On T-Norms for Type-2 Fuzzy Sets

Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets. The degree of membership of an element for T2FSs is a fuzzy set in [0, 1], that is, a T2FS is determined by a membership function from the universe of discourse X to M, where M is the set of functions from [0, 1] to [0, 1]. Walker and Walker extended the definitions oft-norm (triangular norm) and t-conorm to L (subset of normal and convex functions of M), establishing the tr-norms and tr-conorms (according to the “restrictive axioms” given by them), and defined two families of binary operations on M and found that, under certain conditions, these operations are t_r.-norms or t_r.-conorms on L. In this paper, we introduce more general binary operations on M than those given by Walker and Walker and study which of the minimum conditions necessary for these operations satisfy each of the axioms of the tr-norm and tr-conorm. In particular, interesting results about the closure properties are obtained, and the main result of the paper provides sufficient conditions for the given operations to be t_r.-norms or t_r-conorms on L.