Skip to main content
SpringerLink
Log in

MV-semirings and their Sheaf Representations

  • Published:
Order Aims and scope Submit manuscript

Abstract

In this paper we show that the classes of MV-algebras and MV-semirings are isomorphic as categories. This approach allows one to keep the inspiration and use new tools from semiring theory to analyze the class of MV-algebras. We present a representation of MV-semirings by MV-semirings of continuous sections in a sheaf of commutative semirings whose stalks are localizations of MV-semirings over prime ideals. Using the categorical equivalence, we obtain a representation of MV-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

  1. Belluce, L.P.: Semisimple algebras of infinite valued logic and bold fuzzy set theory. Canad. J. Math. 38(6), 1356–1379 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Belluce, L.P., Di Nola, A.: Commutative rings whose ideals form an MV-algebra. Math. Log. Q. 55(5), 468–486 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belluce, L.P., DiNola, A., Lettieri, A.: On some lattice quotients of MV-algebras. Ric. Mat. 39, 41–59 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Birkhoff, G.: Lattice Theory. American Mathematical Society (1967)

  5. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. 88, 467–490 (1958)

    Article  MATH  Google Scholar 

  6. Chermnykh, V.V.: Sheaf representations of semirings. Russ. Math. Surv. 5, 169–170 (1993)

    Article  MathSciNet  Google Scholar 

  7. Cignoli, R., D’Ottaviano, I.M., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht-Boston-London (2000)

    Book  MATH  Google Scholar 

  8. Di Nola, A., Gerla, B.: Algebras of Lukasiewicz’s logic and their semiring reducts. Contemp. Math. 377, 131–144 (2005)

    Article  Google Scholar 

  9. Dubuc, E.J., Poveda, Y.A.: Representation theory of MV-algebras. Ann. Pure Appl. Logic 161(8), 1024–1046 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerla, B.: Many valued logics and semirings. Neural Netw. World 5, 467–480 (2003)

    Google Scholar 

  11. Golan, J.S.: The theory of semirings with applications in mathematics and theoretical computer science. Longman Scientific & Technical (1992)

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Yang, Y.C.: l-groups and Bézout domains. PHD thesis (2006)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, British Columbia University, Vancouver, British Columbia, Canada

    Lawrence Peter Belluce

  2. Dipartimento di Matematica, Università di Salerno, Via Ponte don Melillo, 84084, Fisciano, Salerno, Italy

    Antonio Di Nola & Anna Rita Ferraioli

Authors
  1. Lawrence Peter Belluce
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antonio Di Nola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna Rita Ferraioli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Rita Ferraioli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belluce, L.P., Di Nola, A. & Ferraioli, A.R. MV-semirings and their Sheaf Representations. Order 30, 165–179 (2013). https://doi.org/10.1007/s11083-011-9234-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-011-9234-0

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation