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

Abstract

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 Łukasiewicz 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). © 2002 Elsevier Science B.V. All rights reserved.