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Title: Generated triangular norms (English)
Author: Klement, Erich Peter
Author: Mesiar, Radko
Author: Pap, Endre
Language: English
Journal: Kybernetika
ISSN: 0023-5954
Volume: 36
Issue: 3
Year: 2000
Pages: [363]-377
Summary lang: English
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Category: math
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Summary: An overview of generated triangular norms and their applications is presented. Several properties of generated $t$-norms are investigated by means of the corresponding generators, including convergence properties. Some applications are given. An exhaustive list of relevant references is included. (English)
Keyword: triangular norm
MSC: 03E72
MSC: 54A40
MSC: 54E70
idZBL: Zbl 1249.03100
idMR: MR1773510
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Date available: 2009-09-24T19:33:33Z
Last updated: 2015-03-27
Stable URL: http://hdl.handle.net/10338.dmlcz/135356
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Reference: [1] Abel N. H.: Untersuchungen der Funktionen zweier unabhängigen veränderlichen Groessen $x$ und $y$ wie $f(x,y)$, welche die Eigenschaft haben, dass $f(z,\,f(x,y))$ eine symmetrische Funktion von $x,\,y$ und $z$ ist.J. Reine Angew. Math. 1 (1928), 11–15
Reference: [2] Aczél J.: Sur les opérations definies pour des nombres réels.Bull. Soc. Math. France 76 (1949), 59–64
Reference: [3] Aczél J.: Lectures on Functional Equations and their Applications.Academic Press, New York 1966 MR 0208210
Reference: [4] Aczél J., Alsina C.: Characterization of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgments.Methods Oper. Res. 48 (1984), 3–22 MR 0736352
Reference: [5] Arnold V.: Concerning the representability of functions of two variables in the form $X[\Phi (x)+\Psi (y)]$.Uspekhi Mat. Nauk 12 (1957), 119–121 MR 0090623
Reference: [6] Bezdek J. C., Harris J. D.: Fuzzy partitions and relations: An axiomatic basis for clustering.Fuzzy Sets and Systems 1 (1978), 111–127 Zbl 0442.68093, MR 0502319
Reference: [7] Calvo T., Mesiar R.: Continuous generated associative aggregation operators.Submitted Zbl 0996.03034
Reference: [8] Baets B. De, Mesiar R.: Pseudo–metrics and $T$–equivalences.J. Fuzzy Math. 5 (1997), 471–481 MR 1457163
Reference: [9] Baets B. De, Mesiar R.: ${\mathcal T}$-partitions.Fuzzy Sets and Systems 97 (1998), 211–223 MR 1645614
Reference: [10] Dombi J.: Basic concepts for a theory of evaluation: The aggregative operator.Europ. J. Oper. Research 10 (1982), 282–293 Zbl 0488.90003, MR 0665480, 10.1016/0377-2217(82)90227-2
Reference: [11] Dombi J.: A general class of fuzzy operators, De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators.Fuzzy Sets and Systems 8 (1982), 149–163 MR 0666628, 10.1016/0165-0114(82)90005-7
Reference: [12] Dubois D., Prade H.: Fuzzy numbers: An overview.In: Analysis of Fuzzy Information, Vol. I: Mathematics and Logic (J. C. Bezdek, ed.), CRC Press, Boca Raton 1987, pp. 3–39 Zbl 0663.94028, MR 0910312
Reference: [13] Dubois D., Kerre E. E., Mesiar R., Prade H.: Fuzzy interval analysis.In: Mathematics and Fuzzy Sets. Basic Principles. The Handbook of Fuzzy Sets Series (D. Dubois and H. Prade, eds.), Kluwer Acad. Publ., Boston 2000, pp. 483–582 Zbl 0988.26020, MR 1890240
Reference: [14] Féron R.: Sur les tableaux de corrélation dont les marges sont donneés.Publ. Inst. Statist. Univ. Paris 5 (1956), 3–12 Zbl 0074.14205, MR 0082246
Reference: [15] Frank M. J.: On the simultaneous associativity of $F(x,y)$ and $x+y-F(x,y)$.Aequationes Math. 19 (1979), 194–226 Zbl 0444.39003, MR 0556722, 10.1007/BF02189866
Reference: [16] Fullér R., Keresztfalvi T.: $t$-norm based addition of fuzzy intervals.Fuzzy Sets and Systems 51 (1992), 155–159 MR 1188307, 10.1016/0165-0114(92)90188-A
Reference: [17] Fung L. W., Fu K. S.: An axiomatic approach to rational decision making in a fuzzy environment.In: Fuzzy Sets and Their Applications to Cognitive and Decision Processes (L. A. Zadeh et al, eds.), Academic Press, New York 1975, pp. 227–256 Zbl 0366.90003, MR 0398652
