Skip to main content
×
Home
    • Aa
    • Aa
  • Access
  • Cited by 13
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Kudryavtseva, Ganna and Lawson, Mark V. 2016. Boolean sets, skew Boolean algebras and a non-commutative Stone duality. Algebra universalis, Vol. 75, Issue. 1, p. 1.


    Lawson, Mark V. and Scott, Philip 2016. AF inverse monoids and the structure of countable MV-algebras. Journal of Pure and Applied Algebra,


    Lawson, Mark V. 2016. Subgroups of the group of homeomorphisms of the Cantor space and a duality between a class of inverse monoids and a class of Hausdorff étale groupoids. Journal of Algebra, Vol. 462, p. 77.


    DONSIG, ALLAN P. and MILAN, DAVID 2014. JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT -ALGEBRAS. Bulletin of the Australian Mathematical Society, Vol. 90, Issue. 01, p. 121.


    Jones, David G. and Lawson, Mark V. 2014. Graph inverse semigroups: Their characterization and completion. Journal of Algebra, Vol. 409, p. 444.


    Kudryavtseva, Ganna and Lawson, Mark V. 2014. The structure of generalized inverse semigroups. Semigroup Forum, Vol. 89, Issue. 1, p. 199.


    Bauer, Andrej Cvetko-Vah, Karin Gehrke, Mai van Gool, Samuel J. and Kudryavtseva, Ganna 2013. A non-commutative Priestley duality. Topology and its Applications, Vol. 160, Issue. 12, p. 1423.


    KUDRYAVTSEVA, GANNA 2013. A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY. Journal of the Australian Mathematical Society, Vol. 95, Issue. 03, p. 383.


    Lawson, Mark V. and Lenz, Daniel H. 2013. Pseudogroups and their étale groupoids. Advances in Mathematics, Vol. 244, p. 117.


    LAWSON, M. V. MARGOLIS, S. W. and STEINBERG, B. 2013. THE ÉTALE GROUPOID OF AN INVERSE SEMIGROUP AS A GROUPOID OF FILTERS. Journal of the Australian Mathematical Society, Vol. 94, Issue. 02, p. 234.


    Jones, D. G. and Lawson, M. V. 2012. Strong representations of the polycyclic inverse monoids: Cycles and atoms. Periodica Mathematica Hungarica, Vol. 64, Issue. 1, p. 53.


    Kudryavtseva, Ganna 2012. A refinement of Stone duality to skew Boolean algebras. Algebra universalis, Vol. 67, Issue. 4, p. 397.


    LAWSON, M. V. 2012. NON-COMMUTATIVE STONE DUALITY: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C*-ALGEBRAS. International Journal of Algebra and Computation, Vol. 22, Issue. 06, p. 1250058.


    ×
  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society, Volume 88, Issue 3
  • June 2010, pp. 385-404

A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

  • M. V. Lawson (a1)
  • DOI: http://dx.doi.org/10.1017/S1446788710000145
  • Published online: 01 May 2010
Abstract
Abstract

We prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY
      Your Kindle email address
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]J. Cuntz , ‘Simple C*-algebras generated by isometries’, Comm. Math. Phys. 57 (1977), 173185.

[4]R. Exel , ‘Inverse semigroups and combinatorial C*-algebras’, Bull. Braz. Math. Soc. (N.S.) 39 (2008), 191313.

[5]R. Exel , ‘Tight representations of semilattices and inverse semigroups’, Semigroup Forum 79 (2009), 159182.

[8]B. Hughes , ‘Trees and ultrametric spaces: a categorical equivalence’, Adv. Math. 189 (2004), 148191.

[11]K. Kawamura , ‘Polynomial endomorphisms of the Cuntz algebras arising from permutations I. General theory’, Lett. Math. Phys. 71 (2005), 149158.

[12]K. Kawamura , ‘Branching laws for polynomial endomorphisms of the Cuntz algebras arising from permutations’, Lett. Math. Phys. 77 (2006), 111126.

[13]J. Kellendonk , ‘The local structure of tilings and their integer group of coinvariants’, Commun. Math. Phys. Soc. 187 (1997), 115157.

[14]J. Kellendonk , ‘Topological equivalence of tilings’, J. Math. Phys. 38 (1997), 18231842.

[16]M. V. Lawson , Inverse Semigroups (World Scientific, Singapore, 1998).

[17]M. V. Lawson , ‘Orthogonal completions of the polycyclic monoids’, Comm. Algebra 35 (2007), 16511660.

[18]M. V. Lawson , ‘The polycyclic monoids Pn and the Thompson groups Vn,1’, Comm. Algebra 35 (2007), 40684087.

[19]M. V. Lawson , ‘Primitive partial permutation representations of the polycyclic monoids and branching function systems’, Period. Math. Hungar. 58 (2009), 189207.

[24]A. L. T. Paterson , Groupoids, Inverse Semigroups, and their Operator Algebras (Birkhäuser, Boston, MA, 1999).

[25]J. Renault , A Groupoid Approach to C*-Algebras (Springer, Berlin, 1980).

[27]P. Resende , ‘Étale groupoids and their quantales’, Adv. Math. 208 (2007), 147209.

[29]M. H. Stone , ‘Applications of the theory of Boolean rings to general topology’, Trans. Amer. Math. Soc. 41 (1937), 375481.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords: