Characterization of t-norms for type-2 fuzzy sets (2511.17640v1)

Abstract: Type-2 fuzzy set (T2 FS) were introduced by Zadeh in 1965, and the membership degrees of T2 FSs are type-1 fuzzy sets (T1 FSs). Owing to the fuzziness of membership degrees, T2 FSs can better model the uncertainty of real life, and thus, type-2 rule-based fuzzy systems (T2 RFSs) become hot research topics in recent decades. In T2 RFS, the compositional rule of inference is based on triangular norms (t-norms) defined on complete lattice (L, \le ) ( L is the set of all convex normal functions from [0,1] to [0,1], and , \le is the so-called convolution order). Hence, the choice of t-norm on (L,\le) may influence the performance of T2 RFS. Therefore, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on (L,\le), the mainstream method is convolution which is induced by two operators on the unit interval [0,1]. A key problem appears naturally, when convolution is a t-norm on (L,\le). This paper gives the necessary and sufficient conditions under which convolution is a t-norm on (L,\le). Moreover, note that the computational complexity of operators prevent the application of T2 RFSs. This paper also provides one kind of convolutions which are t-norms on (L,\le) and extremely easy to calculate.