Rough Semiring-Valued Fuzzy Sets with Application

Many of the new fuzzy structures with complete MV-algebras as value sets, such as hesitant, intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into one type of fuzzy set with values in special complete algebras, called AMV-algebras. The category of complete AMV-algebras is isomorphic to the category of special pairs (R,R∗) of complete commutative semirings and the corresponding fuzzy sets are called (R,R∗)-fuzzy sets. We use this theory to define (R,R∗)-fuzzy relations, lower and upper approximations of (R,R∗)-fuzzy sets by (R,R∗)-relations, and rough (R,R∗)-fuzzy sets, and we show that these notions can be universally applied to any fuzzy type structure that is transformable to (R,R∗)-fuzzy sets. As an example, we also show how this general theory can be used to determine the upper and lower approximations of a color segment corresponding to a particular color.