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The topological modification of the \(L\)-fuzzy unit interval. (English) Zbl 0766.54006

Applications of category theory to fuzzy subsets, Mat. 11th Int. Semin. Fuzzy Set Theory, Linz/Austria 1989, Theory Decis. Libr., Ser. B 14, 276-305 (1992).
[For the entire collection see Zbl 0741.00078.]
This paper first reexamines compactness notions of Lowen, Gantner and Zhao with the aim of generalizing them for certain lattices which are not chains. While this question has been addressed previously, the results obtained have been restricted to certain linear subsets of the lattice, a restriction which does not go much beyond assuming a chain. The author bases his redefinition on replacing \(<\) by the \(\not\geq\) relation, which was first used by Liu. Secondly, the author discusses a generalization of Lowen’s \(\omega\)-functor previously suggested by Gierz and Warner, in which the unit interval is replaced by a continuous frame with its Scott topology. The author introduces a compatible topological modification functor, \(\iota_ L\), and shows that, if \(L\) is completely distributive, then \(\omega_ L\) is a left adjoint right inverse of \(\iota_ L\) and \(\omega_ L(T)\) is the stratification of the \(L\)-fuzzy topology consisting of characteristic functions of open sets in \(T\). Lastly, the author shows that the topological modification of the \(L\)-fuzzy unit interval is the quotient topology of the Helly space of \(L\)-valued functions from \(I\). Several \(L\)-fuzzy topological properties follow immediately.

MSC:

54A40 Fuzzy topology

Citations:

Zbl 0741.00078
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