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On the convex combination of left-continuous t-norms. (English) Zbl 1101.39010
Summary: A conjecture of C. Alsina, M. J. Frank and B. Schweizer concerning the convex combinations of t-norms [ibid. 66, No. 1–2, 128–140 (2003; Zbl 1077.39021)] is proved for certain left-continuous t-norms. It is shown that a nontrivial convex combination of two left-continuous t-norms is never a t-norm (in fact, the associativity property of t-norms is violated) provided that the two t-norms have the same involutive \(u\)-level set for some \(u \in [0,1[\). The proof is motivated by a geometrical understanding of associativity.

MSC:
39B22 Functional equations for real functions
39B52 Functional equations for functions with more general domains and/or ranges
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