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The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics.

We introduce the notion of groupoid grading, give some nontrivial examples and prove that groupoid gradings on simple commutative or anti-commutative algebras are necessarily group gradings. We also take advantage of the structure of groupoids to prove some results about groupoid gradings and certain coarsenings of these which turn out to be group gradings. We also study set gradings on arbitrary algebras, by characterizing their homogeneous semisimplicity and their homogeneous simplicity in terms of a property satisfied by the supports of the gradings, and also relate set gradings with groupoid gradings via coarsenings. Finally we study a class of set gradings on M n ( C ) , the orthogonal gradings, and show that all of them which are fine are necessarily groupoid gradings.

In the framework of Banach differential geometry we investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W ∗ -algebra (von Neumann algebra) M . The main role in this theory is played by the complex Banach–Lie groupoid G ( M ) ⇉ L ( M ) of partially invertible elements of M over the lattice L ( M ) of orthogonal projections of M . The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson V B -groupoids with G ( M ) ⇉ L ( M ) as the side groupoid.

The notion of travel groupoids was introduced by Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set V and a binary operation $$*$$ on V satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph G has a travel groupoid if the graph associated with the travel groupoid is equal to G. Nebeský gave a characterization for finite graphs to have a travel groupoid. In this paper, we introduce the notion of T-neighbor systems on a graph and give a characterization of travel groupoids on a graph in terms of T-neighbor systems.

Let $S|_{R}$ be a groupoid Galois extension with Galois groupoid $G$ such that $E_{g}^{G_{r(g)}}\subseteq C1_{g}$, for all $g\in G$, where $C$ is the centre of $S$, $G_{r(g)}$ is the principal group associated to $r(g)$ and $\{E_{g}\}_{g\in G}$ are the ideals of $S$. We give a complete characterisation in terms of a partial isomorphism groupoid for such extensions, showing that $G=\dot{\bigcup }_{g\in G}\text{Isom}_{R}(E_{g^{-1}},E_{g})$ if and only if $E_{g}$ is a connected commutative algebra or $E_{g}=E_{g}^{G_{r(g)}}\oplus E_{g}^{G_{r(g)}}$, where $E_{g}^{G_{r(g)}}$ is connected, for all $g\in G$.

To each supersimple $2-(n,4,\lambda)$ design $\mathcal{D}$ one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid associated to $M_{13}$ which is constructed from $\mathbb{P}_3$. We show that $\operatorname{Sp}_{2m}(2)$ and $2^{2m}.\operatorname{Sp}_{2m}(2)$ naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive $\mathbb{F}_2$-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction to a previously known family of completely transitive codes. We also give a new characterization of $M_{13}$ and prove that, for a fixed $\lambda > 0,$ there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating or symmetric group., Comment: 31 pages

The purpose of the paper is to consider the covering morphism and the action of a double groupoid on a groupoid; and then characterize these notions for double group- groupoids. We also give a categorical equivalence between the actions and covering morphisms of double group-groupoids.

A framework for higher gauge theory based on a 2-group is presented, by constructing a groupoid of connections on a manifold acted on by a 2-group of gauge transformations, following previous work by the authors where the general notion of the action of a 2-group on a category was defined. The connections are discretized, given by assignments of 2-group data to 1- and 2-cells coming from a given cell structure on the manifold, and likewise the gauge transformations are given by 2-group assignments to 0-cells. The 2-cells of the manifold are endowed with a bigon structure, matching the 2-dimensional algebra of squares which is used for calculating with 2-group data. Showing that the action of the 2-group of gauge transformations on the groupoid of connections is well-defined is the central result. The effect, on the groupoid of connections, of changing the discretization is studied, and partial results and conjectures are presented around this issue. The transformation double category that arises from the action of a 2-group on a category, as defined in previous work by the authors, is described for the case at hand, where it becomes a transtormation double groupoid. Finally, examples of the construction are given for simple choices of manifold: the circle, the 2-sphere and the torus., 30 pages, 10 figures

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C ∗ -algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C ∗ -algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.

A Poisson structure on a homogeneous space of a Poisson groupoid is homogeneous if the action of the Lie groupoid on the homogeneous space is compatible with the Poisson structures. According to a result of Liu, Weinstein and Xu, Poisson homogeneous spaces of a Poisson groupoid are in correspondence with suitable Dirac structures in the Courant algebroid defined by the Lie bialgebroid of the Poisson groupoid. We show that this correspondence result fits into a more natural context: the one of Dirac groupoids, which are objects generalizing Poisson groupoids and multiplicative closed 2-forms on groupoids.

We obtain a characterization in terms of dynamical systems of those r-discrete groupoids for which the groupoid C*-algebra is approximately finite-dimensional (AF). These ideas are then used to compute the K-theory for AF algebras by utilizing the actions of these partial homeomorphisms, and these K-theoretic calculations are applied to some specific examples of AF algebras. Finally, we show that, for a certain class of dimension groups, a groupoid can be obtained directly from the dimension group's structure whose associated C*-algebra has its dimension group isomorphic to the original dimension group.

In this paper, we consider the multiplicative groupoid of matrices with elements in a lattice with 0 and 1. Examples of such groupoids are the semigroup of binary relations and semigroups of minimax (fuzzy) relations. It is shown that every automorphism of a groupoid is the composition of an inner automorphism and the automorphism defined by an automorphism of the lattice. Despite the fact that, in general, the groupoid is not associative, it satisfies the UA-property: Every multiplicative automorphism is an additive automorphism. Earlier, the realization of the UA-property has been considered mainly for associative rings and semirings. We describe the invertible matrices that define inner automorphisms.

