Completions of cut systems in $Q$-sets

Abstract

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