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 F-2 (a set of all functions defined from [0,1] into itself). Based on it, we first construct some type-2 t-norms on the fuzzy truth values F-2 with the ordinary partial order <= and the partial order (sic) 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 (F-2, <=) and (F-2, (sic)) are respectively represented. Finally, we briefly discuss the compositional rule of inference based on type-2 t-norms and their residual operators. (C) 2015 Elsevier Inc. All rights reserved.