semiring-valued fuzzy sets, semiring-valued fuzzy relations, semiring-valued rough fuzzy sets, applications to color segmenetation

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 common type of fuzzy sets with values in special semirings. We use this transformation of fuzzy structures to unify the theory of $\R$-fuzzy rough sets with these new fuzzy structures. For this purpose, we use the $({\cal R}_2,{\cal R}_2^*)$-fuzzy rough set defined for fuzzy soft set and $({\cal R}_1,{\cal R}_1^*)$-fuzzy rough sets defined for intuitionistic fuzzy sets. 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.