, Volume 30, Issue 1, pp 65–83 | Cite as

Relational Representation of Groupoid Quantales

  • Alessandra Palmigiano
  • Riccardo Re
Open Access


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).


Unital involutive quantale Strongly Gelfand quantale Set groupoid Representation theorem 

Mathematics Subject Classifications (2010)

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


  1. 1.
    Brown, C., Gurr, D.: A representation theorem for quantales. J. Pure Appl. Algebra 85(1), 27–42 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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. 3.
    Jónsson, B., Tarski, A.: Boolean algebras with operators, part II. Am. J. Math. 74(2), 127–162 (1952)zbMATHCrossRefGoogle Scholar
  4. 4.
    Kruml, D., Paseka, J.: Embeddings of quantales into simple quantales. J. Pure Appl. Algebra 148(2), 209–424 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Johnstone, P.T.: Stone Spaces. Cambridge Studies in Adv. Math., vol. 3. Cambridge University Press, Cambridge, UK (1982)zbMATHGoogle Scholar
  6. 6.
    Mulvey, C.J.: &. Rend. Circ. Mat. Palermo 12, 99–104 (1986)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Mulvey, C.J., Resende, P.: A noncommutative theory of Penrose tilings. Int. J. Theor. Phys. 44, 655–689 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Palmigiano, A., Re, R.: Groupoid quantales: a non-étale setting. J. Pure Appl. Algebra 215(8), 1945–1957 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 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)Google Scholar
  10. 10.
    Paterson, A.L.T.: Groupoids, Inverse Semigroups, and their Operator Algebras. Progress in Mathematics, vol. 170. Birkhäuser, Boston (1999)zbMATHCrossRefGoogle Scholar
  11. 11.
    Protin, M.C., Resende, P.: Quantales of Open Groupoids. arXiv:0811.4539v2
  12. 12.
    Resende, P.: Lectures on Étale Groupoids, Inverse Semigroups and Quantales. Lecture Notes for the GAMAP IP Meeting, Antwerp (2006)Google Scholar
  13. 13.
    Resende, P.: Étale groupoids and their quantales. Adv. Math. 208, 147–209 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rosenthal, K.: Quantales an their Applications. Pitman Research Notes in Math., vol. 234. Longman (1990)Google Scholar
  15. 15.
    Takesaki, M.: Theory of Operator Algebras I, EMS Series, vol. 124. Springer, New York (2002)zbMATHGoogle Scholar
  16. 16.
    Valentini, S.: Representation Theorems for Quantales. Math. Log. Q. 40, 182–190 (1994)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Dipartimento di Matematica e InformaticaUniversità di CataniaCataniaItaly

Personalised recommendations