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

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• 作者：Mark V. Lawson¹ ; ^{M.V.Lawson@hw.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20M18 ; 18B40 ; 06E15
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：462
• 期：Complete
• 页码：77-114
• 全文大小：652 K

文摘

Under non-commutative Stone duality, there is a correspondence between second countable Hausdorff étale groupoids which have a Cantor space of identities and what we call Tarski inverse monoids: that is, countable Boolean inverse ∧-monoids with semilattices of idempotents which are countable and atomless. Tarski inverse monoids are therefore the algebraic counterparts of the étale groupoids studied by Matui and provide a natural setting for many of his calculations. Under this duality, we prove that natural properties of the étale groupoid correspond to natural algebraic properties of the Tarski inverse monoid: effective groupoids correspond to fundamental Tarski inverse monoids and minimal groupoids correspond to 0-simplifying Tarski inverse monoids. Particularly interesting are the principal groupoids which correspond to Tarski inverse monoids where every element is a finite join of infinitesimals and idempotents. Here an infinitesimal is simply a non-zero element with square zero. The groups of units of fundamental Tarski inverse monoids generalize the finite symmetric groups and include amongst their number the Thompson groups G_n,1$G_{n,1}$ as well as the groups of units of AF inverse monoids, Krieger's ample groups being examples.