Associativity of discretized triangular norms

Abstract

In this article, we mainly investigate when the binary operation $T^{[n]}:[0, 1]^2\rightarrow [0, 1]$ satisfies the associative law and thus becomes a triangular norm where \begin{equation*} T^{[n]}(x,y)=\left\{\begin{array}{ll} f_{n}(T(f_{n}(x),f_{n}(y))), &\text{if}\,\, (x,y)\in{[0,1)^{2}}, \\ \min\{x,y\}, &\text{otherwise,} \end{array}\right. \end{equation*} $f_{n}:[0,1]\rightarrow\left\{0, \frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\right\}$ given by $f_{n}(x)=\frac{1}{n}{\lfloor nx \rfloor}$ with $n$ a positive integer and $T:[0, 1]^2\rightarrow [0, 1]$ is a triangular norm. We present some sufficient conditions for $T^{[n]}$ to be triangular norms, and for $T^{[n]}$ to be never associative when $T$ is strictly monotonic.