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

  • Authors:

  • Erich Peter Klement;Radko Mesiar;Endre Pap

  • Affiliations:

  • Department of Algebra, Stochastics and Knowledge-Based Mathematical Systems, Johannes Kepler University, 4040 Linz, Austria;Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak Technical University, 81 368 Bratislava, Slovakia;Institute of Mathematics, University of Novi Sad, 21000 Novi Sad, Yugoslavia

  • Venue:
  • Fuzzy Sets and Systems - Mathematics
  • Year:
  • 2002

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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 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).