Skip to main content
Download PDF

Relational Representation of Groupoid Quantales

Order volume 30, pages 65–83 (2013)Cite this article

Abstract

In Palmigiano and Re (J Pure Appl Algebra 215(8):1945–1957, 2011), spatial SGF-quantales are axiomatically introduced and proved to be representable as sub unital involutive quantales of quantales arising from set groupoids. In the present paper, spatial SGF-quantales of this class are shown to be optimally representable as unital involutive quantales of relations. The results of the present paper have several aspects in common with Jónsson and Tarski’s representation theory for relation algebras (Jónsson and Tarski, Am J Math 74(2):127–162, 1952).

References

  1. Brown, C., Gurr, D.: A representation theorem for quantales. J. Pure Appl. Algebra 85(1), 27–42 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  2. Gelfand, I.M., Naimark, M.A.: On the embedding of normed rings into the ring of operators in Hilbert space. Mat. Sb. 12, 197–213 (1943)

    Google Scholar 

  3. Jónsson, B., Tarski, A.: Boolean algebras with operators, part II. Am. J. Math. 74(2), 127–162 (1952)

    MATH  Article  Google Scholar 

  4. Kruml, D., Paseka, J.: Embeddings of quantales into simple quantales. J. Pure Appl. Algebra 148(2), 209–424 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  5. Johnstone, P.T.: Stone Spaces. Cambridge Studies in Adv. Math., vol. 3. Cambridge University Press, Cambridge, UK (1982)

    MATH  Google Scholar 

  6. Mulvey, C.J.: &. Rend. Circ. Mat. Palermo 12, 99–104 (1986)

    MathSciNet  MATH  Google Scholar 

  7. Mulvey, C.J., Resende, P.: A noncommutative theory of Penrose tilings. Int. J. Theor. Phys. 44, 655–689 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  8. Palmigiano, A., Re, R.: Groupoid quantales: a non-étale setting. J. Pure Appl. Algebra 215(8), 1945–1957 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  9. Palmigiano, A., Re, R.: Topological Groupoid Quantales. In: Aerts, D., Smets, S., van Bendegem, J.P. (eds.) Special Issue: The Contributions of Logic to the Foundations of Physics. Studia Logica, vol. 95, pp. 125–137 (2010)

  10. Paterson, A.L.T.: Groupoids, Inverse Semigroups, and their Operator Algebras. Progress in Mathematics, vol. 170. Birkhäuser, Boston (1999)

    MATH  Book  Google Scholar 

  11. Protin, M.C., Resende, P.: Quantales of Open Groupoids. arXiv:0811.4539v2

  12. Resende, P.: Lectures on Étale Groupoids, Inverse Semigroups and Quantales. Lecture Notes for the GAMAP IP Meeting, Antwerp (2006)

  13. Resende, P.: Étale groupoids and their quantales. Adv. Math. 208, 147–209 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  14. Rosenthal, K.: Quantales an their Applications. Pitman Research Notes in Math., vol. 234. Longman (1990)

  15. Takesaki, M.: Theory of Operator Algebras I, EMS Series, vol. 124. Springer, New York (2002)

    MATH  Google Scholar 

  16. Valentini, S.: Representation Theorems for Quantales. Math. Log. Q. 40, 182–190 (1994)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

  1. Institute for Logic, Language and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE, Amsterdam, The Netherlands

    Alessandra Palmigiano

  2. Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria, 6, 95125, Catania, Italy

    Riccardo Re

Authors
  1. Alessandra Palmigiano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Riccardo Re
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandra Palmigiano.

Additional information

The research of the first author was made possible by the VENI grant 639.031.726 of the Netherlands Organization for Scientific Research (NWO).

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Palmigiano, A., Re, R. Relational Representation of Groupoid Quantales. Order 30, 65–83 (2013). https://doi.org/10.1007/s11083-011-9227-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-011-9227-z

Keywords

  • Unital involutive quantale
  • Strongly Gelfand quantale
  • Set groupoid
  • Representation theorem

Mathematics Subject Classifications (2010)

  • 06D05
  • 06D22
  • 06D50
  • 06F07
  • 18B40
  • 20L05
  • 22A22
  • 54D10
  • 54D30
  • 54D80