# Article

Full entry | PDF   (0.9 MB)
Keywords:
BL-algebra; prime filters; maximal filters; pure filters; stable topology; F-topology
Summary:
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($A$)\,$\vert\,F\nsubseteq P\}$.
References:
[1] L. P. Belluce, A. Di Nola, and S. Sessa: The prime spectrum of an MV-algebra. Math. Logic Quart. 40 (1994), 331–346. MR 1283500
[2] L. P. Belluce and S. Sessa: The stable topology for MV-algebras. Quaestiones Math. 23 (2000), 3, 269–277. MR 1809936
[3] D. Busneage and D. Piciu: On the lattice of deductive system of a BL-algebra. Central European Journal of Mathematics 2 (2003), 221–237. MR 1993450
[4] A. Di Nola, G. Geurgescu, and A. Iorgulescu: Pseudo-BL-algebra, Part II. Multiple Valued Logic 8 (2002), 717–750. MR 1948854
[5] G. Georgescu and L. Leustean: Semilocal and maximal BL-algebras. Preprint.
[6] P. Hájek: Metamathematics of Fuzzy Logic, Trends in Logic. (Studia Logica Library 4.) Kluwer Academic Publishers, Dordrecht 1998. MR 1900263
[7] P. T. Johnstone: Stone Spaces. (Cambridge Studies in Advanced Mathematics.) Cambridge University Press, Cambridge 1982. MR 0698074 | Zbl 0586.54001
[8] L. Leustean: The prime and maximal spectra and the reticulation of BL-algebras. Central European Journal of Mathematics 1 (2003), 382–397. MR 1992899 | Zbl 1039.03052
[9] L. Leustean: Representations of Many-Valued Algebras. PhD. Thesis, University of Bucharest 2004.
[10] E. Turunen: Mathematics Behind Fuzzy Logic. Advances in Soft Computing. Physica-Verlag, Heidelberg 1999. MR 1716958 | Zbl 0940.03029
[11] E. Turunen and S. Sessa: Local BL-algebras. Multi-Valued Log. 6 (2001), 1–2, 229–249. MR 1817445

Partner of