Abstract
Many of the new types of fuzzy sets, such as intuitionistic, neutrosophic or fuzzy soft sets, can be transformed into one common type of fuzzy sets, called (R,R⁎)-fuzzy sets, with values in a set R that is a common underlying set of complete commutative idempotent semirings R and R⁎. For (R,R⁎)-fuzzy sets, the theory of lower and upper approximations by (R,R⁎)-relations is defined and properties of these approximations are investigated. The transformation of new types of fuzzy sets into (R,R⁎)-fuzzy sets is used to show the possibility of introducing approximation operators in new types of fuzzy sets without having to define these operators separately for each new type of fuzzy set.
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