Affiliations: School of Mathematical Sciences, Sichuan Normal University, Chengdu, People’s Republic of China
Correspondence:
[*]
Corresponding author. Xue-Ping Wang, School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, People’s Republic of China. Tel.: +86 28 84761502; Fax: +86 28 84761502; E-mail: xpwang1@hotmail.com.
Note: [1] Supported by the National Natural Science Foundation of China (No. 12071325).
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)=limn→∞xT(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 = TM.
Keywords: Triangular norm, the limit property, the contrary limit property, Archimedean property, continuity
