On the convex combination of TD and continuous triangular norms

The problem of when T-lambda Delta// (1-lambda)T-D + lambda T lambda is an element of (0, 1) is a triangular norm, where T-D is the drastic product and T is a continuous triangular norm, is studied. It is shown that T-lambda cannot be a triangular norm when T is nilpotent. It is also shown that T-lambda is a triangular norm if T is strict and its additive generator f satisfies f (lambda x) = f (x) + f (lambda) for all x is an element of [0, 1]. The cases that T = T-M 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. (c) 2007 Elsevier Inc. All rights reserved.