Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups

Wehrung, Friedrich

doi:10.1007/978-3-319-61599-8

Cited by 37 publications

(54 citation statements)

References 46 publications

(96 reference statements)

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“…In much of the groupoid literature, these are simply called bisections, or sometimes slices 2. Indeed, since G is Hausdorff, S G is a Boolean inverse meet-semigroup (see, for example,[31]) with meet given by intersection. But since the additional Boolean meet-semigroup structure comes from the concrete realisation of S G as a collection of subsets of G, we will not introduce the formal axiomatisation of a Boolean inverse meet-semigroup here, but work concretely with the usual set operations when manipulating elements of S G .…”

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confidence: 99%

Reconstruction of graded groupoids from graded Steinberg algebras

et al. 2016

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Abstract. We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies C * -isomorphism of C * -algebras for graphs E and F in which every cycle has an exit.

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confidence: 99%

Reconstruction of graded groupoids from graded Steinberg algebras

et al. 2016

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“…Recall that a Boolean inverse semigroup is an inverse semigroup S such that E(S) is a Boolean ring (a ring with x = x 2 for all x), and such that every pair x, y ∈ S satisfying x ⊥ y has a supremum, denoted x ⊕ y ∈ S (see [43, Definition 3-1.6] for further details). For the following definition, we refer to [43] for ease of accessibility; but it did not originate there (see for example [42,23,27]). If G is second countable, then the type semigroup Typ(G) is a countable conical refinement monoid.…”

Section: Amenability Of G Tight (S(e C))mentioning

confidence: 99%

The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

Ara

Bosa

Pardo

et al. 2020

International Mathematics Research Notices

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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 Hausdorffétale 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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“…Connections between inverse monoids or inverse semigroups in general, and Boolean structures (often with intersections) has been a subject of study in recent years. For an extended exposition of these and related matters see Friedrich Wehrung's 2017 Springer monograph [60].…”

Section: Some General Factsmentioning

confidence: 99%