Isomorphisms and functors of fuzzy sets and cut systems

Abstract

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.