Answering An Open Problem on $T$-Norms for Type-2 Fuzzy Sets

Abstract

This paper proves that a binary operation $\star$ on $[0,1]$, ensuring that the binary operation $⋏$ is a $t$-norm or $⋎$ is a $t$-conorm, is a $t$-norm, where $⋏$ and $⋎$ are special convolution operations defined by $$(f⋏g)(x)=\sup \left\{f(y)\star g(z):y△z=x\right\},$$ $$(f⋎g)(x)=\sup \left\{f(y)\star g(z):y▽z=x\right\},$$ for any $f,g\in Map([0,1],[0,1])$, where $△$ and $▽$ are a continuous $t$-norm and a continuous $t$-conorm on $[0,1]$, answering negatively an open problem posed in \cite{HCT2015}. Besides, some characteristics of $t$-norm and $t$-conorm are obtained in terms of the binary operations $⋏$ and $⋎$.