Operations in 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 α-cuts. Analogical results were proved for sets with similarity relations, with values in Q (e.g. Q-sets), which are objects of two special categories of Q-sets, and for fuzzy sets defined as special morphisms in these categories. These fuzzy sets can be defined equivalently as special cut systems, called f-cuts. In the paper, we are interested in relationships between sets of fuzzy sets and sets of f-cuts in an Q-set (A, δ), in corresponding categories, 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.