The Extended Sup-Star Composition for Type-2 Fuzzy Sets Made Simple

In this paper we introduce embedded type-2 fuzzy relations and provide an alternative proof of the extended sup-star composition that makes use of these embedded type-2 fuzzy relations. The paper commences by giving a background to some terms and definitions used in type-2 fuzzy logic. Definitions are given for a type-2 fuzzy set, a secondary membership function and an embedded set. Additionally the representation theorem is described along with an explanation of type-2 fuzzy relations. These preliminary sections provide the basis for the remainder of the paper, which introduces the concept of embedded fuzzy relations and makes use of this in order to provide an alternative proof of the extended sup-star composition.