Note on the limit property of triangular norms1

Abstract

In this article, we characterize triangular norms that have not the limit property, which are applied for exploring the characterizations of function f : [0, 1] → [0, 1] with f ( x ) = lim n → ∞ x T ( n ) for a triangular norm T when the function f is continuous. In particular, we prove that a continuous t-norm T satisfies that f (x) >0 for all x ∈ (0, 1) if and only if 0 is an accumulation point of its non-trivial idempotent elements, and the function f is continuous on [0,1] if and only if T = T M .