The Groupoids of Adaptable Separated Graphs and Their Type Semigroups. (10th April 2020)

• Record Type: Journal Article
• Authors: Ara, Pere; Bosa, Joan; Pardo, Enrique; Sims, Aidan
• Is Part Of: International mathematics research notices. Volume 2021:Number 20(2021)
• Page Start: 15444
• Page End: 15496
• Publication Date: 2020-04-10
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa022
• Languages: English
• ISSNs: 1073-7928

Abstract

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 étale 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.