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Pure filters and stable topology on BL-algebras. (English) Zbl 1177.03069
Summary: We introduce stable topology and \(F\)-topology on the set of all prime filters of a BL-algebra \(A\) and show that the set of all prime filters of \(A\), namely Spec\((A)\), with the stable topology is a compact space but not \(T_0\). Then, by means of stable topology, we define and study pure filters of a BL-algebra \(A\) and obtain a one-to-one correspondence between pure filters of \(A\) and closed subsets of Max\((A)\), the set of all maximal filters of \(A\), as a subspace of Spec\((A)\). We also show that for any filter \(F\) of BL-algebra \(A\) if \(\sigma(F)=F\) then \(U(F)\) is stable and \(F\) is a pure filter of \(A\), where \(\sigma(F)=\{a\in A\mid y\wedge z = 0\) for some \(z\in F\) and \(y\in a^\perp\}\) and \(U(F) = \{P\in\text{Spec}(A)\mid F\nsubseteq P\}\).

03G25 Other algebras related to logic
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