On type-2 fuzzy relations and interval-valued type-2 fuzzy sets

This paper introduces new operations on the algebra of fuzzy truth values, extended supremum and extended infimum, which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type-2 fuzzy sets, respectively. Using these new operations, the properties of type-2 fuzzy relations are discussed, especially the compositions of type-2 fuzzy relations. On this basis, this paper introduces interval-valued type-2 fuzzy sets and interval-valued type-2 fuzzy relations, and discusses their properties.