Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
uninorm; Contour line; Orthosymmetry; Portation law; Exchange principle; Contrapositive symmetry; Rotation invariance; Self quasi-inverse property
uninorm; Contour line; Orthosymmetry; Portation law; Exchange principle; Contrapositive symmetry; Rotation invariance; Self quasi-inverse property
Summary:
Any given increasing $[0,1]^2\rightarrow [0,1]$ function is completely determined by its contour lines. In this paper 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.
References:
[1] Birkhoff G.: Lattice Theory. Third edition. (AMS Colloquium Publications, Vol. 25.) American Mathematical Society, Providence, Rhode Island 1967 MR 0227053 | Zbl 0537.06001
[2] Baets B. De: Coimplicators, the forgotten connectives. Tatra Mt. Math. Publ. 12 (1997), 229–240 MR 1607142 | Zbl 0954.03029
[3] Baets B. De: Idempotent uninorms. European J. Oper. Res. 118 (1999), 631–642 DOI 10.1016/S0377-2217(98)00325-7 | Zbl 0933.03071
[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 MR 1685816 | Zbl 0928.03060
[6] Baets B. De, Mesiar R.: Metrics and T-equalities. J. Math. Anal. Appl. 267 (2002), 331–347 MR 1888022
[7] Dombi J.: Basic concepts for the theory of evaluation: the aggregative operator. European J. Oper. Res. 10 (1982), 282–293 DOI 10.1016/0377-2217(82)90227-2 | MR 0665480
[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 DOI 10.1142/S0218488597000312 | MR 1471619 | Zbl 1232.03015
[10] Golan J.: The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science. Addison–Wesley Longman Ltd., Essex 1992 MR 1163371 | 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 MR 1774255
[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 DOI 10.1080/11663081.2000.10510989 | MR 1826844 | Zbl 1050.03505
[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 DOI 10.3166/jancl.11.351-366 | MR 1916884 | Zbl 1050.03505
[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 MR 1908426 | Zbl 1050.03505
[15] Jenei S.: How to construct left-continuous triangular norms – state of the art. Fuzzy Sets and Systems 143 (2004), 27–45 MR 2060271 | Zbl 1040.03021
[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 DOI 10.1007/s00010-004-2759-1 | MR 2167993 | Zbl 1083.39023
[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 MR 1685803 | Zbl 0953.26008
[18] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. (Trends in Logic, Vol. 8.) Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[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 DOI 10.1142/S179300570500010X | MR 2158962 | Zbl 1081.26024
[20] Maes K. C., Baets B. De: Orthosymmetrical monotone functions. B. Belg. Math. Soc.-Sim., to appear MR 2327329 | Zbl 1142.26007
[21] Ruiz D., Torrens J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40 (2004), 21–38 MR 2068596
[22] Schweizer B., Sklar A.: Probabilistic Metric Spaces. Elsevier Science, New York 1983 MR 0790314 | Zbl 0546.60010
[23] Yager R., Rybalov A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80 (1996), 111–120 DOI 10.1016/0165-0114(95)00133-6 | MR 1389951 | Zbl 0871.04007
Search
Partner of
