Semirings, dual pair of semirings, transformation of new fuzzy sets, approximation of new fuzzy sets

Many of the new types of fuzzy sets, such as intuitionistic, neutrosophic, multi-level or fuzzy soft sets and their combinations, can be transformed into one common type of fuzzy sets, called $\R$-fuzzy sets, with values in a set $R$ that is a common underlying set of complete commutative idempotent semirings $\cal R$ and $\cal R^*$. For $\R$-fuzzy sets, the theory of lower and upper approximations by $\R$-relations is defined and the basic properties of these approximations are presented. Using examples of the transformation of some new types of $MV$-valued fuzzy sets and corresponding fuzzy relations into R-fuzzy sets and R-fuzzy relations, examples of approximation of these new types of fuzzy sets through their fuzzy relations are presented, without having to define these operators separately for each new type of fuzzy set.