Reflective categories of cut systems and fuzzy sets in Ω-sets

We investigate relationships between fuzzy sets and some variants of α-cuts (so called f-cuts) in sets with similarity relations, with values in a complete residuated lattice Ω (so called Ω-sets), which are objects of two special categories Set(Ω) and SetR(Ω). We prove that these relationships can be expressed as natural isomorphisms between covariant and contravariant functors representing fuzzy sets and f-cuts, respectively. Moreover, some relationships between sets of fuzzy sets and sets of f-cuts in an Ω-set (A; δ), which are endowed with binary operations extended either from binary operations in the lattice Ω, or from binary operations defined in a set A by the generalized Zadeh's extension principle will be investigated. We prove that the final binary structures are (under some conditions) isomorphic.