Fuzzy sets with values in a complete lattice \(L\) can be defined either by means of membership functions, or, equivalently, by means of nested systems of \(\alpha\)-cuts. This paper is focused on finding some analogies between the second approach and some nested systems of subsets defined in sets with similarity relations. Special \(f\)-cut systems are introduced and their properties and category-theoretic isomorphic objects are discussed. Moreover, a first-order fuzzy logic interpretations in cut systems are discussed, proving, among others, some relationships between classical interpretations in sets with similarity relations and interpretations using nested systems.