Algunas condiciones para la obtención de negaciones sobre los Conjuntos Tipo 2

En Hernández et. al. (2014) se presentó, a partir del "Principio de Extensión" de Zadeh, un conjunto de operadores en [0,1], exponiendo algunas condiciones bajo las que dichas operaciones son negaciones en L (conjunto de las funciones normales y convexas de [0,1] en [0,1]. Además, L es un subconjunto del conjunto de los grados de pertenencia de los conjuntos borrosos de tipo 2). En el presente trabajo, se define un conjunto de operaciones más general que el estudiado en el referido artículo, y se establecen algunas condiciones suficientes para que sean negaciones y negaciones fuertes sobre L

About t-norms on type-2 fuzzy sets.

Walker et al. defined two families of binary operations on M (set of functions of [0,1] in [0,1]), and they determined that, under certain conditions, those operations are t-norms (triangular norm) or t-conorms on L (all the normal and convex functions of M). We define binary operations on M, more general than those given by Walker et al., and we study many properties of these general operations that allow us to deduce new t-norms and t-conorms on both L, and M

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