Coverages on inverse semigroups

• DOI: 10.1007/s00233-020-10134-1
• Publication Date: 2020-09-30T20:02:55Z
• Authors: Gilles G. de Castro
• Keywords: Semilattice; Inverse semigroup; Rank (graph theory)

First we give a definition of a coverage on a inverse semigroup that is weaker than the one gave by a 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.