Fuzzy sets and cut systems in a category of sets with similarity relations.

*(English)*Zbl 1260.03096Given 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.

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.

Reviewer: Sergejs Solovjovs (Brno)

##### 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 |

##### Keywords:

category of relations; closed cut system; complete residuated lattice; fuzzy set; naturally isomorphic functors; similarity relation; system of \(\alpha\)-cuts; category of sets
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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 |

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[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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