α-CUTS AND MODELS OF FUZZY LOGIC

Let Omega be a complete residuated lattice. Let (A, delta) be a set with a similarity relation delta with values in Omega. We prove that any fuzzy set in (A, delta), i.e. a morphism f : (A, delta) -> (Omega, <->) in a special category Set(Omega) can be defined as a system (called f-cut system) of subsets (C-alpha)(alpha is an element of Omega) in a set A with some special properties. In (A, delta) f-cut systems represent an analogy of alpha-cuts of a classical Omega-valued fuzzy set in A. We define an interpretation of a first order fuzzy logic in models based on these f-cut systems and we prove that there is a natural relationships between classical interpretations in sets with similarity relations and interpretations in f-cut systems.