On the Existence of $t_r$-Norm and $t_r$-Conorm not in Convolution Form

Abstract

This paper constructs a $t_{r}$-norm and a $t_{r}$-conorm on the set of all normal and convex functions from ${[0, 1]}$ to ${[0, 1]}$, which are not obtained by using the following two formulas on binary operations ${\curlywedge}$ and ${\curlyvee}$: $$ {(f\curlywedge g)(x)=\sup\left\{f(y)\ast g(z)\mid y\vartriangle z=x\right\},} $$ $$ {(f\curlyvee g)(x)=\sup\left\{f(y)\ast g(z)\mid y\ \triangledown\ z=x\right\},} $$ where ${f, g\in Map([0, 1], [0, 1])}$, ${\vartriangle}$ and ${\triangledown}$ are respectively a ${t}$-norm and a ${t}$-conorm on ${[0, 1]}$, and ${\ast}$ is a binary operation on ${[0, 1]}$. {\color{blue}This result answers affirmatively an open problem posed in \cite{HCT2015}. Moreover, the duality between $t_r$-norms and $t_r$-conorms is obtained by the introduction of operations dual to binary operations on ${Map([0, 1], [0, 1])}$.}