The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations

Abstract

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 S is a Boolean inverse semigroup presented by the abstract versions of the Cuntz-Krieger relations. We call this Boolean inverse semigroup the Exel completion of S and show that it arises from Exel's tight groupoid under non-commutative Stone duality. Crucial to our thinking, was the work of Ruy Exel [7] and Daniel Lenz [30]; our main theorem (Theorem 1.4) is analogous to a result of Benjamin Steinberg [41, Corollary 5.3] but we work, of course, with Boolean inverse semigroups. Our use of covers and cover-to-join maps goes back to [28] although they also play a role in [7]; see also [6], a paper tightly linked to this one; in particular, the authors would like to thank Allan Donsig for answering some of their questions. In the first version of this paper, the authors proved Theorem 1.4 under the assumption that the inverse semigroup was a 'weak semilattice' in the sense of Steinberg [40,41]. The authors would like to thank Enrique Pardo for pointing out that this was unnecessary and supplying a result, suggested by Lisa Orloff Clark, that enabled us to prove the more general version of the theorem. Finally, the authors would like to thank Ruy

Higher dimensional generalizations of the Thompson groups

“…The above results means that R(S) is the distributive completion of Σ(S). By Proposition 14.2 and Proposition 14.4, it follows that the Boolean inverse monoid C(S) is what we termed in [36] the tight completion of Σ(S) and the groupoid G(S) is therefore the tight groupoid of Σ(S). Remark 14.5.…”

Zappa-Szép products for partial actions of groupoids on Left Cancellative Small Categories

“…Now we will see that Zappa-Szép products of LCSC arises naturally among LCSC. In [11] they defined the generalized higher rank k-graphs categories and described them as Zappa-Szép products. Here we slightly generalize their arguments to a more general class of LCSC.…”

Quotients of the Booleanization of an inverse semigroup

“…Tight representations of inverse semigroups [7,8,9] are useful in the study of various C *algebras generated by partial isometries, see, e.g., [1,6,8,10,11,23,27] and are closely related to cover-to-join representations [6,17,20] which are more simpler defined but equally useful: universal objects based on them are isomorphic to those defined on tight representations [9]. The tight Booleanization B tight (S) of an inverse semigroup S is the universal Boolean inverse semigroup with respect to tight representations (or, equivalently, with respect to cover-tojoin representations) of S in Boolean inverse semigroups.…”

“…Under the non-commutative Stone duality, it is dual to Exel's tight groupoid G tight (S) of S. The latter groupoid is the restriction of G(S) to the tight spectrum E(S) tight of the semilattice of idempotents E(S) of S which is a closed and invariant subset of the spectrum E(S). Equivalently, B tight (S) is a quotient of B(S) by certain relations, which can be looked at as abstract Cuntz-Krieger relations, see [20].…”