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Soft Computing

, Volume 12, Issue 5, pp 419–423 | Cite as

Filter theory of BL algebras

  • Michiro KondoEmail author
  • Wiesław A. Dudek
Original Paper

Abstract

In this paper we consider fundamental properties of some types of filters (Boolean, positive implicative, implicative and fantastic filters) of BL algebras defined in Haveshki et al. (Soft Comput 10:657–664, 2006) and Turunen (Arch Math Logic 40:467–473, 2001). It is proved in Haveshki et al. (2006) that if F is a maximal and (positive) implicative filter then it is a Boolean filter. In that paper there is an open problem

Under what condition are Boolean filters positive implicative filters?

One of our results gives an answer to the problem, that is, we need no more conditions. Moreover, we give simple characterizations of those filters by an identity form ∀ x, y(t(x, y) ∈ F), where t(x, y) is a term containing x, y.

Keywords

(Positive) implicative filter Boolean filter Fantastic filter BL algebra 

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References

  1. Cignoli R, Esteva F, Godo L, Torrens A (2000) Basic fuzzy logic is the logic of continuous t-norm and their residua. Soft Comput 4:106–112CrossRefGoogle Scholar
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  3. Haveshki M, Saeid AB, Eslami E (2006) Some types of filters in BL algebras. Soft Comput 10:657–664zbMATHCrossRefGoogle Scholar
  4. Turunen E (1999) BL-algebras of basic fuzzy logic. Mathw soft comput 6:49-61zbMATHMathSciNetGoogle Scholar
  5. Turunen E (2001) Boolean deductive systems of BL-algebras. Arch Math Logic 40:467–473zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Information EnvironmentTokyo Denki UniversityInzaiJapan
  2. 2.Institute of MathematicsWroclaw University of TechnologyWroclawPoland

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