@article{KYB_2000_36_3_a6,
author = {Klement, Erich Peter and Mesiar, Radko and Pap, Endre},
title = {Generated triangular norms},
journal = {Kybernetika},
pages = {363--377},
year = {2000},
volume = {36},
number = {3},
mrnumber = {1773510},
zbl = {1249.03100},
language = {en},
url = {http://geodesic.mathdoc.fr/item/KYB_2000_36_3_a6/}
}
Klement, Erich Peter; Mesiar, Radko; Pap, Endre. Generated triangular norms. Kybernetika, Tome 36 (2000) no. 3, pp. 363-377. http://geodesic.mathdoc.fr/item/KYB_2000_36_3_a6/
[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
[2] Aczél J.: Sur les opérations definies pour des nombres réels. Bull. Soc. Math. France 76 (1949), 59–64
[3] Aczél J.: Lectures on Functional Equations and their Applications. Academic Press, New York 1966 | MR
[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
[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
[6] Bezdek J. C., Harris J. D.: Fuzzy partitions and relations: An axiomatic basis for clustering. Fuzzy Sets and Systems 1 (1978), 111–127 | MR | Zbl
[7] Calvo T., Mesiar R.: Continuous generated associative aggregation operators. Submitted | Zbl
[8] Baets B. De, Mesiar R.: Pseudo–metrics and $T$–equivalences. J. Fuzzy Math. 5 (1997), 471–481 | MR
[9] Baets B. De, Mesiar R.: ${\mathcal T}$-partitions. Fuzzy Sets and Systems 97 (1998), 211–223 | MR
[10] Dombi J.: Basic concepts for a theory of evaluation: The aggregative operator. Europ. J. Oper. Research 10 (1982), 282–293 | DOI | MR | Zbl
[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 | DOI | MR
[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 | MR | Zbl
[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 | MR | Zbl
[14] Féron R.: Sur les tableaux de corrélation dont les marges sont donneés. Publ. Inst. Statist. Univ. Paris 5 (1956), 3–12 | MR | Zbl
[15] Frank M. J.: On the simultaneous associativity of $F(x,y)$ and $x+y-F(x,y)$. Aequationes Math. 19 (1979), 194–226 | DOI | MR | Zbl
[16] Fullér R., Keresztfalvi T.: $t$-norm based addition of fuzzy intervals. Fuzzy Sets and Systems 51 (1992), 155–159 | DOI | MR
[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 | MR | Zbl
[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 | MR | Zbl
[19] Hamacher H.: Uber logische Aggregationen nicht-binär explizierter Entscheidungskriterien. Rita G. Fischer Verlag, Frankfurt 1978
[20] Hoele U.: Fuzzy equalities and indistinguishability. In: Proc. EUFIT’93, Aachen 1993, pp. 358–363
[21] Jenei S.: On Archimedean triangular norms. Fuzzy Sets and Systems 99 (1998), 179–186 | DOI | MR | Zbl
[22] Jenei S., Fodor J. C.: On continuous triangular norms. Fuzzy Sets and Systems 100 (1998), 273–282 | DOI | MR | Zbl
[23] Klement E. P.: Construction of fuzzy $\sigma $-algebras using triangular norms. J. Math. Anal. Appl. 85 (1982), 543–565 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[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
[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 | DOI | MR | Zbl
[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 | MR | Zbl
[28] Klement E. P., Mesiar R., Pap E.: Triangular Norms. Kluwer Acad. Publ., Dordrecht 2000 | MR | Zbl
[29] Kolesárová A.: Triangular norm–based addition of linear fuzzy numbers. Tatra Mt. Math. Publ. 6 (1995), 75–81 | MR | Zbl
[30] Kolesárová A.: Similarity preserving $t$-norm–based addition of fuzzy numbers. Fuzzy Sets and Systems 91 (1997), 215–229 | DOI | MR
[31] Kolesárová A.: Comparison of quasi–arithmetic means. In: Proc. EUROFUSE–SIC’98, Budapest 1999, pp. 237–240
[32] Kolesárová A.: Limit properties of quasi–arithmetic means. In: Proc. EUFIT’99, Aachen, 1999 (CD–rom)
[33] Kolesárová A., Komorníková M.: Triangular norm–based iterative compensatory operators. Fuzzy Sets and Systems 104 (1999), 109–120 | DOI | Zbl
