Convex combinations of continuous t-norms with the same diagonal function (Q949711)

The authors study nontrivial convex combinations of triangular norms with the same diagonal functions, and present the following results. If the nontrivial convex combination of two nilpotent \(t\)-norms with the same diagonal function is itself a \(t\)-norm, then the two nilpotent \(t\)-norms are coincide. Analogous statement remains true for strict \(t\)-norms. It is also shown that a nontrivial convex combination of two continuous \(t\)-norms with the same diagonal function is a \(t\)-norm if and only if the two continuous \(t\)-norms are coincide. An example for a convex class of non-continuous \(t\)-norms is also presented.