Completions of cut systems in -sets

Let be a complete residuated lattice. Let be the category of sets with similarity relations with values in (called -sets), which is an analogy of the category of classical sets with relations as morphisms. A cut in an -set is a system , where are subsets of . It is well known that in the category , there is a close relation between special cuts (called f-cuts) in an -set on one hand and fuzzy sets in the same -set, on the other hand. Moreover, there exists a completion procedure according to which any cut can be extended onto an f-cut . In the paper, 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.