of continuous uninorms (English) |
| Author:
|
Drygaś, Paweł |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
183-196 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation $U$ in the unit interval with the neutral element $e\in [0,1]$. If operation $U$ is continuous, then $e=0$ or $e=1$. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element $e\in (0,1)$, which is continuous in the open unit square may be given in $[0,1)^2$ or $(0,1]^2$ as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7]. (English) |
| Keyword:
|
uninorms |
| Keyword:
|
continuity |
| Keyword:
|
$t$-norms |
| Keyword:
|
$t$-conorms |
| Keyword:
|
ordinal sum of semigroups |
| MSC:
|
03B52 |
| MSC:
|
03E72 |
| MSC:
|
06F05 |
| idZBL:
|
Zbl 1132.03349 |
| idMR:
|
MR2343394 |
| . |
| Date available:
|
2009-09-24T20:22:44Z |
| Last updated:
|
2012-06-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135765 |
| . |
| Reference:
|
[1] Clifford A. H.: Naturally totally ordered commutative semigroups.Amer. J. Math. 76 (1954), 631–646 MR 0062118 |
| Reference:
|
[2] Climescu A. C.: Sur l’équation fonctionelle de l’associativité.Bull. Ecole Polytechn. 1 (1946), 1–16 |
| Reference:
|
[3] Czogała E., Drewniak J.: Associative monotonic operations in fuzzy set theory.Fuzzy Sets and Systems 12 (1984), 249–269 Zbl 0555.94027, MR 0740097 |
| Reference:
|
[4] Dombi J.: Basic concepts for a theory of evaluation: The aggregative operators.European J. Oper. Res. 10 (1982), 282–293 MR 0665480 |
| Reference:
|
[5] Drewniak J., Drygaś P.: Ordered semigroups in constructions of uninorms and nullnorms.In: Issues in Soft Computing Theory and Applications (P. Grzegorzewski, M. Krawczak, and S. Zadrożny, eds.), EXIT, Warszawa 2005, pp. 147–158 |
| Reference:
|
[6] Fodor J., Yager, R., Rybalov A.: Structure of uninorms.Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 5 (1997), 411–427 Zbl 1232.03015, MR 1471619 |
| Reference:
|
[7] Hu S.-K., Li Z.-F.: The structure of continuous uninorms.Fuzzy Sets and Systems 124 (2001), 43–52 Zbl 1132.03349, MR 1859776 |
| Reference:
|
[8] Jenei S.: A note on the ordinal sum theorem and its consequence for the construction of triangular norm.Fuzzy Sets and Systems 126 (2002), 199–205 MR 1884686 |
| Reference:
|
[9] Klement E. P., Mesiar, R., Pap E.: Triangular Norms.Kluwer Academic Publishers, Dordrecht 2000 Zbl 1087.20041, MR 1790096 |
| Reference:
|
[10] Li Y.-M., Shi Z.-K.: Remarks on uninorm aggregation operators.Fuzzy Sets and Systems 114 (2000), 377–380 Zbl 0962.03052, MR 1775275 |
| Reference:
|
[11] Mas M., Monserrat, M., Torrens J.: On left and right uninorms.Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 9 (2001), 491–507 Zbl 1045.03029, MR 1852342 |
| Reference:
|
[12] Sander W.: Associative aggregation operators.In: Aggregation Operators (T. Calvo, G. Mayor, and R. Mesiar, eds), Physica–Verlag, Heidelberg 2002, pp. 124–158 Zbl 1025.03054, MR 1936386 |
| Reference:
|
[13] Yager R., Rybalov A.: Uninorm aggregation operators.Fuzzy Sets and Systems 80 (1996), 111–120 Zbl 0871.04007, MR 1389951 |
| . |
