Abstract
Many of the new MV-valued fuzzy structures, including intuitionistic, neutrosophic, or fuzzy soft sets, can be transformed into so-called almost MV-valued fuzzy sets, or, equivalently, fuzzy sets with values in dual pair of semirings (in symbols, (R,R*)-fuzzy sets). This transformation allows any construction of almost MV-valued fuzzy sets to be retransformed into an analogous construction for these new fuzzy structures. In that way, approximation theories for (R,R*)-fuzzy sets, rough (R,R*)-fuzzy sets theories, or F-transform theories for (R,R*)-fuzzy sets have already been created and then retransformed for these new fuzzy structures. In this paper, we continue this trend and define, on the one hand, the theory of extensional (R,R*)-fuzzy sets defined on sets with fuzzy similarity relations with values in dual pair of semirings and power sets functors related to this theory and, at the same time, the theory of cuts with relational morphisms of these structures. Illustratively, the reverse transformations of some of these concepts into new fuzzy structures are presented.
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