Skip to main content
SpringerLink
Log in

Dualities Induced by Topological Semirings

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In this paper we establish a general duality theorem for compact Hausdorff spaces being recognizable over certain pairs consisting of a commutative unital topological semiring and a closed proper prime ideal. Indeed, we utilize the concept of blueprints and their localization to prove that the category of compact Hausdorff spaces generated by such a pair can be dually embedded into the category of commutative unital semirings if the pair possesses sufficiently many covering polynomials.

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. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories: The joy of cats. Repr. Theory Appl. Categ. 17, 1–507 (2006). (electronic), Reprint of the 1990 original [Wiley, New York; MR1051419]

    MathSciNet  MATH  Google Scholar 

  2. Banaschewski, B.: On lattices of continuous functions. In: Proceedings of the Symposium on Categorical Algebra and Topology (Cape Town, 1981), vol. 6, pp 1–12 (1983)

  3. Gelfand, I., Kolmogoroff, A.N.: On rings of continuous functions on a topological space, C. R. (Doklady). Acad. Sci. URSS 22, 11–15 (1939)

    MATH  Google Scholar 

  4. Herrlich, H.: Topologische Reflexionen und Coreflexionen, Lecture Notes in Mathematics, No. 78. Springer-Verlag, Berlin (1968)

    MATH  Google Scholar 

  5. Hewitt, E.: Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, 45–99 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hochster, M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43–60 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Johnstone, P.T.: Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1982)

    Google Scholar 

  8. Kaplansky, I.: Lattices of continuous functions. Bull. Amer. Math. Soc. 53, 617–623 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  9. Krull, W.: Idealtheorie in Ringen ohne Endlichkeitsbedingung. Math. Ann. 101 (1), 729–744 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lorscheid, O.: The geometry of blueprints. I: Algebraic background and scheme theory. Adv. Math. 229(3), 1804–1846 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Milgram, A.N.: Multiplicative semigroups of continuous functions. Duke Math. J. 16, 377–383 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  12. Peña, A.J., Ruza, L.M., Vielma, J.: P-spaces and the prime spectrum of commutative semirings. Int. Math. Forum 3(33–36), 1795–1802 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Ruza, L.M., Vielma, J.: Gelfand semirings, m-semirings and the Zariski topology. Int. J. Algebra 3(17–20), 981–991 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Shirota, T.: A generalization of a theorem of I. Kaplansky. Osaka Math. J. 4, 121–132 (1952)

    MathSciNet  MATH  Google Scholar 

  15. Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41(3), 375–481 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  16. Stone, M.H.: A general theory of spectra II. Proc. Nat. Acad. Sci. U.S.A. 27, 83–87 (1941)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technische Universität, Dresden, Germany

    Sebastian Kerkhoff & Friedrich Martin Schneider

Authors
  1. Sebastian Kerkhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Friedrich Martin Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Friedrich Martin Schneider.

Additional information

Friedrich Martin Schneider is supported by funding of the Excellence Initiative by the German Federal and State Governments.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kerkhoff, S., Schneider, F.M. Dualities Induced by Topological Semirings. Appl Categor Struct 24, 315–329 (2016). https://doi.org/10.1007/s10485-015-9398-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-015-9398-7

Keywords

Search

Navigation