Revisiting type-2 triangular norms on normal convex fuzzy truth values

TL;DR

This paper investigates the properties and axioms of type-2 t-norms on normal convex fuzzy truth values, revealing their relationships and necessary conditions for convolution operations.

Contribution

It characterizes the axioms and relationships among different types of t-norms on fuzzy truth values, clarifying their structural properties and constraints.

Findings

Only one non-convolution type-2 t-norm satisfies distributivity.

Walker and Walker's t-norm is stronger than t_r-norm on .

Necessary conditions for t_r-(co)norm convolutions are established.

Abstract

This paper studies t-norms on the space $\mathbf{L}$ of all normal and convex fuzzy truth values. We first prove that the only non-convolution form type-2 t-norm constructed by Wu et al. satisfies the distributivity law for meet-convolution and show that t-norm in the sense of Walker and Walker is strictly stronger than t$_{r}$-norm on $\mathbf{L}$, which is strictly stronger than t-norm on $\mathbf{L}$. Furthermore, we characterize some restrictive axioms of t$_{r}$-norms for convolution operations on $\mathbf{L}$ and obtain some necessary conditions for t$_{r}$-(co)norm convolution operations on $\mathbf{L}$ .