{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:38:03Z","timestamp":1772447883374,"version":"3.50.1"},"reference-count":41,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:p>We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse \u2227-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse \u2227-semigroups arise as completions of inverse \u2227-semigroups we call pre-Boolean. An inverse \u2227-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson\u2013Higman groups G<jats:sub>n, r<\/jats:sub>. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz\u2013Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse \u2227-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.<\/jats:p>","DOI":"10.1142\/s0218196712500580","type":"journal-article","created":{"date-parts":[[2012,8,13]],"date-time":"2012-08-13T00:37:01Z","timestamp":1344818221000},"page":"1250058","source":"Crossref","is-referenced-by-count":58,"title":["NON-COMMUTATIVE STONE DUALITY: INVERSE SEMIGROUPS, TOPOLOGICAL GROUPOIDS AND C*-ALGEBRAS"],"prefix":"10.1142","volume":"22","author":[{"given":"M. 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