# zbMATH — the first resource for mathematics

Discussion of the structure of uninorms. (English) Zbl 1249.03093
Summary: The paper deals with binary operations in the unit interval. We investigate connections between families of triangular norms, triangular conorms, uninorms and some decreasing functions. It is well known that every uninorm is built up using some triangular norm and some triangular conorm. If we assume that uninorm fulfils additional assumptions, then this triangular norm and this triangular conorm have to be ordinal sums. The intervals in ordinal sums depend on the set of values of a decreasing function.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 03B52 Fuzzy logic; logic of vagueness
Full Text:
##### References:
 [1] Czogała E., Drewniak J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets and Systems 12 (1984), 249-269 · Zbl 0555.94027 [2] Baets B. De: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631-642 · Zbl 0933.03071 [3] Drewniak J., Drygaś P.: On a class of uninorms. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 10 (2002), Supplement, 5-10 · Zbl 1053.03510 [4] Fodor J., Baets, B. De, Calvo T.: Characterization of uninorms with given underlying t-norms and t-conorms, submitte. [5] Fodor J., Yager, R., Rybalov A.: Structure of uninorms. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997), 411-427 · Zbl 1232.03015 [6] 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 · Zbl 0996.03508 [7] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 · Zbl 1087.20041 [8] Li Y.-M., Shi Z.-K.: Remarks on uninorm aggregation operators. Fuzzy Sets and Systems 114 (2000), 377-380 · Zbl 0962.03052 [9] Martin J.: On a theorem of Czogała and Drewniak (1984). Proc. EUROFUSE PM’01, Granada 2001, pp. 49-54 [10] Martin J., Mayor G., and, Torrens J.: On locally internal monotonic operations. Fuzzy Sets and Systems 137 (2003), 27-42 · Zbl 1022.03038 [11] Yager R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111-120 · Zbl 0871.04007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.