The relationships between type-2 t-norms on normal convex fuzzy truth values

Recent literature has mainly focused on four forms of type-2 t-norms on $F_{NC}$ composed of all normal convex fuzzy truth values: t-norms defined on the complete lattice, t_r-norms defined by Hernández et al., t${}_{lo}$-norms defined by Harding et al., and t${}_{lor}$-norms defined by Wu et al. The paper studies the relationships among these four types of type-2 t-norms constructed by generalized extended t-norms that come from the generalization of Zadeh's extension principle. Firstly, we sequentially characterize the conditions under which generalized extended t-norms satisfy each restrictive axiom in the definitions of type-2 t-norms, particularly closure properties. Then, we prove that on $F_{NC}$, generalized extended t-norms being t-norms (resp. t${}_{lo}$-norms) is equivalent to them being t_r-norms (resp. t${}_{lor}$-norms). Finally, through examples, we demonstrate that on $F_{NC}$, t${}_{lo}$-norms (resp. t${}_{lor}$-norms) are strictly stronger than t-norms (resp. t_r-norms) even if all of them are constructed by generalized extended t-norms.