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Every Linear Pseudo BL-Algebra Admits a State

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Abstract

We show that every linear pseudo BL-algebra, hence every representable one, admits a state and is good. This solves positively the problem on the existence of states raised in Dvurečenskij and Rachůnek (Probabilistic averaging in bounded communitative residuated ℓ-monoids, 2006), and gives a partial answer to the problem on good pseudo BL-algebras from [Di Nola, Georgescu and Iorgulescu (Multiple Val Logic 8:715–750, 2002) Problem 3.21]. Moreover, we present that every saturated linear pseudo BL-algebra can be expressed as an ordinal sum of Hájek’s type of irreducible pseudo linear pseudo BL-algebras.

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References

  1. Agliano P, Montagna F (2003) Varieties of BL-algebras I: general properties. J Pure Appl Algebra 181:105–129

    Article  MathSciNet  MATH  Google Scholar 

  2. Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo-BL algebras I. Multiple Val Logic 8:673–714

    MathSciNet  MATH  Google Scholar 

  3. Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo-BL algebras II. Multiple Val Logic 8:715–750

    Google Scholar 

  4. Dvurečenskij A (2001) States on pseudo MV-algebras. Stud Logica 68:301–327

    Article  MATH  Google Scholar 

  5. Dvurečenskij A (2002) Pseudo MV-algebras are intervals in ℓ-groups. J Aust Math Soc 70:427–445

    Article  Google Scholar 

  6. Dvurečenskij A, Hyčko M (2005) On the existence of states for linear pseudo BL-algebras. Atti Sem Mat Fis Univ Modena e Reggio Emilia 53:93–110

    MATH  Google Scholar 

  7. Dvurečenskij A, Rachůnek J (2006) Probabilistic averaging in bounded commutative residuated ℓ-monoids. Soft Comput 10:212–218

    Article  MATH  Google Scholar 

  8. Dvurečenskij A, Rachůnek J (2006) Probabilistic averaging in bounded residuated ℓ-monoids. Semigroup Forum (to appear)

  9. Fuchs L (1963) Partially Ordered Algebraic Systems. Pergamon, Oxford

    MATH  Google Scholar 

  10. Georgescu G (2004) Bosbach states on fuzzy structures. Soft Comput 8:217–230

    MathSciNet  MATH  Google Scholar 

  11. Georgescu G, Iorgulescu A (2001) Pseudo-MV algebras. Multiple Val Logic 6:95–135

    MathSciNet  MATH  Google Scholar 

  12. Hájek P (1998) Basic fuzzy logic and BL-algebras. Soft Comput 2:124–128

    Google Scholar 

  13. Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44:21–34

    MathSciNet  MATH  Google Scholar 

  14. Kühr J (2003) Pseudo BL-algebras and DRℓ-monoids. Math Bohemica 128:199–202

    MATH  Google Scholar 

  15. Łukasiewicz J (1920) O logice trójwartościowej. Ruch Filozoficzny 6:170–171. Translated by Wojtasiewicz O. In: Borkowski L, Selected works of Łukasiewicz J, North-Holland, Amsterdam, 1970

    Google Scholar 

  16. Mundici D (1995) Averaging the truth-value in Łukasiewicz logic. Stud Logica 55:113–127

    Article  MathSciNet  MATH  Google Scholar 

  17. Rachůnek J (2002) A non-commutative generalization of MV-algebras. Czechoslovak Math J 52:255–273

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Correspondence to Anatolij Dvurečenskij.

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The paper has been supported by the Center of Excellence SAS—Physics of Information—I/2/2005, the grant VEGA no. 2/3163/23 SAV and by Science and Technology Assistance Agency under the contract no. APVT-51-032002. Bratislava, Slovakia.

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Dvurečenskij, A. Every Linear Pseudo BL-Algebra Admits a State. Soft Comput 11, 495–501 (2007). https://doi.org/10.1007/s00500-006-0078-2

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