[34] Komorníková M.: Generated aggregation operators. In: Proc. EUSFLAT’99, Palma de Mallorca 1999, pp. 355–357
[35] Komorníková M.: Smoothly generated discrete aggregation operators. BUSEFAL 80 (1999), 35–39
[36] Ling C. M.: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189–212 | MR
[37] Mareš M.: Computation over Fuzzy Quantities. CRC Press, Boca Raton 1994 | MR | Zbl
[38] Marko V., Mesiar R.: A note on a nilpotent lower bound of nilpotent triangular norms. Fuzzy Sets and Systems 104 (1999), 27–34 | MR | Zbl
[39] Marko V., Mesiar R.: Lower and upper bounds of continuous Archimedean $t$-norms. Fuzzy Sets and Systems, to appear | MR
[40] Marková A.: A note on $g$-derivative and $g$-integral. Tatra Mt. Math. Publ. 8 (1996), 71–76 | MR | Zbl
[41] Marková A.: $T$-sum of $L$-$R$-fuzzy numbers. Fuzzy Sets and Systems 85 (1997), 379–384 | DOI | MR
[42] Marková–Stupňanová A.: A note on idempotent functions with respect to pseudo–convolution. Fuzzy Sets and Systems 102 (1999), 417–421 | MR
[43] Menger K.: Statistical metrics. Proc. Nat. Acad. Sci. U. S. A. 8 (1942), 535–537 | DOI | MR | Zbl
[44] Mesiar R.: A note on $T$-sum of $L$-$R$ fuzzy numbers. Fuzzy Sets and Systems 79 (1996), 259–261 | DOI | MR
[45] Mesiar R.: Triangular norm–based addition of fuzzy intervals. Fuzzy Sets and Systems 91 (1997), 231–237 | DOI | MR | Zbl
[46] Mesiar R.: Approximation of continuous $t$-norms by strict $t$-norms with smooth generators. BUSEFAL 75 (1998), 72–79
[47] Mesiar R.: On the pointwise convergence of continuous Archimedean $t$-norms and the convergence of their generators. BUSEFAL 75 (1998), 39–45
[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
[49] Mesiar R., Navara M.: $T_s$-tribes and $T_s$-measures. J. Math. Anal. Appl. 201 (1996), 91–102 | DOI | MR
[50] Mesiar R., Navara M.: Diagonals of continuous triangular norms. Fuzzy Sets and Systems 104 (1999), 35–41 | DOI | MR | Zbl
[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 | DOI | MR
[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 | DOI | MR | Zbl
[53] Pap E.: $g$-calculus. Univ. u Novom Sadu Zb. Rad. Prirod.–Mat. Fak. Ser. Mat. 23 (1993), 145–156 | MR | Zbl
[54] Schweizer B., Sklar A.: Statistical metric spaces. Pacific J. Math. 10 (1960), 313–334 | DOI | MR | Zbl
[55] Schweizer B., Sklar A.: Associative functions and triangle inequalities. Publ. Math. Debrecen 8 (1961), 169–186 | MR
[56] Schweizer B., Sklar A.: Associative functions and abstract semigroups. Publ. Math. Debrecen 10 (1963), 69–81 | MR
[57] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North–Holland, New York 1983 | MR | Zbl
[58] Sklar A.: Fonctions de répartition a $n$-dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8 (1959), 229–231 | MR
[59] Smutná D., Vojtáš P.: Fuzzy resolution with residuation of material implication. In: Proc. EUROFUSE–SIC’99, Budapest 1999, pp. 472–476
[60] Tardiff R. M.: On a generalized Minkowski inequality and its relation to dominates for $t$-norms. Aequationes Math. 27 (1984), 308–316 | DOI | MR | Zbl
[61] Viceník P.: A note on generators of $t$-norms. BUSEFAL 75 (1998), 33–38
[62] Viceník P.: Additive generators and discontinuity. BUSEFAL 76 (1998), 25–28
[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
[64] Viceník P.: Generated $t$-norms and the Archimedean property. In: Proc. EUFIT’99, Aachen 1999 (CD–rom)
[65] Viceník P.: Non–continuous generated $t$-norms. In: Abstracts of Linz’98 “Topological and Algebraic Structures”, Linz 1999, pp. 9–10
[66] Viceník P.: Additive generators of non–continuous triangular norms. Preprint, submitted | Zbl
[67] Yager R. R.: On a general class of fuzzy connectives. Fuzzy Setsna and Systems 4 (1980), 235–242 | DOI | MR | Zbl
[68] Zadeh L. A.: Similarity relations and fuzzy orderings. Inform. Sci. 3 (1971), 177–200 | DOI | MR | Zbl
[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