Describing fuzzy sets using a new concept: fuzzify functor

This paper proposed a fuzzify functor as an extension of the concept of fuzzy sets. The fuzzify functor and the first-order operated fuzzy set are defined. From the theory analysis, it can be observed that when the fuzzify functor acts on a simple crisp set, we get the first order fuzzy set or type-1 fuzzy set. By operating the fuzzify functor on fuzzy sets, we get the higher order fuzzy sets or higher type fuzzy sets and their membership functions. Using the fuzzify functor we can exactly describe the type-1 fuzzy sets, type-2 fuzzy sets and higher type or higher order fuzzy sets. The fuzzify functor makes type-1, type-2 and any fuzzy sets much more accessible to all readers. This work has been supported in part by the Project of Tianjin Key Laboratory of Control Theory & Applications in Complicated Systems and the Key Project of Tianjin Natural Science Foundation (Grant No. 06YFJMJC01)