zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.