On the order of triangular norms-comments on "A triangular norm hierarchy" by E. Cretu

In Cretu (Fuzzy Sets and Systems 120 (2001) 371), the members of the families of Frank, Dubois-Prade, Yager, and Hamacher t-norms, respectively, are compared (in a pointwise way) with the minimum, the product, and the Lukasiewicz t-norm. All these results are well-known and trivial. Moreover, these families of t-norms cannot only be compared with the three basic t-norms above, but all these families are monotone with respect to their index, a fact which is also well known and straightforward to prove (with the exception of the family of Frank t-norms whose monotonicity has been first proven in Butnariu and Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht, 1993).

[1]  Erich-Peter Klement,et al.  Some mathematical aspects of fuzzy sets: Triangular norms, fuzzy logics, and generalized measures , 1997, Fuzzy Sets Syst..

[2]  Robert Lowen.  Fuzzy set theory - basic concepts, techniques and bibliography , 1996 .

[3]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[4]  D. Butnariu,et al.  Triangular Norm-Based Measures and Games with Fuzzy Coalitions , 1993 .

[5]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[6]  Radko Mesiar,et al.  A characterization of the ordering of continuous t-norms , 1997, Fuzzy Sets Syst..

[7]  Erich-Peter Klement,et al.  Operations on fuzzy sets - an axiomatic approach , 1982, Inf. Sci..

[8]  J. Aczél,et al.  Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements , 1982 .

[9]  János Fodor,et al.  A characterization of the Hamacher family of t -norms , 1994 .

[10]  János Fodor,et al.  A remark on constructing t-norms , 1991 .

[11]  Yu Yandong.  Triangular norms and TNF-sigma-algebras , 1985 .

[12]  J. Dombi.  A general class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators , 1982 .

[13]  A. H. Clifford,et al.  Naturally Totally Ordered Commutative Semigroups , 1954 .

[14]  S. Weber,et al.  Fundamentals of a Generalized Measure Theory , 1999 .

[15]  M. J. Frank.  On the simultaneous associativity ofF(x,y) andx +y -F(x,y) , 1979 .

[16]  S. Weber.  ⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥ , 1984 .

[17]  J. M Anthony,et al.  Fuzzy groups redefined , 1979 .

[18]  J. Fodor.  Contrapositive symmetry of fuzzy implications , 1995 .

[19]  B. Schweizer,et al.  Statistical metric spaces. , 1960 .

[20]  S. Weber.  A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms , 1983 .

[21]  E. Klement.  Characterization of fuzzy measures constructed by means of triangular norms , 1982 .

[22]  M. Mizumoto.  Pictorial representations of fuzzy connectives, part I: cases of t-norms, t-conorms and averaging operators , 1989 .

[23]  Liu Dsosu,et al.  Fuzzy random measure and its extension theorem , 1983 .

[24]  Francesc Esteva,et al.  Review of Triangular norms by E. P. Klement, R. Mesiar and E. Pap. Kluwer Academic Publishers , 2003 .

[25]  R. Yager.  On a general class of fuzzy connectives , 1980 .

[26]  K. Menger.  Statistical Metrics. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[27]  János Fodor,et al.  Fuzzy connectives via matrix logic , 1993 .

[28]  Anna Kolesárová,et al.  Similarity preserving t-norm-based additions of fuzzy numbers , 1997, Fuzzy Sets Syst..

[29]  Eugen Cretu.  A triangular norm hierarchy , 2001, Fuzzy Sets Syst..

[30]  D. Butnariu,et al.  On triangular norm-based propositional fuzzy logics , 1995 .

[31]  E. Klement.  Construction of Fuzzy σ-algebras using triangular norms , 1982 .

[32]  J. Fodor.  On fuzzy implication operators , 1991 .

[33]  Joan Torrens,et al.  On a family of t-norms , 1991 .

[34]  János Fodor,et al.  A new look at fuzzy connectives , 1993 .

[35]  Anna Kolesárová.  Triangular norm-based addition preserving linearity of T-sums of linear fuzzy intervals , 1998 .

[36]  Siegfried Weber,et al.  Generalized measures , 1991 .

[37]  Siegfried Weber,et al.  Measures of fuzzy sets and measures of fuzziness , 1984 .

[38]  Didier Dubois,et al.  New Results about Properties and Semantics of Fuzzy Set-Theoretic Operators , 1980 .