Fuzzy sets and cut systems in a category of sets with similarity relations

Let Omega be a complete residuated lattice. Let SetR(Omega) be the category of sets with similarity relations with values in Omega (called Omega-sets), which is an analogy of the category of classical sets with relations as morphisms. A fuzzy set in an Omega-set in the category SetR(Omega) is a morphism from Omega-set to a special Omega-set (Omega,<->), where <-> is the biresiduation operation in Omega. In the paper, we prove that fuzzy sets in Omega-sets in the category SetR(Omega) can be expressed equivalently as special cut systems (C-alpha)(alpha is an element of Omega).