International Journal of Theoretical Physics

, Volume 57, Issue 5, pp 1582–1590 | Cite as

The Unitality of Quantum B-algebras



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.


Quantale Quantum B-algebra Unital quantum B-algebra 



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.


  1. 1.
    Blyth, T.S.: Lattices and Ordered Algebraic Structures. Springer-Verlag, London (2005)MATHGoogle Scholar
  2. 2.
    Botur, M., Paseka, J.: Filters on some clases of quantum B-algebras. Int. J. Theor. Phys. 54, 4397–4409 (2015)CrossRefMATHGoogle Scholar
  3. 3.
    Dudek, W.A., Yun, Y.B.: Pseudo-BCI algebras. East Asian Math. J. 24, 187–190 (2008)MATHGoogle Scholar
  4. 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)MATHGoogle Scholar
  5. 5.
    Georgescu, G., Iorgulescu, A.: Pseudo MV-algebras. Mult.-Valued Log. 6, 95–135 (2001)MathSciNetMATHGoogle Scholar
  6. 6.
    Han, S., Zhao, B.: Remark on the unital quantale Q[e]. Appl. Categor. Struct. 20, 239–250 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jipsen, P., Montagna, F.: On the structure of generalized BL-algebras. Algebra Univers. 55, 226–237 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kruml, D., Paseka, J.: Algebraic and categorical aspects of quantales. Handbook of Algebra, vol. 5, pp. 323–362. Elsevier, Amsterdam (2008)MATHGoogle Scholar
  9. 9.
    Kühr, J.: Pseudo-BCK Algebras and Related Structures. Habilitation Thesis, University of Olomouc, Olomouc (2007)MATHGoogle Scholar
  10. 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)Google Scholar
  11. 11.
    Paseka, J., Kruml, D.: Embeddings of quantales into simple quantales. J. Pure Appl. Algebra 148, 209–216 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Rosenthal, K.I.: Quantales and their applications. Longman Scientific & Technical, London (1990)MATHGoogle Scholar
  13. 13.
    Rump, W.: Quantum B-algebras. Cent. Eur. J. Math. 11, 1881–1899 (2013)MathSciNetMATHGoogle Scholar
  14. 14.
    Rump, W., Yang, Y.C.: Non-commutative logical algebras and algebraic quantales. Ann. Pure Appl. Logic 165, 759–785 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rump, W.: The completion of a quantum B-algebra. Cah. Topol. Géom. Différ. Catég. 57, 203–228 (2016)MathSciNetMATHGoogle Scholar
  16. 16.
    Rump, W.: Multi-posets in algebraic logic, group theory, and non-commutative topology. Ann. Pure Appl. Logic 167, 1139–1160 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rump, W.: Quantum B-algebras: their omnipresence in algebraic logic and beyond. Soft Compt. 21, 2521–2529 (2017)CrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShaanxi Normal UniversityXi’anPeople’s Republic of China
  2. 2.College of Mathematics and Information ScienceJiangxi Normal UniversityNanchangPeople’s Republic of China

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