The Polycyclic Inverse Monoids and the Thompson Groups Revisited

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Abstract

We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group Gn1 is the group of units of a Boolean inverse monoid Cn called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid Pn. The étale topological groupoid associated with Cn under non-commutative stone duality is the usual groupoid associated with the corresponding Cuntz C -algebra. We then show that the group Gn 1 is also the group of automorphisms of a specific n-ary Cantor algebra: this n-ary Cantor algebra is constructed first as the monoid of total maps of a restriction semigroup à la Statman and then in terms of labelled trees à la Higman.

Original languageEnglish
Title of host publicationSemigroups, Categories, and Partial Algebras. ICSAA 2019
EditorsP. G. Romeo, Mikhail V. Volkov, A. R. Rajan
PublisherSpringer
Pages179-214
Number of pages36
ISBN (Electronic)9789813348424
ISBN (Print)9789813348417
DOIs
Publication statusPublished - 27 Mar 2021
EventInternational Conference on Semigroups and Applications 2019 - Kochi, India
Duration: 9 Dec 201912 Dec 2019

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume345
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceInternational Conference on Semigroups and Applications 2019
Abbreviated titleICSAA 2019
CountryIndia
CityKochi
Period9/12/1912/12/19

Keywords

  • Cantor algebras
  • Free monoids
  • Polycyclic inverse monoids
  • Thompson groups
  • étale groupoids

ASJC Scopus subject areas

  • Mathematics(all)

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