Type-2 triangular norms and their residual operators

In this paper, we mainly investigate T-extension operations of any t-norms and t  -conorms on type-2 fuzzy sets’ truth values F2F2 (a set of all functions defined from [0,1][0,1] into itself). Based on it, we first construct some type-2 t  -norms on the fuzzy truth values F2F2 with the ordinary partial order ⩽⩽ and the partial order ⊑⊑, respectively. The algebraic properties of these type-2 t-norms are then studied. Moreover, the residual operators of some special type-2 t  -norms on (F2,⩽)(F2,⩽) and (F2,⊑)(F2,⊑) are respectively represented. Finally, we briefly discuss the compositional rule of inference based on type-2 t-norms and their residual operators.