semiring valued fuzzy sets; semirings valued relations; upper and lower approximations; semirings valued rough sets

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 sets, called $\R$-fuzzy sets, with values in special pairs of semirings. We use this theory to define $\R$-fuzzy relations, lower and upper approximations of $\R$-fuzzy sets by $\R$-relations and rough $\R$-fuzzy sets and we show that these notions can be universally applied to any fuzzy type structure that is transformable to $\R$-fuzzy sets. As an example we also show how this general theory can be used to determine the upper and lower approximations of a colour segment corresponding to a particular colour.