Some observations about the convex combination of continuous triangular norms Original Research Article

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 View the MathML sourceCi(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.