On the convex combination of left-continuous t-norms

A conjecture of C. Alsina M. J. Frank and B. Schweizer concerning the convex combinations of t-norms [4] 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 [0 1[. The proof is motivated by a geometrical understanding of associativity. © Birkhäuser Verlag, Basel, 2006.