# zbMATH — the first resource for mathematics

Structure of $$n$$-uninorms. (English) Zbl 1122.03044
Summary: We study binary operators on $$[0,1]$$ which are associative, monotone non-decreasing in both variables and commutative (AMC) with neutral element. In this work, we generalize the concept of neutral element and this generalization gives rise to a new class of AMC binary operators on $$[0,1]$$ called $$n$$-uninorms. $$n$$-uninorms are denoted as $$U^{n}$$, where $$n$$ comes from the generalization of the neutral element. We study the structure of $$n$$-uninorms. The structure resembles an ordinal sum structure made up of $$n$$ uninorms. We characterize some special cases of them based on some continuity considerations and show that t-norms, t-conorms, uninorms and nullnorms (t-operators) are special cases of $$n$$-uninorms. We also show that given $$n$$ there are $$n+1$$ classes of operators in $$U^{n}$$ and each of them has many subclasses. We also study the Frank equation involving $$n$$-uninorms and show that we need to consider only $$n$$-uninorms for the study. Finally, we show that the total number of subclasses of operators in $$U^n$$ follows the famous series called Catalan numbers.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
Full Text:
##### References:
 [1] Calvo, T.; De Baets, B.; Fodor, J., The functional equations of Frank and alsina for uninorms and nullnorms, Fuzzy sets and systems, 120, 385-394, (2001) · Zbl 0977.03026 [2] Czogala, E.; Drewniak, J., Associative monotonic operations in fuzzy set theory, Fuzzy sets and systems, 12, 249-269, (1984) · Zbl 0555.94027 [3] De Baets, B., Uninorms: the known classes, (), 21-28 [4] De Baets, B., Idempotent uninorms, Eur. J. oper. res., 118, 631-642, (1999) · Zbl 0933.03071 [5] De Baets, B.; Fodor, J.C., Residual operators of uninorms, Soft comput., 3, 89-100, (1999) [6] Drewniak, J.; Drygas, P., On a class of uninorms, Internat. J. uncertain. fuzziness knowledge-based systems, 10, 5-10, (2002) · Zbl 1053.03510 [7] Drygas, P., Discussion of the structure of uninorms, Kybernetika, 41, 2, 213-226, (2005) · Zbl 1249.03093 [8] Fodor, J.C.; Yager, R.; Rybalov, A., Structure of uninorms, Internat. J. uncertain. fuzziness knowledge-based systems, 5, 411-427, (1997) · Zbl 1232.03015 [9] Frank, M.J., On the simultaneous associativity of $$F(x, y)$$ and $$x + y - F(x, y)$$, Aequationes math., 19, 194-226, (1979) · Zbl 0444.39003 [10] Hu, S.-K.; Li, Z.-F., The structure of continuous uni-norms, Fuzzy sets and systems, 123, 43-52, (2001) · Zbl 0989.03058 [11] Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0972.03002 [12] Li, Y.-M.; Shi, Z.-K., Weak uninorm aggregation operators, Inform. sci., 124, 317-323, (2000) · Zbl 0954.03058 [13] Li, Y.-M.; Shi, Z.-K., Remarks on uninorm aggregation operators, Fuzzy sets and systems, 114, 377-380, (2000) · Zbl 0962.03052 [14] Mas, M.; Mayor, G.; Torrens, J., $$t$$-operators, Internat. J. uncertain. fuzziness knowledge-based systems, 7, 31-50, (1999) · Zbl 1087.03515 [15] Mas, M.; Mayor, G.; Torrens, J., The modularity condition for uninorms and t-operators, Fuzzy sets and systems, 126, 2, 207-218, (2002) · Zbl 0996.03038 [16] Mas, M.; Mayor, G.; Torrens, J., The distributivity condition for uninorms and t-operators, Fuzzy sets and systems, 128, 2, 209-225, (2002) · Zbl 1005.03047 [17] Mas, M.; Mesiar, R.; Monserrat, M.; Torrens, J., Aggregation operators with annihilator, Internat. J. gen. systems, 34, 17-38, (2005) · Zbl 1081.68107 [18] Mas, M.; Monserrat, M.; Torrens, J., On left and right uninorms, Int. J. uncertain. fuzziness knowledge-based systems, 9, 491-507, (2001) · Zbl 1113.03341 [19] Monserrat, M.; Torrens, J., On the reversibility of uninorms and t-operators, Fuzzy sets and systems, 131, 3, 303-314, (2002) · Zbl 1012.03034 [20] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. math., 10, 313-334, (1960) · Zbl 0091.29801 [21] Stanley, R.P., Enumerative combinatorics, Vol. 2, (1999), Cambridge University Press Cambridge · Zbl 0928.05001 [22] 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. 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.