Cut systems in sets with similarity relations

The aim of the paper is to find some analogies between α-cuts systems in a classical fuzzy set theory and some nested systems of subsets defined in sets with similarity relations. We prove that any set A with similarity relation δ with values in complete residuated lattice Ω can be uniquely defined by a special system of nested subsets in A×A. Using some nested systems we can also define fuzzy sets in sets with similarity relations and we show that these fuzzy sets correspond to fuzzy sets classically defined in (A,δ) (in corresponding categories of sets with similarities). Moreover, we also define an interpretation of a first order fuzzy logic in models based on these nested systems and we prove that there are some relationships between classical interpretations in sets with similarity relations and interpretations using nested systems.