Advertisement

Order

pp 1–14 | Cite as

Tensor Products of Rings \(\mathfrak {Z}L\) of Zero-Dimensional Frames

  • Taewon YangEmail author
Article
  • 5 Downloads

Abstract

Let L and M be zero-dimensional frames. It is shown that concerning the Banaschewski compactification ζ, a necessary and sufficient condition for the canonically induced frame homomorphism hL,M : ζLζMζ(LM) to be an isomorphism is given in terms of the tensor product of the rings of all bounded integer-valued continuous functions on L and M, respectively. This provides the integral counterpart of A. Hager’s work (Math. Zeitschr. 92, 210–224, 1966, Section 2.2) in the setting of frames.

Keywords

Banaschewski compactification Zero-dimensional frames Integer-valued continuous functions on a frame Frame coproducts Tensor products of rings 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank J. Harding for his valuable suggestions and B. Olberding for his helpful discussions on the spatial frames in Corollary 4.8. He also wishes to express his warmest thanks to the referee for a number of detailed helpful comments.

References

  1. 1.
    Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories. Repr. Theory Appl. Categ. 17, 1–507 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Banaschewski, B., Holgate, D., Sioen, M.: Some new characterizations of pointfree pseudocompactness. Quaest. Math. 36(4), 589–599 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banaschewski, B.: On the function rings of pointfree topology. Kyungpook Math. J. 48(2), 195–206 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Banaschewski, B., Hong, S.S.: Completeness properties of function rings in pointfree topology. Comment. Math. Univ. Carolin. 44(2), 245–359 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Banaschewski, B.: Integral Gelfand duality: notes for a talk at Tehran, the 12th Algebra Seminar Iranian Mathematical Society (2000)Google Scholar
  6. 6.
    Banaschewski, B.: Gelfand and exchange rings: their spectra in pointfree topology. The Arabian J. for Sci. and Eng 25(2C), 3–22 (2000)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Banaschewski, B.: The real numbers in pointfree topology. Mathematical texts, no. 12 the University of Coimbra (1997)Google Scholar
  8. 8.
    Banaschewski, B.: Compactification of frames. Math. Nachr. 149, 105–116 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Banaschewski, B.: Universal zero-dimensional compactifications. In: Categorical Topology and its Relation to Analysis, Algebra, and Combinatorics, Prague, 1988, pp 257–269. World Scientific, Teaneck (1989)Google Scholar
  10. 10.
    Banaschewski, B.: Another look at the localic Tychonoff theorem. Comment. Math. Carolin. 29(4), 647–656 (1988)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Birkhoff, G.: Lattice theory. Colloquium Publications, vol. XXV, American Mathematical Society (1979)Google Scholar
  12. 12.
    Borceux, F.: Handbook of categorical algebra I. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  13. 13.
    Glicksberg, I.: Stone-Čech compactifications of products. Trans. Amer. Math. Soc. 90, 369–382 (1959)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hager, A.: Some remarks on the tensor product of function rings. Math. Zeitschr. 92, 210–224 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Johnstone, P.T.: Stone Spaces Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1982)Google Scholar
  16. 16.
    Mac Lane, S.: Categories for the working mathematician graduate texts in mathematics, vol. 5. Springer, New York (1978)CrossRefGoogle Scholar
  17. 17.
    Picado, J., Pultr, A.: Frames and locales: topology without Points. Frontier in Mathematics. Springer, Basel (2012)CrossRefzbMATHGoogle Scholar
  18. 18.
    Pierce, R.S.: Rings of integer-valued continuous functions. Trans. Amer. Math. Soc. 100, 371–394 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yang, T.: Rings of Integer-Valued Functions in Pointfree Topology. MSc Thesis, McMaster University (2004)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Gunpo-siKorea

Personalised recommendations