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...

Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and $O^u$ equipped with naturally defined holomorphic structures, where u, v are finite sequences of Weyl elements, were studied by Jiang Hua Lu the author. We prove in this paper that $G^{u,u}$ is groupoid over $O^u$, extending result from aforementioned authors about \pi_{st})$. O...

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...

Many important C∗-algebras, such as AF-algebras, Cuntz-Krieger algebras, graph algebras and foliation C∗-algebras, are the C∗-algebras of r-discrete groupoids. These C∗-algebras are often associated with inverse semigroups through the C∗-algebra of the inverse semigroup [HR90] or through a crossed product construction as in Kumjian’s localization [Kum84]. Nica [Nic94] connects groupoid C∗-algeb...

The purpose of these notes is to discuss the problem of moduli for curves of genus g ≥ 3 1 and outline the construction of the (coarse) moduli scheme of stable curves due to Gieseker. The notes are broken into 4 parts. In Section 1 we discuss the general problem of constructing a moduli “space” of curves. We will also state results about its properties, some of which will be discussed in the se...

Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds i...

The focus of this thesis is the study of nuclearity and exactness for groupoid crossed product C∗-algebras. In particular, we present generalizations of two well-known facts from group dynamical systems and crossed products to the groupoid setting. First, we show that if G is an amenable groupoid acting on an upper-semicontinuous C∗-bundle A with nuclear section algebra A, then the associated g...