Associativity of discretized triangular norms

Abstract

In this article, we mainly investigate when the binary operation $T^{[n]}:[0,1{]}^2\to [0,1]$ 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*} $f_n:[0,1]\to \left\{0,\frac{1}{n},\frac{2}{n},\dots ,\frac{n-1}{n},1\right\}$ given by $f_n(x)=\frac{1}{n}\lfloor nx\rfloor$ with $n$ a positive integer and $T:[0,1{]}^2\to [0,1]$ 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.

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