Skip to main content
SpringerLink
Log in

Effective equivalence relations and principal quantales

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

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 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. 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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. 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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Book  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Book  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisboa, Portugal

    Juan Pablo Quijano & Pedro Resende

Authors
  1. Juan Pablo Quijano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Resende
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pedro Resende.

Additional information

Communicated by Mark V. Lawson.

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. 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

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-019-09994-z

Keywords

Search

Navigation