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 $⋏$ and $⋎$: $$(f⋏g)(x)=\sup \left\{f(y)\ast g(z)\mid y△z=x\right\},$$ $$(f⋎g)(x)=\sup \left\{f(y)\ast g(z)\mid y▽z=x\right\},$$ where $f,g\in Map([0,1],[0,1])$, $△$ and $▽$ 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 $Map([0,1],[0,1])$.}

arXiv admin note: text overlap with arXiv:1907.12394