Geodesic
On the structure of continuous uninorms
Kybernetika, Tome 43 (2007) no. 2, pp. 183-196 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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].
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].
Classification : 03B52, 03E72, 06F05
Keywords: uninorms; continuity; $t$-norms; $t$-conorms; ordinal sum of semigroups 
@article{KYB_2007_43_2_a5,
     author = {Dryga\'s, Pawe{\l}},
     title = {On the structure of continuous uninorms},
     journal = {Kybernetika},
     pages = {183--196},
     year = {2007},
     volume = {43},
     number = {2},
     mrnumber = {2343394},
     zbl = {1132.03349},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a5/}
}
TY  - JOUR
AU  - Drygaś, Paweł
TI  - On the structure of continuous uninorms
JO  - Kybernetika
PY  - 2007
SP  - 183
EP  - 196
VL  - 43
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a5/
LA  - en
ID  - KYB_2007_43_2_a5
ER  - 
%0 Journal Article
%A Drygaś, Paweł
%T On the structure of continuous uninorms
%J Kybernetika
%D 2007
%P 183-196
%V 43
%N 2
%U http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a5/
%G en
%F KYB_2007_43_2_a5
Drygaś, Paweł. On the structure of continuous uninorms. Kybernetika, Tome 43 (2007) no. 2, pp. 183-196. http://geodesic.mathdoc.fr/item/KYB_2007_43_2_a5/

[1] Clifford A. H.: Naturally totally ordered commutative semigroups. Amer. J. Math. 76 (1954), 631–646 | MR

[2] Climescu A. C.: Sur l’équation fonctionelle de l’associativité. Bull. Ecole Polytechn. 1 (1946), 1–16

[3] Czogała E., Drewniak J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets and Systems 12 (1984), 249–269 | MR | Zbl

[4] Dombi J.: Basic concepts for a theory of evaluation: The aggregative operators. European J. Oper. Res. 10 (1982), 282–293 | MR

[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

[6] Fodor J., Yager, R., Rybalov A.: Structure of uninorms. Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 5 (1997), 411–427 | MR | Zbl

[7] Hu S.-K., Li Z.-F.: The structure of continuous uninorms. Fuzzy Sets and Systems 124 (2001), 43–52 | MR | Zbl

[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

[9] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 | MR | Zbl

[10] Li Y.-M., Shi Z.-K.: Remarks on uninorm aggregation operators. Fuzzy Sets and Systems 114 (2000), 377–380 | MR | Zbl

[11] Mas M., Monserrat, M., Torrens J.: On left and right uninorms. Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 9 (2001), 491–507 | MR | Zbl

[12] Sander W.: Associative aggregation operators. In: Aggregation Operators (T. Calvo, G. Mayor, and R. Mesiar, eds), Physica–Verlag, Heidelberg 2002, pp. 124–158 | MR | Zbl

[13] Yager R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111–120 | MR | Zbl