semirings; semiring-valued fuzzy sets; power set theories; relational morphisms

Semiring-valued fuzzy sets represent fuzzy objects with new value-set structures, which make it possible to unify part of new and frequently used fuzzy structures. The theory of semiring-valued fuzzy sets uses some basic tools of category theory and can be directly applied to these new fuzzy structures. For this reason, the development of this theory appears to be useful not only for the theory of semirings-valued fuzzy sets, but also for the theory of new fuzzy structures. In this paper, we will therefore develop another part of this theory, namely, we define two basic types of relational power set theory for these structures and examine the basic relationships between these theories.