Effective equivalence relations and principal quantales


Stably supported quantales generalize pseudogroups and provide an algebraic context in which to study the correspondences between inverse semigroups and étale groupoids. Here we study a further generalization where a non-unital version of supported quantale carries the algebraic content of such correspondences to the setting of open groupoids. A notion of principal quantale is introduced which, in the case of groupoid quantales, corresponds precisely to effective equivalence relations.

This is a preview of subscription content, log in to check access.


  1. 1.

    Bunge, M.: An application of descent to a classification theorem for toposes. Math. Proc. Camb. Philos. Soc. 107(1), 59–79 (1990). https://doi.org/10.1017/S0305004100068365

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Kock, A.: Moerdijk, Ieke, Presentations of étendues, English, with French summary. Cah. Topol. Géom. Différ. Catég. 32(2), 145–164 (1991)

    Google Scholar 

  3. 3.

    Kumjian, A.: On localizations and simple \(C^{\ast } \)-algebras. Pac. J. Math. 112(1), 141–192 (1984)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Kudryavtseva, G., Lawson, M.V.: A perspective on non-commutative frame theory. Adv. Math. 311(378–468), 0001–8708 (2017). https://doi.org/10.1016/j.aim.2017.02.028

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Lawson, M.V., Lenz, D.H.: Pseudogroups and their étale groupoids. Adv. Math. 244(117–170), 0001–8708 (2013). https://doi.org/10.1016/j.aim.2013.04.022

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Marcelino, S., Resende, P.: An algebraic generalization of Kripke structures. Math. Proc. Camb. Philos. Soc. 145(3), 549–577 (2008)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Matsnev, D., Resende, P.: Étale groupoids as germ groupoids and their base extensions. Proc. Edinb. Math. Soc. (2) 53(3), 765–785 (2010). https://doi.org/10.1017/S001309150800076X

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Moerdijk, I.: The classifying topos of a continuous groupoid. Trans. Am. Math. Soc. 310(2), 629–668 (1988)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Moerdijk, I.: The classifying topos of a continuous groupoid. II, Cah. Topol. Géom. Différ. Catég. 31(2), 137–168 (1990)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Moerdijk, I., Mrčun, J.: Introduction to Foliations and Lie Groupoids. Cambridge Studies in Advanced Mathematics, vol. 91, p. x+173. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511615450

    Google Scholar 

  11. 11.

    Mulvey, C.J., Pelletier, J.W.: On the quantisation of points. J. Pure Appl. Algebra 159(2–3), 231–295 (2001)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Paterson, A.L.T.: Groupoids, Inverse Semigroups, and Their Operator Algebras. Progress in Mathematics, vol. 170, p. xvi+274. Birkhäuser Boston Inc., Boston (1999)

    Google Scholar 

  13. 13.

    Protin, M.C., Resende, P.: Quantales of open groupoids. J. Noncommut. Geom 6(2), 199–247 (2012). https://doi.org/10.4171/JNCG/90

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Quijano, J.P.: Sheaves and functoriality of groupoid quantales, Univ. Lisboa, Doctoral Thesis (2018)

  15. 15.

    Renault, J.: A Groupoid Approach to \(C^{\ast } \)-algebras. Lecture Notes in Mathematics, vol. 793. Springer, Berlin (1980)

    Google Scholar 

  16. 16.

    Renault, J.: Cartan subalgebras in \(C^*\)-algebras. Ir. Math. Soc. Bull. 61(29–63), 0791–5578 (2008)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Resende, P.: Étale groupoids and their quantales. Adv. Math. 208(1), 147–209 (2007)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Resende, P.: Groupoid sheaves as quantale sheaves. J. Pure Appl. Algebra 216(1), 41–70 (2012). https://doi.org/10.1016/j.jpaa.2011.05.002

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Pedro Resende.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Pedro Resende: Work funded by FCT/Portugal through the LisMath program and projects EXCL/MAT-GEO/0222/2012 and PEst-OE/EEI/LA0009/2013.

Communicated by Mark V. Lawson.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Quijano, J.P., Resende, P. Effective equivalence relations and principal quantales. Semigroup Forum 99, 754–787 (2019). https://doi.org/10.1007/s00233-019-09994-z

Download citation


  • Open localic groupoids
  • Effective equivalence relations
  • Supported quantales