A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G×X→X that defines a group action is replaced by a partial function; in addition, the existence of g·(h·x) implies the existence of (gh)·x, but not necessarily conversely. Such partial actions are extremely widespread in mathematics, and the main aim of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal inverse semigroup [Formula: see text], which has the property that partial actions of the group G give rise to actions of the inverse semigroup [Formula: see text]. We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result to group theory and the theory of E-unitary inverse semigroups.
A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C * -algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category ofétale groupoids from which we are able to set up a duality between spatial pseudogroups and soberétale groupoids. As a corollary to this duality, we deduce a non-commutative version of Stone duality involving what we call boolean inverse semigroups and booleań etale groupoids, as well as a generalization of this duality to distributive inverse semigroups. Non-commutative Stone duality has important applications in the theory of C * -algebras: it is the basis for the construction of Cuntz and Cuntz-Krieger algbras and in the case of the Cuntz algebras it can also be used to construct the Thompson groups. We then define coverages on inverse semigroups and the resulting presentations of pseudogroups. As applications, we show that Paterson's universal groupoid is an example of a booleanization, and reconcile Exel's recent work on the theory of tight maps with the work of the second author.1991 Mathematics Subject Classification. 20M18, 18B40, 06E15.
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