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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.

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          Most cited references 14

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          Nonstable K-theory for Graph Algebras

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            Graphs, Groupoids, and Cuntz–Krieger Algebras

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              Inverse semigroups and combinatorial C*-algebras

               Ruy Exel* (2008)
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                Author and article information

                Journal
                10 April 2019
                Article
                1904.05197

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math.OA math.RA

                Algebra

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