×

On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums. (English) Zbl 1191.03039

The study is concerned with binary operations \(U: [0,1]^2\to [0,1]\) which are increasing, associative, and have a neutral element \(e\) in the unit interval. Particular cases include uninorms (where in addition to the properties listed above, we also require commutativity) and t-norms and t-conorms where the neutral element is 1 and 0, respectively.
Presented are some properties of operations for which the underlying components are given as ordinal sums. It is demonstrated that if such components are given by ordinal sums, then the Cartesian product of the union of two arbitrary intervals (each coming from a domain of different ordinal sums) is closed under the uninorm. Furthermore, a description of uninorm-like operations is provided when the underlying operations are pseudo t-norms and pseudo t-conorms and one of them is idempotent.

MSC:

03E72 Theory of fuzzy sets, etc.
03B52 Fuzzy logic; logic of vagueness
Full Text: DOI

References:

[1] De Baets, B., Idempotent uninorms, European J. Oper. Res., 118, 631-642 (1999) · Zbl 0933.03071
[2] B. De Baets, J. Fodor, T. Calvo, The characterization of uninorms with continuous underlying t-norms and t-conorms, Fuzzy Sets and Systems, submitted for publication.; B. De Baets, J. Fodor, T. Calvo, The characterization of uninorms with continuous underlying t-norms and t-conorms, Fuzzy Sets and Systems, submitted for publication.
[3] Clifford, A. H., Naturally totally ordered commutative semigroups, Amer. J. Math., 76, 631-646 (1954) · Zbl 0055.01503
[4] Czogała, E.; Drewniak, J., Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems, 12, 249-269 (1984) · Zbl 0555.94027
[5] J. Drewniak, P. Drygaś, Ordered semigroups in constructions of uninorms and nullnorms, in: P. Grzegorzewski, M. Krawczak, S. Zadrożny, (Eds.), Issues in Soft Computing Theory and Applications, EXIT, Warszawa, 2005, pp. 147-158.; J. Drewniak, P. Drygaś, Ordered semigroups in constructions of uninorms and nullnorms, in: P. Grzegorzewski, M. Krawczak, S. Zadrożny, (Eds.), Issues in Soft Computing Theory and Applications, EXIT, Warszawa, 2005, pp. 147-158.
[6] Drygaś, P., Discussion of the structure of uninorms, Kybernetika, 41, 213-226 (2005) · Zbl 1249.03093
[7] Drygaś, P., Remarks about idempotent uninorms, J. Electrical Eng., 57, 92-94 (2006) · Zbl 1118.03315
[8] Drygaś, P., On the structure of continuous uninorms, Kybernetika, 43, 183-196 (2007) · Zbl 1132.03349
[9] Fodor, J.; Yager, R.; Rybalov, A., Structure of uninorms, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst., 5, 411-427 (1997) · Zbl 1232.03015
[10] Gottwald, S., A Treatise on Many-valued Logic (2001), Research Studies Press: Research Studies Press Baldock · Zbl 1048.03002
[11] Hu, S.-K.; Li, Z.-F., The structure of continuous uninorms, Fuzzy Sets and Systems, 124, 43-52 (2001) · Zbl 0989.03058
[12] Jenei, S., A note on the ordinal sum theorem and its consequence for the construction of triangular norm, Fuzzy Sets and Systems, 126, 199-205 (2002) · Zbl 0996.03508
[13] Klement, E. P.; Mesiar, R.; Pap, E., Triangular Norms (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0972.03002
[14] Li, Y.-M.; Shi, Z.-K., Remarks on uninorm aggregation operators, Fuzzy Sets and Systems, 114, 377-380 (2000) · Zbl 0962.03052
[15] Maes, K. C.; De Baets, B., On the structure of left-continuous t-norms that have a continuous contour line, Fuzzy Sets and Systems, 158, 843-860 (2007) · Zbl 1122.03051
[16] Maes, K. C.; De Baets, B., Rotation-invariant t-norms: the rotation invariance property revisited, Fuzzy Sets and Systems, 160, 44-51 (2009) · Zbl 1183.03051
[17] Maes, K. C.; De Baets, B., Rotation-invariant t-norms: where triple rotation and rotation-annihilation meet, Fuzzy Sets and Systems, 160, 1998-2016 (2009) · Zbl 1182.03052
[18] Martín, J.; Mayor, G.; Torrens, J., On locally internal monotonic operations, Fuzzy Sets and Systems, 137, 27-42 (2003) · Zbl 1022.03038
[19] Sander, W., Associative aggregation operators, (Calvo, T.; Mayor, G.; Mesiar, R., Aggregation Operators (2002), Physica: Physica Heidelberg), 124-158 · Zbl 1025.03054
[20] Yager, R.; Rybalov, A., Uninorm aggregation operators, Fuzzy Sets and Systems, 80, 111-120 (1996) · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.