This paper is a contribution to the theory of what might be termed 0-dimensional non-commutative spaces. We prove that associated with each inverse semigroup is a Boolean inverse semigroup presented by the abstract versions of the Cuntz–Krieger relations. We call this Boolean inverse semigroup the tight completion of and show that it arises from Exel's tight groupoid under non-commutative Stone duality.
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Mark V. Lawson, Alina Vdovina, The universal Boolean inverse semigroup presented by the abstract Cuntz–Krieger relations. J. Noncommut. Geom. 15 (2021), no. 1, pp. 279–304DOI 10.4171/JNCG/400