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

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 $\mathbf{L}$ .