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Topology on the set of \(R_{0}\) semantics for \(R_{0}\) algebras. (English) Zbl 1138.03048

Summary: This paper introduces the concepts of \(R_{0}\) valuation, \(R_{0}\) semantics, countable \(R_{0}\) category \(\mathcal{C}R_0\), \(R_{0}\) fuzzy topological category \(\mathcal{R}CG\), etc. We establish in a natural way the fuzzy topology \(\delta\) and its cut topology on the set \(\Omega_{M}\) consisting of all \(R_{0}\) valuations of an \(R_{0}\) algebra \(M\), and some properties of the fuzzy topology \(\delta\) and its cut topology are investigated carefully. Moreover, the representation theorem for \(R_{0}\) algebras by means of fuzzy topology is given, that is to say the category \(\mathcal{C}R_0\) is equivalent to the category \(\mathcal{R}CG^{\text{op}}\). By studying the relation between valuations and filters, the Loomis-Sikorski theorem for \(R_{0}\) algebras is obtained. As an application, K-compactness of the \(R_{0}\) logic \(\mathcal{L}^{*}\) is discussed.

MSC:

03G25 Other algebras related to logic
03B52 Fuzzy logic; logic of vagueness
54A40 Fuzzy topology
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References:

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