×

A contour view on uninorm properties. (English) Zbl 1249.26022

Summary: Any given increasing \([0,1]^2\rightarrow [0,1]\) function is completely determined by its contour lines. We show how each individual uninorm property can be translated into a property of contour lines. In particular, we describe commutativity in terms of orthosymmetry and we link associativity to the portation law and the exchange principle. Contrapositivity and rotation invariance are used to characterize uninorms that have a continuous contour line.

MSC:

26B40 Representation and superposition of functions
06F05 Ordered semigroups and monoids
03B52 Fuzzy logic; logic of vagueness
PDFBibTeX XMLCite
Full Text: EuDML Link

References:

[1] Birkhoff G.: Lattice Theory. Third edition. (AMS Colloquium Publications, Vol. 25.) American Mathematical Society, Providence, Rhode Island 1967 · Zbl 0537.06001
[2] Baets B. De: Coimplicators, the forgotten connectives. Tatra Mt. Math. Publ. 12 (1997), 229-240 · Zbl 0954.03029
[3] Baets B. De: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631-642 · Zbl 0933.03071 · doi:10.1016/S0377-2217(98)00325-7
[4] Baets B. De, Fodor J.: Residual operators of uninorms. Soft Computing 3 (1999), 89-100 · doi:10.1007/s005000050057
[5] Baets B. De, Fodor J.: van Melle’s combining function in MYCIN is a representable uninorm: An alternative proof. Fuzzy Sets and Systems 104 (1999), 133-136 · Zbl 0928.03060 · doi:10.1016/S0165-0114(98)00265-6
[6] Baets B. De, Mesiar R.: Metrics and T-equalities. J. Math. Anal. Appl. 267 (2002), 331-347 · Zbl 0996.03035 · doi:10.1006/jmaa.2001.7786
[7] Dombi J.: Basic concepts for the theory of evaluation: the aggregative operator. European J. Oper. Res. 10 (1982), 282-293 · Zbl 0488.90003 · doi:10.1016/0377-2217(82)90227-2
[8] Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht 1994 · Zbl 0827.90002
[9] Fodor J., Yager, R., Rybalov A.: Structure of uninorms. Internat. J. Uncertain Fuzz. 5 (1997), 411-427 · Zbl 1232.03015 · doi:10.1142/S0218488597000312
[10] Golan J.: The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Addison-Wesley Longman Ltd., Essex 1992 · Zbl 0780.16036
[11] Jenei S.: Geometry of left-continuous t-norms with strong induced negations. Belg. J. Oper. Res. Statist. Comput. Sci. 38 (1998), 5-16 · Zbl 1010.03520
[12] Jenei S.: Structure of left-continuous triangular norms with strong induced negations. (I) Rotation construction. J. Appl. Non-Classical Logics 10 (2000), 83-92 · Zbl 1050.03505 · doi:10.1016/S0165-0114(01)00061-6
[13] Jenei S.: Structure of left-continuous triangular norms with strong induced negations. (II) Rotation-annihilation construction. J. Appl. Non-Classical Logics 11 (2001), 351-366 · Zbl 1050.03505 · doi:10.1016/S0165-0114(01)00061-6
[14] Jenei S.: Structure of left-continuous triangular norms with strong induced negations. (III) Construction and decomposition. Fuzzy Sets and Systems 128 (2002), 197-208 · Zbl 1050.03505 · doi:10.1016/S0165-0114(01)00061-6
[15] Jenei S.: How to construct left-continuous triangular norms - state of the art. Fuzzy Sets and Systems 143 (2004), 27-45 · Zbl 1040.03021 · doi:10.1016/j.fss.2003.06.006
[16] Jenei S.: On the determination of left-continuous t-norms and continuous Archimedean t-norms on some segments. Aequationes Math. 70 (2005), 177-188 · Zbl 1083.39023 · doi:10.1007/s00010-004-2759-1
[17] Klement E. P., Mesiar, R., Pap E.: Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets and Systems 104 (1999), 3-13 · Zbl 0953.26008 · doi:10.1016/S0165-0114(98)00252-8
[18] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. (Trends in Logic, Vol. 8.) Kluwer Academic Publishers, Dordrecht 2000 · Zbl 1087.20041 · doi:10.1017/S1446788700008065
[19] Klement E. P., Mesiar, R., Pap E.: Different types of continuity of triangular norms revisited. New Mathematics and Natural Computation 1 (2005), 195-211 · Zbl 1081.26024 · doi:10.1142/S179300570500010X
[20] Maes K. C., Baets B. De: Orthosymmetrical monotone functions. B. Belg. Math. Soc.-Sim., to appear · Zbl 1142.26007
[21] Ruiz D., Torrens J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40 (2004), 21-38 · Zbl 1249.94095
[22] Schweizer B., Sklar A.: Probabilistic Metric Spaces. Elsevier Science, New York 1983 · Zbl 0546.60010
[23] Yager R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111-120 · Zbl 0871.04007 · doi:10.1016/0165-0114(95)00133-6
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.