Pure filters and stable topology on BL-algebras

Eslami, Esfandiar ; Haghani, Farhad Kh.

Kybernetika, Tome 45 (2009) no. 3, pp. 491-506

Résumé

In this paper 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 $\sigma(F)=\{a\in A|\,y\wedge z=0$ for some $z\in F$ and $y\in a^\perp\}$ and $U(F)=\{P\in $ Spec($A$)\,$\vert\,F\nsubseteq P\}$.

Détail

Classification : 03G25, 06F35, 06F99, 08A72
Keywords: BL-algebra; prime filters; maximal filters; pure filters; stable topology; F-topology