Skip to main content
SpringerLink
Log in

The Unitality of Quantum B-algebras

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blyth, T.S.: Lattices and Ordered Algebraic Structures. Springer-Verlag, London (2005)

    MATH  Google Scholar 

  2. Botur, M., Paseka, J.: Filters on some clases of quantum B-algebras. Int. J. Theor. Phys. 54, 4397–4409 (2015)

    Article  MATH  Google Scholar 

  3. Dudek, W.A., Yun, Y.B.: Pseudo-BCI algebras. East Asian Math. J. 24, 187–190 (2008)

    MATH  Google Scholar 

  4. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructral Logics. Stud. Logic Found Math., vol. 51. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Han, S., Zhao, B.: Remark on the unital quantale Q[e]. Appl. Categor. Struct. 20, 239–250 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Kruml, D., Paseka, J.: Algebraic and categorical aspects of quantales. Handbook of Algebra, vol. 5, pp. 323–362. Elsevier, Amsterdam (2008)

    MATH  Google Scholar 

  9. Kühr, J.: Pseudo-BCK Algebras and Related Structures. Habilitation Thesis, University of Olomouc, Olomouc (2007)

    MATH  Google Scholar 

  10. Mulvey, C.J.: . In: Second Topology Conference, Taormina, April 4-7, 1984. Suppl. Rend. Circ. Math. Palermo ser, vol. II 12, pp. 99–104 (1986)

  11. Paseka, J., Kruml, D.: Embeddings of quantales into simple quantales. J. Pure Appl. Algebra 148, 209–216 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rosenthal, K.I.: Quantales and their applications. Longman Scientific & Technical, London (1990)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Rump, W.: The completion of a quantum B-algebra. Cah. Topol. Géom. Différ. Catég. 57, 203–228 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Rump, W.: Multi-posets in algebraic logic, group theory, and non-commutative topology. Ann. Pure Appl. Logic 167, 1139–1160 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rump, W.: Quantum B-algebras: their omnipresence in algebraic logic and beyond. Soft Compt. 21, 2521–2529 (2017)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We wish to express our sincere thanks to the anonymous referee for careful reading of the manuscript, and for useful suggestions and valuable comments which helped to improve the presentation of the results.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shaanxi Normal University, Xi’an, 710062, People’s Republic of China

    Shengwei Han & Xiaoting Xu

  2. College of Mathematics and Information Science, Jiangxi Normal University, 330022, Nanchang, People’s Republic of China

    Feng Qin

Authors
  1. Shengwei Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoting Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shengwei Han.

Additional information

This work was supported by the National Natural Science Foundation of China (11531009, 61563020) and the Fundamental Research Funds for the Central Universities (GK201501001).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, S., Xu, X. & Qin, F. The Unitality of Quantum B-algebras. Int J Theor Phys 57, 1582–1590 (2018). https://doi.org/10.1007/s10773-018-3683-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-018-3683-9

Keywords

Search

Navigation