The groupoids of adaptable separated graphs and their type semigroup
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 -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.
Keywords
Cite
@article{arxiv.1904.05197,
title = {The groupoids of adaptable separated graphs and their type semigroup},
author = {Pere Ara and Joan Bosa and Enrique Pardo and Aidan Sims},
journal= {arXiv preprint arXiv:1904.05197},
year = {2020}
}
Comments
v2: minor update in introduction to acknowledge Scarparo's resolution of Matui's conjecture; v3: a number of improvements in exposition, and typos corrected - this version to appear in International Mathematics Research Notices