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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.

The question of determining uniquely either a continuous Archimedean t-norm or a left-continuous t-norm on some vertical segments and/or on some level sets of its graph is investigated.