On the convex combination of T-D 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.
convex combination continuous triangular norm triangular norm drastic product nilpotent triangular norm strict triangular norm