Some observations about the convex combination of continuous triangular norms

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