Some observations about the convex combination of continuous triangular norms

The convex combination of continuous triangular norms is discussed. Let C1C1 be the class of all nilpotent tt-norms, C2C2 be the class of all strict tt-norms and C3C3 be the class of all continuous non-Archimedean tt-norms. It is shown that for any tt-norms T1,T2T1,T2 which are taken from two different classes of Ci(i=1,2,3), the convex combination T=(1−λ)T1+λT2T=(1−λ)T1+λT2 is not associative for any λ∈(0,1)λ∈(0,1). The case of two continuous non-Archimedean tt-norms is also investigated. By our results, the problem of whether or not two continuous tt-norms can be combined with each other is reduced to that of whether two strict (or two nilpotent) tt-norms can be combined with each other or not.