Note on the limit property of triangular norms1

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 .