Abstract
Under noncommutative 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 0simplifying 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 nonzero 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 Gn,1 as well as the groups of units of AF inverse monoids, Krieger's ample groups being examples.
Original language  English 

Pages (fromto)  77–114 
Number of pages  38 
Journal  Journal of Algebra 
Volume  462 
Early online date  1 Jun 2016 
DOIs  
Publication status  Published  15 Sep 2016 
Keywords
 Inverse semigroups
 Etale groupoid
 Stone duality
 Thompson groups
 The Cantor space
ASJC Scopus subject areas
 Mathematics(all)
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Mark Lawson
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)