Skip to main content
SpringerLink
Log in

Dual quantum B-algebras

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Quantum B-algebras as implicational subreducts of quantales were introduced by Rump and Yang. They cover the majority of implicational algebras and provide a unified semantics for a wide class of algebraic logics. Some concepts for quantales survive in the framework of quantum B-algebras. In this paper, we first introduce the concept of dual quantum B-algebras (Girard quantum B-algebras). Next, we prove that every dual quantum B-algebra is a residuated poset and that complete dual quantum B-algebras and dual quantales are equivalent to each other. Further, we consider the construction of Girard quantum B-algebras from dual 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

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

    Article  MathSciNet  MATH  Google Scholar 

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logic. In: Hazewinkel M (ed) Studies in logic and the foundations of mathematics, vol 151. Elsevier, Amsterdam

  • Grätzer G (1978) General lattice theory. Academic, New York

    Book  MATH  Google Scholar 

  • Han SW, Zhao B (2009) The quantic conuclei on quantales. Algebra Univ 61:97–114

    Article  MathSciNet  MATH  Google Scholar 

  • Han SW, Zhao B (2011) Nuclei and conuclei on residuated lattices. Fuzzy Sets Syst 172:51–70

    Article  MathSciNet  MATH  Google Scholar 

  • Han SW, Xu XT, Qin F (2018a) The unitality of quantum B-algebras. Int J Theor Phys 57:1582–1590

    Article  MathSciNet  MATH  Google Scholar 

  • Han SW, Wang RR, Xu XT (2018b) On the injective hulls of quantum B-algebras. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2018.05.011

  • Kruml D, Paseka J (2008) Algebraic and categorical aspects of quantales. In: Suppes P, Abramsky S, Artemov S, Gabbay DM, Kechris A, Pillay A, Shore RA (eds) Handbook of algebra, vol 5. Elsevier, Amsterdam, pp 323–362

  • Li YM (2001) Noncommutative linear logic and its quantale semantics. J Shaanxi Norm Univ (Nat Sci Ed) 29:1–5 (in Chinese)

    MathSciNet  MATH  Google Scholar 

  • Rosenthal KI (1990) Quantales and their applications. Longman Scientific & Technical, New York

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Rump W, Yang YC (2014) Non-commutative logical algebras and algebraic quantales. Ann Pure Appl Log 165:759–785

    Article  MathSciNet  MATH  Google Scholar 

  • Yetter DN (1990) Quantales and (noncommutative) linear logic. J Symb Log 55:41–64

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I first express my gratitude to the Natural Science Program for Basic Research of Shaanxi Province, China (Grant No. 2017JM1015), and I also would like to thank the referees for some of their comments and suggestions for the improvement of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Xi’an University of Posts and Telecommunications, Xi’an, 710121, People’s Republic of China

    Fangfang Pan

Authors
  1. Fangfang Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fangfang Pan.

Ethics declarations

Conflict of interest

The author declares that their is no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, F. Dual quantum B-algebras. Soft Comput 23, 6813–6817 (2019). https://doi.org/10.1007/s00500-018-03708-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-03708-3

Keywords

Search

Navigation