Given a set \(A\) and a complete residuated lattice \(\Omega\) [V. Novák et al., Mathematical principles of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1999; Zbl 0940.03028)], a nested system of \(\alpha\)-cuts in \(A\) is a family \(\mathcal{C}=(C_{\alpha})_{\alpha\in\Omega}\) of subsets of \(A\) which has the following two properties: firstly, \(C_{\alpha}\subseteq C_{\beta}\) for every \(\beta\leqslant\alpha\), and, secondly, the set \(\{\alpha\in\Omega\,|\,a\in C_{\alpha}\}\) has a unique greatest element for every \(a\in A\). In particular, every such system gives rise to a lattice-valued set \(\mu:A\rightarrow\Omega\), where \(\mu(a)=\bigvee\{\alpha\in\Omega\,|\,a\in C_{\alpha}\}\). Conversely, every lattice-valued set \(\mu:A\rightarrow\Omega\) induces a nested system of \(\alpha\)-cuts in \(A\) in which \(C_{\alpha}=\{a\in A\,|\,\alpha\leqslant\mu(a)\}\). A thorough study of the properties of the above passages between lattice-valued sets and systems of \(\alpha\)-cuts was done, e.g., in [R. Bělohlávek, Fuzzy relational systems. Foundations and principles. New York, NY: Kluwer Academic Publishers (2002; Zbl 1067.03059); R. Bělohlávek and V. Vychodil, Fuzzy equational logic. Berlin: Springer (2005; Zbl 1083.03030)].
In [Fuzzy Sets Syst. 161, No. 24, 3127–3140 (2010; Zbl 1225.03070)], the author of the paper under review introduced an extension of this machinery to the category Set\((\Omega)\) whose objects \((A,\delta)\) are sets equipped with an \(\Omega\)-valued similarity relation (in the sense of, e.g., [U. Höhle, Theory Decis. Libr., Ser. B 14, 34–72 (1992; Zbl 0766.03037)]), and whose morphisms are maps which preserve these similarity relations. Notice that lattice-valued sets in this setting are Set\((\Omega)\)-morphisms \(s:(A,\delta)\rightarrow(\Omega,\leftrightarrow)\), where \(\alpha\leftrightarrow\beta= (\alpha\rightarrow\beta)\wedge(\beta\rightarrow\alpha)\). It is the main purpose of the current paper to extend this technique even further, namely, to the category SetR\((\Omega)\) whose morphisms are no longer maps, but \(\Omega\)-valued relations between sets.
The author begins by introducing two suitable analogues of systems of \(\alpha\)-cuts for the category SetR\((\Omega)\), and shows their equivalence. He then constructs a functor \(\mathcal{F}:\mathbf{SetR}(\Omega)\rightarrow\text\textbf{Set}\) (where Set is the category of sets and maps) the value of which on an object \((A,\delta)\) is the family of lattice-valued sets in \((A,\delta)\), i.e., the set of \(\mathbf{SetR}(\Omega)\)-morphisms \(s:(A,\delta)\rightarrow(\Omega,\leftrightarrow)\). Additionally, the author defines a functor \(\mathcal{C}:\mathbf{SetR}(\Omega)\rightarrow\text\textbf{Set}\) whose values on objects are families of generalized systems of \(\alpha\)-cuts. The main result of the paper is given in Theorem 4.2, stating that the functors \(\mathcal{F}\) and \(\mathcal{C}\) are naturally isomorphic (which is then an extension of the above representation of lattice-valued sets through systems of \(\alpha\)-cuts).
The paper is well written (almost no typos), conveniently self-contained, and will certainly be of interest to the researchers studying categories of lattice-valued sets.