Granular fuzzy sets and three-way approximations of fuzzy sets

A Pawlak approximation space is a pair of a ground set/space and a quotient set/space, where the latter is induced by an equivalence relation on the former. With this two-space understanding, it is possible to lift any concepts and notions from the ground space to the quotient space. The results are granular versions that approximate the original concepts and notions. In this paper, we investigate the problem of lifting a fuzzy set in the ground space to granular fuzzy sets in the quotient space. By applying the principles of three-way decision, we introduce the idea of three-way granular approximations of fuzzy sets in terms of three granular fuzzy sets that represent the two extremes and one middle. The two extremes are given by granular rough fuzzy sets. We present several different ways to interpret and construct a middle.