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.