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# The groupoids of adaptable separated graphs and their type semigroup

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### Abstract

Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $$E^*$$-unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorff \'etale groupoid. We show that this groupoid is always amenable, and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the Realization Problem for von Neumann regular rings, in the finitely generated case.

### Most cited references14

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### Nonstable K-theory for Graph Algebras

(2007)
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### Graphs, Groupoids, and Cuntz–Krieger Algebras

(1997)
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• Record: found

### Inverse semigroups and combinatorial C*-algebras

(2008)
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### Author and article information

###### Journal
10 April 2019
###### Article
1904.05197