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Filters on Some Classes of Quantum B-Algebras

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Abstract

In this paper, we continue the study of quantum B-algebras with emphasis on filters on integral quantum B-algebras. We then study filters in the setting of pseudo-hoops. First, we establish an embedding of a cartesion product of polars of a pseudo-hoop into itself. Second, we give sufficient conditions for a pseudohoop to be subdirectly reducible. We also extend the result of Kondo and Turunen to the setting of noncommutative residuated ∨-semilattices that, if prime filters and ∨-prime filters of a residuated ∨-semilattice A coincide, then A must be a pseudo MTL-algebra.

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Notes

  1. M ⊥⋅M ⊥⊥≅M ⊥×M ⊥⊥

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosbach, B.: Komplementäre Halbgruppen. Axiomatik und Arithmetik. Fund. Math. 64, 257–287 (1966)

    MathSciNet  Google Scholar 

  3. Bosbach, B.: Komplementäre Halbgruppen. Kongruenzen and Quotienten. Fund. Math. 69, 1–14 (1970)

    MathSciNet  MATH  Google Scholar 

  4. Bosbach, B.: Residuation groupoids - again. Results Math. 53, 27–51 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bosbach, B.: Divisibility groupoids - again. Results Math. 57, 257–285 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Botur, M., Dvurečenskij, A. , Kowalski, T.: On normal-valued basic pseudo-hoops. Soft Comput. 16, 635–644 (2012)

    Article  MATH  Google Scholar 

  7. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc 88, 467–490 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ciungu, L.C.: Non-commutative Multiple-Valued Logic Algebras Springer Monographs in Mathematics (2014)

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  11. Dvurečenskij, A.: Aglianò–Montagna type decomposition of linear pseudo-hoops and its applications. J. Pure Appl. Algebra 211, 851–861 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dvurečenskij, A., Giuntini, R., Kowalski, T.: On the structure of pseudo BL-algebras and pseudo-hoops in quantum logics. Found. Phys. 40, 1519–1542 (2010). doi:10.1007/s10701-009-9342-5

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices An Algebraic Glimpse at Substructural Logics Elsevier Studies in Logic and Foundations (2007)

  14. Galatos, N., Tsinakis, C.: Generalized MV-algebras. J. Algebra 283, 254–291 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gasse, B. Van, Deschrijver, G., Cornelis, C., Kerre, E.E.: Filters of residuated lattices and triangle algebras. Inf. Sci. 180, 3006–3020 (2010)

    Article  MATH  Google Scholar 

  16. Georgescu, G., Iorgulescu, A.: Pseudo MV algebras. Mult.-Valued Log. 6, 95–135 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Georgescu, G., Leuştean, L., Preoteasa, V.: Pseudo-hoops. J. Mult.-Val. Log. Soft Comput. 11, 153–184 (2005)

    MATH  Google Scholar 

  18. Girard, J.-Y.: Linear logic, Theoretical Computer Science 50, 1–102 (1997)

    Article  Google Scholar 

  19. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)

    Book  MATH  Google Scholar 

  20. Jipsen, P., Montagna, F.: On the structure of generalized BL-algebras. Algebra Universalis 55, 226–237 (2006)

    Article  MathSciNet  Google Scholar 

  21. Kondo, M., Turunen, E.: Prime Filters on Residuated Lattices In IEEE 43rd International Symposium on Multiple-Valued Logic 2012, pp 89–91 (2013)

  22. Mesiar, R., Nánásiová, O., Riečanová, Z., Paseka, J.: Special issue - Quantum structures: Theory and applications. Inform. Sci. 179, 475–477 (2009)

    Article  MathSciNet  Google Scholar 

  23. Mulvey, C. J.: §. Suppl. Rend. Circ. Mat. Palermo ser II 12, 99–104 (1986)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  25. Rosenthal, K.J.: Quantales and their applications, Pitman Research Notes in Mathematics Series 234 Longman Scientific & Technical (1990)

  26. Rump, W.: Quantum B-algebras. Cent. Eur. J. Math. 11, 1881–1899 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Rump, W., Yang, Y. C.: Non-commutative logical algebras and algebraic quantales. Ann. Pure Appl. Logic 165, 759–785 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhu, Y., Xu, Y.: On filter theory of residuated lattices. Inform. Sci. 180, 3614–3632 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank the anonymous referee for the careful reading of the paper and the suggestions on improving its presentation.

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Authors and Affiliations

  1. Palacký University Olomouc, Faculty of Sciences, tř. 17.listopadu 1192/12, Olomouc, 771 46, Czech Republic

    Michal Botur

  2. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jan Paseka

Authors
  1. Michal Botur
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  2. Jan Paseka
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Correspondence to Michal Botur.

Additional information

Both authors acknowledge the support by a bilateral project I 1923-N25 New Perspectives on Residuated Posets financed by Austrian Science Fund (FWF) and the Czech Science Foundation (GACR) and by ESF Project CZ.1.07/2.3.00/20.0051 Algebraic methods in Quantum Logic of the Masaryk University.

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Botur, M., Paseka, J. Filters on Some Classes of Quantum B-Algebras. Int J Theor Phys 54, 4397–4409 (2015). https://doi.org/10.1007/s10773-015-2608-0

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