Reference: [18] Gottwald S.: Approximate solutions of fuzzy relational equations and a characterization of $t$-norms that define metrics for fuzzy sets.Fuzzy Sets and Systems 75 (1995), 189–201 Zbl 0856.04006, MR 1358221
Reference: [19] Hamacher H.: Uber logische Aggregationen nicht-binär explizierter Entscheidungskriterien.Rita G. Fischer Verlag, Frankfurt 1978
Reference: [20] Hoele U.: Fuzzy equalities and indistinguishability.In: Proc. EUFIT’93, Aachen 1993, pp. 358–363
Reference: [21] Jenei S.: On Archimedean triangular norms.Fuzzy Sets and Systems 99 (1998), 179–186 Zbl 0938.03083, MR 1646173, 10.1016/S0165-0114(97)00021-3
Reference: [22] Jenei S., Fodor J. C.: On continuous triangular norms.Fuzzy Sets and Systems 100 (1998), 273–282 Zbl 0938.03084, MR 1663745, 10.1016/S0165-0114(97)00063-8
Reference: [23] Klement E. P.: Construction of fuzzy $\sigma $-algebras using triangular norms.J. Math. Anal. Appl. 85 (1982), 543–565 Zbl 0491.28003, MR 0649189, 10.1016/0022-247X(82)90015-4
Reference: [24] Klement E. P., Mesiar R., Pap E.: On the relationship of associative compensatory operators to triangular norms and conorms.Internat. J. Uncertain. Fuzziness Knowledge–Based Systems 4 (1996), 129–144 Zbl 1232.03041, MR 1390899, 10.1142/S0218488596000081
Reference: [25] Klement E. P., Mesiar R., Pap E.: Additive generators of $t$-norms which are not necessarily continuous.In: Proc. EUFIT’98, Aachen 1996, pp. 70–73
Reference: [26] Klement E. P., Mesiar R., Pap E.: A characterization of the ordering of continuous $t$-norms.Fuzzy Sets and Systems 86 (1997), 189–195 Zbl 0914.04006, MR 1437918, 10.1016/0165-0114(95)00407-6
Reference: [27] 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, MR 1685803
Reference: [28] Klement E. P., Mesiar R., Pap E.: Triangular Norms.Kluwer Acad. Publ., Dordrecht 2000 Zbl 1087.20041, MR 1790096
Reference: [29] Kolesárová A.: Triangular norm–based addition of linear fuzzy numbers.Tatra Mt. Math. Publ. 6 (1995), 75–81 Zbl 0851.04005, MR 1363985
Reference: [30] Kolesárová A.: Similarity preserving $t$-norm–based addition of fuzzy numbers.Fuzzy Sets and Systems 91 (1997), 215–229 MR 1480047, 10.1016/S0165-0114(97)00142-5
Reference: [31] Kolesárová A.: Comparison of quasi–arithmetic means.In: Proc. EUROFUSE–SIC’98, Budapest 1999, pp. 237–240
Reference: [32] Kolesárová A.: Limit properties of quasi–arithmetic means.In: Proc. EUFIT’99, Aachen, 1999 (CD–rom)
Reference: [33] Kolesárová A., Komorníková M.: Triangular norm–based iterative compensatory operators.Fuzzy Sets and Systems 104 (1999), 109–120 Zbl 0931.68123, 10.1016/S0165-0114(98)00263-2
Reference: [34] Komorníková M.: Generated aggregation operators.In: Proc. EUSFLAT’99, Palma de Mallorca 1999, pp. 355–357
Reference: [35] Komorníková M.: Smoothly generated discrete aggregation operators.BUSEFAL 80 (1999), 35–39
Reference: [36] Ling C. M.: Representation of associative functions.Publ. Math. Debrecen 12 (1965), 189–212 MR 0190575
Reference: [37] Mareš M.: Computation over Fuzzy Quantities.CRC Press, Boca Raton 1994 Zbl 0859.94035, MR 1327525
Reference: [38] Marko V., Mesiar R.: A note on a nilpotent lower bound of nilpotent triangular norms.Fuzzy Sets and Systems 104 (1999), 27–34 Zbl 0930.03075, MR 1685806
Reference: [39] Marko V., Mesiar R.: Lower and upper bounds of continuous Archimedean $t$-norms.Fuzzy Sets and Systems, to appear MR 1685806
Reference: [40] Marková A.: A note on $g$-derivative and $g$-integral.Tatra Mt. Math. Publ. 8 (1996), 71–76 Zbl 0918.28023, MR 1475263
Reference: [41] Marková A.: $T$-sum of $L$-$R$-fuzzy numbers.Fuzzy Sets and Systems 85 (1997), 379–384 MR 1428314, 10.1016/0165-0114(95)00370-3
Reference: [42] Marková–Stupňanová A.: A note on idempotent functions with respect to pseudo–convolution.Fuzzy Sets and Systems 102 (1999), 417–421 MR 1676908
Reference: [43] Menger K.: Statistical metrics.Proc. Nat. Acad. Sci. U. S. A. 8 (1942), 535–537 Zbl 0063.03886, MR 0007576, 10.1073/pnas.28.12.535
