The Polycyclic Inverse Monoids and the Thompson Groups Revisited

Mark V. Lawson

doi:10.1007/978-981-33-4842-4_12

The Polycyclic Inverse Monoids and the Thompson Groups Revisited

Mark V. Lawson^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

2 Citations (Scopus)

52 Downloads (Pure)

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 G_n₁ is the group of units of a Boolean inverse monoid C_n called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid P_n. The étale topological groupoid associated with C_n under non-commutative stone duality is the usual groupoid associated with the corresponding Cuntz C^∗ -algebra. We then show that the group G_n 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 language	English
Title of host publication	Semigroups, Categories, and Partial Algebras. ICSAA 2019
Editors	P. G. Romeo, Mikhail V. Volkov, A. R. Rajan
Publisher	Springer
Pages	179-214
Number of pages	36
ISBN (Electronic)	9789813348424
ISBN (Print)	9789813348417
DOIs	https://doi.org/10.1007/978-981-33-4842-4_12
Publication status	Published - 27 Mar 2021
Event	International Conference on Semigroups and Applications 2019 - Kochi, India Duration: 9 Dec 2019 → 12 Dec 2019

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	345
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Conference on Semigroups and Applications 2019
Abbreviated title	ICSAA 2019
Country/Territory	India
City	Kochi
Period	9/12/19 → 12/12/19

Keywords

Cantor algebras
Free monoids
Polycyclic inverse monoids
Thompson groups
étale groupoids

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-981-33-4842-4_12

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This is a post-peer-review, pre-copyedit version of an article published in Springer Proceedings in Mathematics and Statistics. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-981-33-4842-4_12
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title = "The Polycyclic Inverse Monoids and the Thompson Groups Revisited",

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 {\'e}tale 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 {\`a} la Statman and then in terms of labelled trees {\`a} la Higman.",

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Lawson, MV 2021, The Polycyclic Inverse Monoids and the Thompson Groups Revisited. in PG Romeo, MV Volkov & AR Rajan (eds), Semigroups, Categories, and Partial Algebras. ICSAA 2019. Springer Proceedings in Mathematics and Statistics, vol. 345, Springer, pp. 179-214, International Conference on Semigroups and Applications 2019, Kochi, India, 9/12/19. https://doi.org/10.1007/978-981-33-4842-4_12

The Polycyclic Inverse Monoids and the Thompson Groups Revisited. / Lawson, Mark V.
Semigroups, Categories, and Partial Algebras. ICSAA 2019. ed. / P. G. Romeo; Mikhail V. Volkov; A. R. Rajan. Springer, 2021. p. 179-214 (Springer Proceedings in Mathematics and Statistics; Vol. 345).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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AU - Lawson, Mark V.

PY - 2021/3/27

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N2 - 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.

AB - 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.

KW - Cantor algebras

KW - Free monoids

KW - Polycyclic inverse monoids

KW - Thompson groups

KW - étale groupoids

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ER -

The Polycyclic Inverse Monoids and the Thompson Groups Revisited

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