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.
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Notes
Usually, for a set A to be an order ideal we also ask for it to be non-empty and directed, that is for every \(a,b\in A\), there exists \(c\in A\) such that \(a\le c\) and \(b\le c\). For this paper, we use the same terminology as in [12].
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Communicated by Mark V. Lawson.
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de Castro, G.G. Coverages on inverse semigroups. Semigroup Forum 102, 375–396 (2021). https://doi.org/10.1007/s00233-020-10134-1
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DOI: https://doi.org/10.1007/s00233-020-10134-1