Topology on the set of R ₀ semantics for R ₀ algebras

Abstract

This paper introduces the concepts of R ₀ valuation, R ₀ semantic, countable R ₀ category $\mathcal{C}R_0$, R ₀ fuzzy topological category $\mathcal{R}CG$, etc. It is established in a natural way that the fuzzy topology δ and its cut topology on the set Ω _M consisting of all R ₀ valuations of an R ₀ algebra M, and some properties of fuzzy topology δ 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 $\mathcal{R}CG^{op}$. 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.