residuated lattice; set with similarity relation; fuzzy set; cut system

It is well known that any fuzzy set $X$ in a classical set $A$ with values in a complete (residuated) lattice $\Omega$ can be identified with a system of $\alpha$-cuts $X_{\alpha}$, $\alpha\in\Omega$. In this paper we are interested in relationships between sets of fuzzy sets and sets of f-cuts in an $\Omega$-set $(A,\delta)$ in corresponding categories $\Set$ and $\Setr$, endowed with binary operations extended either from binary operations in the lattice $\Omega$, or from binary operations defined on a set $A$ by the generalized Zadeh's extension principle. We prove that the final binary structures are (under some conditions) isomorphic.