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

Let $\Omega$ be a complete residuated lattice. Let $\Setr$ be the category of sets with similarity relations with values in $\Omega$ (called $\Omega$-sets), which is an analogy of the category of classical sets with relations as morphisms. A cut system in an $\Omega$-set $(A,\delta)$ in the category $\Setr$ is a system $(C_{\alpha})_{\alpha\in\Omega}$, where $C_{\alpha}$ are subsets in $A\times\Omega$ with some special properties. It is well known that in the category $\Setr$, there is a close relation between cut systems in an $\Omega$-set on one hand and fuzzy sets in the same $\Omega$-set, on the other hand. Moreover, there exists an extension procedure such that any system $(C_{\alpha})_{\alpha}$ of subsets from $A\times\Omega$ can be extended onto a cut system $(\overline{C_{\alpha}})_{\alpha}$. In the paper we prove that the extension procedure is, in some sense, the best possible.

Let $\Omega$ be a complete residuated lattice. Let $\Setr$ be the category of sets with similarity relations with values in $\Omega$ (called $\Omega$-sets), which is an analogy of the category of classical sets with relations as morphisms. A cut system in an $\Omega$-set $(A,\delta)$ in the category $\Setr$ is a system $(C_{\alpha})_{\alpha\in\Omega}$, where $C_{\alpha}$ are subsets in $A\times\Omega$ with some special properties. It is well known that in the category $\Setr$, there is a close relation between cut systems in an $\Omega$-set on one hand and fuzzy sets in the same $\Omega$-set, on the other hand. Moreover, there exists an extension procedure such that any system $(C_{\alpha})_{\alpha}$ of subsets from $A\times\Omega$ can be extended onto a cut system $(\overline{C_{\alpha}})_{\alpha}$. In the paper we prove that the extension procedure is, in some sense, the best possible. This will be expressed by the theorem which states that the category of such subsets is a full reflective subcategory in the category of cut systems.