fuzzy set, sets with similarity relations, cut systems, reflective categories

There exists a completion procedure according to any cut can be extended to an f-cut. We prove that the completion procedure is, in some sense, the best possible. This will be expressed by the theorem which states that the category of f-cuts is a full reflective subcategory in the category of cuts. Some relationships between sets of fuzzy sets and sets of f-cuts in a set with similarity relation in corresponding categories, which are endowed with binary operations extended either from binary operations in the lattice or from binary operations defined in a set A by the generalized Zadeh extension principle will be investigated. We prove that the final binary structures are (under some conditions) isomorphic