In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group G with dual G⋆ we obtain a suitably connected integrating symplectic double groupoid S. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold P can be integrated to a Poisson actio...

We introduce the concept of vague ideals in a distributive implication groupoid and investigate their properties. The vague ideals of a distributive implication groupoid are also characterized.

We investigate finite commutative groupoids G = 〈G, ◦〉 such that g ◦ h 6= g for all elements g, h of G. First, we show that for any such groupoid, its weak (i.e. partial) subgroupoid lattice uniquely determines its subgroupoid lattice. Next, we characterize the lattice of all weak subgroupoids of such a groupoid. This is a distributive finite lattice satisfying some combinatorial conditions con...

In a recent paper of Tamura, Merkel, and Latimer [2], the following question was raised: Suppose 5 is a groupoid (cf. [l]) which satisfies: (1) There is at least one left identity in S. (2) If y or z is a left identity of 5, then x(yz) =■= (xy)z for all xES. (3) For all a, bES there exists x£5 such that ax = b. Then does S satisfy: (3') For any xES there is a unique left identity e (which may d...

It is shown that a central extension of a Lie groupoid by an Abelian Lie group A has a principal A-bundle structure and the extended Lie groupoid is classified by an Euler esclass. Then we prove that for a symplectic α-connected, αβtransversal or α-simply connected groupoid, there exists at most one central S-extension, the Euler es-class of which corresponds to the Poisson cohomology class of ...

We define a bundle over a totally disconnected set such that each fiber is homeomorphic to a fractal blowup. We prove that there is a natural action of a Renault–Deaconu groupoid on our fractafold bundle and that the resulting action groupoid is a Renault– Deaconu groupoid itself. We also show that when the bundle is locally compact the associated C∗-algebra is primitive and has a densely defin...

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (...

We use the n-globe with its skeletal filtration to define the fundamental globular ω–groupoid of a filtered space; the proofs use an analogous fundamental cubical ω–groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free glob...

Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2–holonomy on line bundles may be formulated using the notion of a connection pair on a double category,...