Isomorphisms and functors of fuzzy sets and cut systems

Any fuzzy set \(X\) in a classical set \(A\) with values in a complete (residuated) lattice \( Q\) can be identified with a system of \(\alpha \)-cuts \(X_{\alpha }\), \(\alpha \in Q\). Analogical results were proved for sets with similarity relations with values in \( Q\) (e.g. \( Q\)-sets), which are objects of two special categories \({\mathbf K}={Set}( Q)\) or \({SetR}( Q)\) of \( Q\)-sets, and for fuzzy sets defined as morphisms from an \( Q\)-set into a special \(Q\)-set \(( Q,\leftrightarrow )\). These fuzzy sets can be defined equivalently as special cut systems \((C_{\alpha })_{\alpha }\), called f-cuts. This equivalence then represents a natural isomorphism between covariant functor of fuzzy sets \(\mathcal{F}_{\mathbf K}\) and covariant functor of f-cuts \(\mathcal{C}_{\mathbf K}\). In this paper, we prove that analogical natural isomorphism exists also between contravariant versions of these functors. We are also interested in relationships between sets of fuzzy sets and sets of f-cuts in an \(Q\)-set \((A,\delta )\) in the corresponding categories \({Set}( Q)\) and \({SetR}( Q)\), which are endowed with binary operations extended either from binary operations in the lattice \(Q\), or from binary operations defined in a set \(A\) by the generalized Zadeh’s extension principle. We prove that the resulting binary structures are (under some conditions) isomorphic.