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Algebra universalis

, 80:52 | Cite as

On the quantale of quantic nuclei

  • Shengwei HanEmail author
Article
  • 56 Downloads

Abstract

It is well known that the set \(\textit{NL}\) of all nuclei on a frame L is again a frame with the pointwise order. As a non-commutative generalization of frames, quantales were introduced by Mulvey in 1986. Niefield and Rosenthal attempted to define a binary operation  & on the set \(\textit{NQ}\) of all quantic nuclei of a quantale Q such that (\(\textit{NQ}\), &) is again a quantale. However, Sun showed that the way of Niefield and Rosenthal failed for a general quantale. Girard quantales are a special class of quantales, which play an important role in the study of linear intuitionistic logic, quantitative domain, lattice-valued topology and enriched category. In this note, we give an example to indicate that \(\textit{NQ}\), together with  & defined by Niefield and Rosenthal, is in general not a quantale even for a Girard quantale Q. However, with respect to  & on a Girard quantale Q, we redefine a binary operation \(\odot \) on \(\textit{NQ}\) such that \((\textit{NQ}^{op}, \odot )\) is a quantale.

Keywords

Quantale Girard quantale Ideal Quantic nucleus 

Mathematics Subject Classification

06F07 06A06 54A10 

Notes

Acknowledgements

I would like to thank the referees for some of their comments and suggestions for the improvement of this paper.

References

  1. 1.
    Grätzer, G.: General Lattice Theory. Academic Press, New York (1978)CrossRefGoogle Scholar
  2. 2.
    Han, S.W., Zhao, B.: The quantic conuclei on quantales. Algebra Univ. 61, 97–114 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Han, S.W., Zhao, B.: The Foundation of Quantale Theory (In Chinese). Science Press, Beijing (2016)Google Scholar
  4. 4.
    Hofmann, D., Waszkiewicz, P.: Approximation in quantale-enriched categories. Topol. Appl. 158, 963–977 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kruml, D., Paseka, J.: Algebraic and Categorical Aspects of Quantales. Handbook of Algebra, vol. 5, pp. 323–362. North-Holland, Amsterdam (2008)zbMATHGoogle Scholar
  6. 6.
    Mulvey, C.J.:&. Rend. Circ. Mat. Palermo 12, 99–104 (1986)Google Scholar
  7. 7.
    Niefield, S.B., Rosenthal, K.I.: Constructing locales from quantales. Math. Proc. Camb. Philos. Soc. 104, 215–234 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Paseka, J.: Simple quantales. In: Proceedings of the Eighth Prague Topological Symposium (Prague 1996), pp. 314–328 (electronic). Topology Atlas, North Bay (1997)Google Scholar
  9. 9.
    Rosenthal, K.I.: Quantales and their Applications. Longman Scientific and Technical, New York (1990)zbMATHGoogle Scholar
  10. 10.
    Sun, S.H.: Remarks on quantic nuclei. Math. Proc. Camb. Philos. Soc. 108, 257–260 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Waszkiewicz, P.: On domain theory over Girard quantale. Fundam. Inf. 92, 169–192 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yetter, D.N.: Quantales and (noncommutative) linear logic. J. Symb. Log. 55, 41–64 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zhang, D.: An enriched category approach to many valued topology. Fuzzy Sets Syst. 158, 349–366 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zhao, B., Wang, S.Q.: Cyclic dualizing elements in Girard quantales. Sci. Magna 5, 72–77 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsShaanxi Normal UniversityXi’anChina

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