On the convex combination of TD and continuous triangular norms

The problem of when Tλ≜(1-λ)TD+λTλ∈(0,1) is a triangular norm, where TD is the drastic product and T is a continuous triangular norm, is studied. It is shown that Tλ cannot be a triangular norm when T is nilpotent. It is also shown that Tλ is a triangular norm if T is strict and its additive generator f   satisfies f(λx)=f(x)+f(λ)f(λx)=f(x)+f(λ) for all x∈[0,1]x∈[0,1]. The cases that T=TMT=TM and T is the ordinal sum of continuous Archimedean summands are also discussed. Some left-continuous t-norms which can be combined with each other are given.