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 ${\curlywedge}$ is a ${t}$-norm or ${\curlyvee}$ is a ${t}$-conorm, is a ${t}$-norm, where ${\curlywedge}$ and ${\curlyvee}$ are special convolution operations defined by $${(f\curlywedge g)(x)=\sup\left\{f(y)\star g(z): y\vartriangle z=x\right\},} $$ $${(f\curlyvee g)(x)=\sup\left\{f(y)\star g(z): y\ \triangledown\ z=x\right\},} $$ for any ${f, g\in Map([0, 1], [0, 1])}$, where ${\vartriangle}$ and ${\triangledown}$ 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 ${\curlywedge}$ and ${\curlyvee}$.