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Under non-commutative Stone duality, there is a correspondence between secondcountable Hausdorff \'etale groupoids which have a Cantor space of identitiesand what we call Tarski inverse monoids: that is, countable Boolean inverse$\wedge$-monoids with semilattices of idempotents which are countable andatomless. Tarski inverse monoids are therefore the algebraic counterparts ofthe \'etale groupoids studied by Matui and provide a natural setting for manyof his calculations. Under this duality, we prove that natural properties ofthe \'etale groupoid correspond to natural algebraic properties of the Tarskiinverse monoid: effective groupoids correspond to fundamental Tarski inversemonoids and minimal groupoids correspond to $0$-simplifying Tarski inversemonoids. Particularly interesting are the principal groupoids which correspondto Tarski inverse monoids where every element is a finite join ofinfinitesimals and idempotents. Here an infinitesimal is simply a non-zeroelement with square zero. Such inverse monoids are natural infinitegeneralizations of finite symmetric inverse monoids. The groups of units offundamental Tarski inverse monoids generalize the finite symmetric groups andinclude amongst their number the Thompson groups $G_{n,1}$ as well as thegroups of units of what we term AF inverse monoids, Krieger's ample groupsbeing examples. We characterize such groups as subgroups of particular kinds ofthe group of homeomorphisms of the Cantor space.

Subgroups of the group of homeomorphisms of the Cantor space and a  duality between a class of inverse monoids and a class of Hausdorff \'etale  groupoids