On Hirata separability and inner partial group action

Partial actions of groups on generalized matrix rings

Actions of inverse semigroups arising from partial actions of groups

In this paper, we study partial group actions on 2-complexes. Our results include a characterization, in terms of generating sets, of when a partial group action on a connected 2-complex has a connected globalization. Using this result, we give a short combinatorial proof that a group acting without fixed points on a connected 2-complex, with finite quotient, is finitely generated. This result is then generalized to characterize finitely generated groups as precisely those groups having a partial action, without fixed points, on a finite tree, with a connected globalization. Finally, using Bass-Serre theory, we determine when a partial group action on a graph has a globalization which is a tree.

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an -partial Galois extension of and a normal subgroup H of G, we prove that induces a unital partial action of G/H on the subalgebra of invariants of S such that is an -partial Galois extension of Second, assuming that G is abelian, we construct a commutative inverse semigroup whose elements are equivalence classes of -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between and T(G, A), where ρ is a congruence on and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of reduces to the case where G is cyclic. The set of idempotents of is also investigated.

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG : X → X / ∼G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y G such that the global group action of G on Y G is continuous and the injection Y into Y G is an open dense equivariant embedding.

In this survey paper we define partial actions of groups on algebras and give several related results. For any partial action a partial skew group ring is defined. This partial skew group ring is not always associative. Conditions under which associativity holds are studied. Several other questions are considered like, for example, enveloping actions and properties of partial skew group rings.

Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Hölder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.

Partial actions of groups on semiprime rings

In this paper, we introduce the concept of partial groupoid actions on [Formula: see text]-semicategories as well as we give criteria for existence of a globalization of it. This point of view is a generalization of the notions of partial groupoid actions on rings and partial group action on an [Formula: see text]-semicategory. We also define the notions of partial skew groupoid category, smash product and describe functorial relations between them, in particular we show that the smash product is a Galois covering of its associated skew groupoid category.

Nuclear dimension of crossed products associated to partial dynamical systems

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product. Based on these notions and properties, then we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra. Finally, we prove that any Hopf partial Galois extension induces a unique partial entwining map compatible with the right partial coaction.

Partial projective representations and partial actions II

We consider a large class of partial actions of groups on rings, called regular, which contains all s-unital partial actions, as well as all partial actions on C∗-algebras. For them the notion of Morita equivalence is introduced, and it is shown that any regular partial action is Morita equivalent to a globalizable one, and the globalization is essentially unique. It is also proved that Morita equivalent s-unital partial actions on rings with orthogonal local units are stably isomorphic. In addition, we show that Morita equivalent s-unital partial actions on commutative rings must be isomorphic, and an analogous result for C∗-algebras is also established. Date: November 8, 2015. This work was partially supported by CNPq of Brazil (Proc. 305975/2013-7, Proc. 300362/20102), Fapesp of Brazil (Proc. Proc. 2009/52665-0), MINECO (Ministerio de Economia y Competitividad), (Fondo Europeo de Desarrollo Regional) project MTM2012-35240, Spain, and Fundacion Seneca of Murcia, Programa Hispano Brasileno de Cooperacion Universitaria PHB2012-0135, Spain. 2000 Mathematics Subject Classification: Primary 16S35; Secondary 16W22, 46L05.

Strong equivalence of graded algebras