Coverages on inverse semigroups
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Coverages on inverse semigroups Gilles G. de Castro1 Received: 18 May 2020 / Accepted: 27 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract First we give a definition of a coverage on an inverse semigroup that is weaker than the one given by Lawson and Lenz and that generalizes the definition of a coverage on a semilattice given by Johnstone. Given such a coverage, we prove that there exists a pseudogroup that is universal in the sense that it transforms cover-to-join idempotent-pure maps into idempotent-pure pseudogroup homomorphisms. Then, we show how to go from a nucleus on a pseudogroup to a topological groupoid embedding of the corresponding groupoids. Finally, we apply the results found to study Exel’s notions of tight filters and tight groupoids. Keywords Inverse semigroup · Coverage · Étale groupoid · Nucleus · Tight filter
1 Introduction Stone’s duality between Boolean algebras and what is now called Stone spaces [22] was one of the first steps in studying the topology on a set in a more algebraical setting. His work was vastly generalized, for example to the duality between sober spaces and spatial frames [8, 17, 23]. In this context, Johnstone’s definition of a coverage on a semilattice [8], used to present frames using generators and relations, can be also used to impose relations on open sets of a topological space. Another result connecting topological spaces with algebras is the Gelfand–Naimark duality between commutative C*-algebras and locally compact Hausdorff spaces [7] by means of a subalgebra of the algebra of all complex valued continuous functions from a topological space. One can then think that a noncommutative C*-algebra represents a complex valued functions of a virtual object thought to be a noncommutative space. This led Connes to the development of Communicated by Mark V. Lawson. * Gilles G. de Castro [email protected] 1
Departamento de Matemática, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, Florianópolis, SC 88040‑970, Brazil
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Noncommutative Geometry [4]. A concrete mathematical object that can play the role of the noncommutative space is a topological groupoid [4, 18], and it is often useful to describe a C*-algebra using groupoids in order to use the toolkit started in [18]. One interesting problem that arises is to generalize the duality between topological spaces and frames by replacing topological spaces with topological groupoids (or even more general objects). One major milestone in this direction is the work of Resende [20], where he relates étale groupoids, quantales and (abstract) pseudogroups. In this paper we are more interested in the relation between étale groupoids and pseudogroups, the latter being a certain subclass of inverse semigroups. Going back to C*-algebras, inverse semigroups already appears in Renault’s monograph [18] and it is one tool in describing a C*-algebras as in Paterson’s book [15]. Paterson’s universal groupoid from an in
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