Convex combinations of continuous t-norms with the same diagonal function

The problem of whether a non-trivial convex combination of two continuous t-norms with the same diagonal function can be a t-norm is studied. It is shown that in both cases–of two nilpotent and of two strict t-norms–a non-trivial convex combination of t-norms with common diagonal function is associative only if the two t-norms involved coincide. For general continuous t-norms a similar result follows. An example of a convex class of non-continuous t-norms is also included.