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The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

Journal Article


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 ��tale 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.

Publication Date


  • 2021

Citation


  • Ara, P., Bosa, J., Pardo, E., & Sims, A. (2021). The Groupoids of Adaptable Separated Graphs and Their Type Semigroups. International Mathematics Research Notices, 2021(20), 15444-15496. doi:10.1093/imrn/rnaa022

Scopus Eid


  • 2-s2.0-85118996001

Web Of Science Accession Number


Start Page


  • 15444

End Page


  • 15496

Volume


  • 2021

Issue


  • 20

Place Of Publication


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 ��tale 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.

Publication Date


  • 2021

Citation


  • Ara, P., Bosa, J., Pardo, E., & Sims, A. (2021). The Groupoids of Adaptable Separated Graphs and Their Type Semigroups. International Mathematics Research Notices, 2021(20), 15444-15496. doi:10.1093/imrn/rnaa022

Scopus Eid


  • 2-s2.0-85118996001

Web Of Science Accession Number


Start Page


  • 15444

End Page


  • 15496

Volume


  • 2021

Issue


  • 20

Place Of Publication