A u-map representation of general type-2 fuzzy sets via concepts from activation detection: Application to constructing type-2 fuzzy set measures

Following the introduction of type-2 fuzzy sets (T2FS) by Zadeh in 1975, the theory of T2FS has been further developed by a series of notable researchers, and it has been widely used in a variety of applications nowadays. While recent theoretical developments have led to simplifications in the theory of T2FS, it can be argued that theoretical difficulties concerning these developments still persist, inhibiting T2FS from reaching its full potential. With this in mind, in this article, we introduce a concept termed the u-map representation of type-2 fuzzy sets, a simple yet powerful tool for T2FS specialists to further develop T2FS theories, and for non-specialists to adapt T2FS for their applications. With the u-map representation, we show that it is extremely simple to translate manipulations involving type-1 fuzzy sets into corresponding manipulations involving type-2 fuzzy sets. This means that a measure for type-1 fuzzy sets could be converted into a corresponding measure for type-2 fuzzy sets in a straightforward and theoretically satisfying manner, overcoming some of the key theoretical difficulties and restrictions of previous approaches. Moreover, we describe a foundational mental model underlying the u-map representation. Such a foundational model empowers one to check the reasonableness of various definitions, propositions, and computation results associated with measures constructed under the u-map representation. We illustrate the utility of the u-map representation and its foundational mental model via the constructions and the interpretations of a T2FS subsethood measure, a T2FS entropy measure, and a T2FS relative entropy measure.