A Study of Modules of Fractions Under Fuzzy Sets and Soft Sets

The primary purpose of this paper is to introduce the concept of the fuzzy module of fractions and investigate a relationship between fuzzy modules of fractions and rough sets (RSs). In this respect, first, the concept of the soft module of fractions is introduced. Then, the idea of fuzzy approximations of the FS of a module of fractions is defined by using a soft module of fractions and obtaining a new hybrid model called multi-granulation soft rough FSs (MGSR-FSs) in a module of fractions. Moreover, the idea of a fuzzy upper rough module of fractions is introduced. However, it is important to note that the fuzzy lower approximation of a fuzzy module of fractions is not a fuzzy lower rough module of fractions. Several examples are provided to elaborate on the proposed notions.