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

Distributivity between extended nullnorms and uninorms on fuzzy truth values

This paper mainly investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. It presents the distributive laws between the extended nullnorm and t-conorm, and the left and right distributive laws between the extended generalization nullnorm and uninorm, where a generalization nullnorm is an operator from the class of aggregation operators with absorbing element that generalizes a nullnorm.Comment: 2