×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML Link
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. · Zbl 0815.06010
[2] L. P. Belluce and S. Sessa: The stable topology for MV-algebras. Quaestiones Math. 23 (2000), 3, 269-277. · Zbl 0974.06005
[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. · Zbl 1040.03047
[4] A. Di Nola, G. Geurgescu, and A. Iorgulescu: Pseudo-BL-algebra, Part II. Multiple Valued Logic 8 (2002), 717-750.
[5] G. Georgescu and L. Leustean: Semilocal and maximal BL-algebras. Preprint. · Zbl 0963.03088
[6] P. Hájek: Metamathematics of Fuzzy Logic, Trends in Logic. (Studia Logica Library 4.) Kluwer Academic Publishers, Dordrecht 1998. · Zbl 0937.03030
[7] P. T. Johnstone: Stone Spaces. (Cambridge Studies in Advanced Mathematics.) Cambridge University Press, Cambridge 1982. · 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. · 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. · Zbl 0940.03029
[11] E. Turunen and S. Sessa: Local BL-algebras. Multi-Valued Log. 6 (2001), 1-2, 229-249.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.