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Fuzzy sets and cut systems in a category of sets with similarity relations. (English) Zbl 1260.03096
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.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 06F05 Ordered semigroups and monoids 18B05 Categories of sets, characterizations 18B10 Categories of spans/cospans, relations, or partial maps
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##### References:
 [1] Bělohlávek R (2002) Fuzzy relational systems, foundations and principles. Kluwer, Dordrecht [2] Bělohlávek R, Vychodil V (2005) Fuzzy equational logic. Springer, Berlin [3] Höhle U (1992) M-valued sets and sheaves over integral, commutative cl-monoids. Applications of category theory to fuzzy subsets. Kluwer, Dordrecht, pp 33–72 [4] Močkoř J (2004) Complete subobjects of fuzzy sets over $$MV$$ -algebras. Czech Math J 129(54):379–392 · Zbl 1080.18001 [5] Močkoř J (2007) Extensional subobjects in categories of $$$$\backslash$$Upomega$$ -fuzzy sets. Czech Math J 57(132):631–645 · Zbl 1174.06320 [6] Močkoř J (2006) Covariant functors in categories of fuzzy sets over MV-algebras. Adv Fuzzy Sets Syst 1(2):83–109 · Zbl 1121.03077 [7] Močkoř J (2010) Cut systems in sets with similarity relations. Fuzzy Sets Syst 161(24):3127–3140 · Zbl 1225.03070 [8] Novák V, Perfilijeva I, Močkoř J (1999) Mathematical principles of fuzzy logic. Kluwer Academic Publishers, Dordrecht [9] Mac Lane S (1971) Categories for the working mathematician. Springer, Berlin · Zbl 0232.18001
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