Reference: [44] Mesiar R.: A note on $T$-sum of $L$-$R$ fuzzy numbers.Fuzzy Sets and Systems 79 (1996), 259–261 MR 1388398, 10.1016/0165-0114(95)00178-6
Reference: [45] Mesiar R.: Triangular norm–based addition of fuzzy intervals.Fuzzy Sets and Systems 91 (1997), 231–237 Zbl 0919.04011, MR 1480048, 10.1016/S0165-0114(97)00143-7
Reference: [46] Mesiar R.: Approximation of continuous $t$-norms by strict $t$-norms with smooth generators.BUSEFAL 75 (1998), 72–79
Reference: [47] Mesiar R.: On the pointwise convergence of continuous Archimedean $t$-norms and the convergence of their generators.BUSEFAL 75 (1998), 39–45
Reference: [48] Mesiar R.: Generated conjunctions and related operators in MV-logic as a basis of AI applications.In: Proc. ECAI’98, Brighton 1998, Workshop WG17
Reference: [49] Mesiar R., Navara M.: $T_s$-tribes and $T_s$-measures.J. Math. Anal. Appl. 201 (1996), 91–102 MR 1397888, 10.1006/jmaa.1996.0243
Reference: [50] Mesiar R., Navara M.: Diagonals of continuous triangular norms.Fuzzy Sets and Systems 104 (1999), 35–41 Zbl 0972.03052, MR 1685807, 10.1016/S0165-0114(98)00256-5
Reference: [51] Mostert P. S., Shields A. L.: On the structure of semigroups on a compact manifold with boundary.Ann. of Math. 65 (1957), 117–143 MR 0084103, 10.2307/1969668
Reference: [52] Nguyen H. T., Kreinovich V., Wojciechowski T.: Strict Archimedean $t$-norms and $t$-conorms as universal approximators.Internat. J. Approx. Reason. 18 (1998), 239–249 Zbl 0955.68108, MR 1661123, 10.1016/S0888-613X(98)00009-7
Reference: [53] Pap E.: $g$-calculus.Univ. u Novom Sadu Zb. Rad. Prirod.–Mat. Fak. Ser. Mat. 23 (1993), 145–156 Zbl 0823.28011, MR 0939298
Reference: [54] Schweizer B., Sklar A.: Statistical metric spaces.Pacific J. Math. 10 (1960), 313–334 Zbl 0096.33203, MR 0115153, 10.2140/pjm.1960.10.313
Reference: [55] Schweizer B., Sklar A.: Associative functions and triangle inequalities.Publ. Math. Debrecen 8 (1961), 169–186 MR 0132939
Reference: [56] Schweizer B., Sklar A.: Associative functions and abstract semigroups.Publ. Math. Debrecen 10 (1963), 69–81 MR 0170967
Reference: [57] Schweizer B., Sklar A.: Probabilistic Metric Spaces.North–Holland, New York 1983 Zbl 0546.60010, MR 0790314
Reference: [58] Sklar A.: Fonctions de répartition a $n$-dimensions et leurs marges.Publ. Inst. Stat. Univ. Paris 8 (1959), 229–231 MR 0125600
Reference: [59] Smutná D., Vojtáš P.: Fuzzy resolution with residuation of material implication.In: Proc. EUROFUSE–SIC’99, Budapest 1999, pp. 472–476
Reference: [60] Tardiff R. M.: On a generalized Minkowski inequality and its relation to dominates for $t$-norms.Aequationes Math. 27 (1984), 308–316 Zbl 0549.26013, MR 0762688, 10.1007/BF02192679
Reference: [61] Viceník P.: A note on generators of $t$-norms.BUSEFAL 75 (1998), 33–38
Reference: [62] Viceník P.: Additive generators and discontinuity.BUSEFAL 76 (1998), 25–28
Reference: [63] Viceník P.: Additive generators of triangular norms with an infinite set of discontinuity points.In: Proc. EUROFUSE–SIC’99, Budapest 1999, pp. 412–416
Reference: [64] Viceník P.: Generated $t$-norms and the Archimedean property.In: Proc. EUFIT’99, Aachen 1999 (CD–rom)
Reference: [65] Viceník P.: Non–continuous generated $t$-norms.In: Abstracts of Linz’98 “Topological and Algebraic Structures”, Linz 1999, pp. 9–10
Reference: [66] Viceník P.: Additive generators of non–continuous triangular norms.Preprint, submitted Zbl 1045.03046
Reference: [67] Yager R. R.: On a general class of fuzzy connectives.Fuzzy Setsna and Systems 4 (1980), 235–242 Zbl 0443.04008, MR 0589241, 10.1016/0165-0114(80)90013-5
Reference: [68] Zadeh L. A.: Similarity relations and fuzzy orderings.Inform. Sci. 3 (1971), 177–200 Zbl 0218.02058, MR 0297650, 10.1016/S0020-0255(71)80005-1
Reference: [69] Zadeh L. A.: The concept of linguistic fuzzy variable and its applications to approximate reasoning.Part I. Inform. Sci. 8 (1975), 199–261 MR 0386369
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