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On the order of triangular norms – comments on “A triangular norm hierarchy” by E. Cretu. (English) Zbl 1012.03033
The authors present a critical overview of E. Cretu’s paper “A triangular norm hierarchy” [Fuzzy Sets Syst. 120, 371-383 (2001; Zbl 0982.03014)]. They make it evident that the results on the order of t-norms (as real functions) presented by Cretu can be found in previous sources and, in a compact form, in their book [Triangular norms. Dordrecht/Boston/London: Kluwer (2000; Zbl 0972.03002), Ex. 3.32 and Chapter 6]. (The book appeared between the submission and publication of the paper by Cretu.) According to the paper under review, the only original results of Cretu are in Section 4, where he compares the strength of a semantics of fuzzy logics based on different triangular norms (in the sense of D. Butnariu, E. P. Klement and S. Zafrany [“On triangular norm-based propositional fuzzy logics”, Fuzzy Sets Syst. 69, 241-255 (1995; Zbl 0844.03011)]). However, these follow also from previous papers [namely M. Navara, “A characterization of triangular norm based tribes”, Tatra Mt. Math. Publ. 3, 161-166 (1993; Zbl 0799.28013) and D. Butnariu, E. P. Klement and M. Navara, “All strict triangular norms are ‘equally strong’ ”, Proc. Conf. on Fuzzy Logic and Applications, Tel Aviv, Israel, 1997, 116-124 (1997), Cor. 2.8], which are not mentioned in both sources. A brief overview of the situation can also be found in the review of Cretu’s paper [loc. cit.]. The paper under review contains an extensive bliography relevant to properties of t-norms.

MSC:
03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
26B99 Functions of several variables
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