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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 $$U(F) = \{P\in\text{Spec}(A)\mid F\nsubseteq P\}$$.

##### MSC:
 03G25 Other algebras related to logic
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##### References:
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