Revisiting T-Norms for Type-2 Fuzzy Sets

Let $\mathbf{L}$ be the set of all normal and convex functions from ${[0, 1]}$ to ${[0, 1]}$. This paper proves that ${t}$-norm in the sense of Walker-and-Walker is strictly stronger that ${t_r}$-norm on $\mathbf{L}$, which is strictly stronger than ${t}$-norm on $\mathbf{L}$. Furthermore, let ${\curlywedge}$ and ${\curlyvee}$ be 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 ${f, g\in Map([0, 1], [0, 1])}$, where ${\vartriangle}$ and ${\triangledown}$ are respectively a ${t}$-norm and a ${t}$-conorm on ${[0, 1]}$ (not necessarily continuous), and ${\star}$ is a binary operation on ${[0, 1]}$. Then, it is proved that if the binary operation ${\curlywedge}$ is a ${t_r}$-norm (resp., ${\curlyvee}$ is a ${t_r}$-conorm), then ${\vartriangle}$ is a continuous ${t}$-norm (resp., ${\triangledown}$ is a continuous ${t}$-conorm) on ${[0, 1]}$, and ${\star}$ is a ${t}$-norm on ${[0, 1]}$.Comment: arXiv admin note: text overlap with arXiv:1908.10532, arXiv:1907.1239