We consider the Deaconu–Renault groupoid of an action of a finitely generated free abelian monoid by local homeomorphisms of a locally compact Hausdorff space. We catalogue the primitive ideals of the associated groupoid C ∗-algebra. For a special class of actions we describe the Jacobson topology.

In the paper we study the algebroid A(M) of the groupoid G(M)⇉L(M) of partially invertible elements over the lattice L(M) of orthogonal projections of a W∗-algebra M. In particular the complex Banach manifold structure of these objects is investigated. The expressions on the algebroid Lie brackets for A(M) and related algebroids are given in noncommutative operator coordinates in the explicit way. We also describe structure of the groupoid of partial isometries U(M)⇉L(M) and the frame groupoid GlinAp0(M)⇉Lp0(M) as well as their algebroids.

Let G be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and K a compact subgroupoid of G with a Haar system too. (G, K) is a Gelfand pair if the algebra of bi-K-invariant functions is commutative under convolution. In this paper, we study spherical functions on groupoids associated to compact Gelfand pairs.

A groupoid is called power-commutative if every mono-generated subgroupoid is commutative. The class Pc of power-commutative groupoids is a variety. A description of free objects in this variety and their characterization by means of injective groupoids in Pc are obtained.

In this article, we study dynamical and $C^{*}$ -algebraic properties of self-similar group actions on finite directed graphs. We investigate the structure of limit dynamical systems induced from group actions on graphs, and we deduce conditions of group actions and graphs for the groupoid $C^{*}$ -algebras defined by limit dynamical systems to be simple, separable, purely infinite, nuclear, and satisfying the universal coefficient theorem.

We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the cotangent functor to the structure maps of a Lie groupoid. We show that such objects are in one-to-one correspondence with symplectic double groupoids, generalizing a result of Zakrzewski concerning symplectic double groups and Hopf algebra objects in the aforementioned category. The proof relies on the fact that one can realize the core of a symplectic double groupoid as a symplectic quotient of the total space. The resulting constructions apply more generally to give a correspondence between double Lie groupoids and groupoid-like objects in the category of smooth manifolds and smooth relations, and we show that the cotangent functor relates the two constructions., 30 pages, to appear in Journal of Geometric Mechanics. arXiv admin note: text overlap with arXiv:1309.1362

The notion of recognition of a language by a finite semigroup can be generalized to recognition by finite groupoids, i.e. sets equipped with a binary operation '?' which is not necessarily associative. It is well known that L can be recognized by a groupoid iff L is context-free. However it is also known that some subclasses of groupoids can only recognize regular languages.A groupoid H is said to be conservative if a ? b ? { a , b } for all a , b ? H . The first result of this paper is that conservative groupoids can only recognize regular languages. This class of groupoids is incomparable with the ones identified so far which share this property, so we are exhibiting a new way in which a groupoid can be too weak to recognize non-regular languages.We also study the class L cons of regular languages that can be recognized in this way and explain how it fits within the well-known Straubing-Therien hierarchy. In particular we show that L cons contains depth 1/2 of the hierarchy and is entirely contained in depth 3/2.

We decompose the full and reducedC∗C^*-algebras of an extension of a groupoid by the circle into a direct sum of twisted groupoidC∗C^*-algebras.

We prove that the C*-algebra of a second-countable, \'etale, amenable groupoid is simple if and only if the groupoid is topologically principal and minimal. We also show that if G has totally disconnected unit space, then the associated complex *-algebra introduced by Steinberg is simple if and only if the interior of the isotropy subgroupoid of G is equal to the unit space and G is minimal., Comment: The introduction has been updated and minor changes have been made throughout. To appear in Semigroup Forum

In this paper we define a monoid called the Brauer semigroup for a locally compact Hausdorff groupoid E E whose elements consist of Morita equivalence classes of E E -dynamical systems. This construction generalizes both the equivariant Brauer semigroup for transformation groups and the Brauer group for a groupoid. We show that groupoid equivalence induces an isomorphism of Brauer semigroups and that this isomorphism preserves the Morita equivalence classes of the respective crossed products, thus generalizing Raeburn’s symmetric imprimitivity theorem.

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid C*-algebras. We show that the C*-algebra is generated by an inverse semigroup T of partial isometries. We show that the groupoid 𝓖tight associated with the inverse semigroup T by Exel is exactly the same as the groupoid considered by Sheu.

Iakovos Androulidakis and Georges Skandalis have defined a holonomy groupoid for any singular foliation. This groupoid, whose topology is usually quite bad, is the starting point for the study of longitudinal pseudodifferential calculus on such foliation and its associated index theory. These studies can be highly simplified under the assumption of the holonomy groupoid being longitudinally smooth. In this note, we rephrase the period bounding lemma that asserts that a vector field on a compact manifold admits a strictly positive lower bound for its periodic orbits in order to prove that the holonomy groupoid is always longitudinally smooth.

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of O_X. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex. The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties., 29 pages. This material was part of arXiv:0905.0488v3, now made into a separate paper with improved presentation

Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

In this paper, we investigate the topologically weak concepts of topological groupoids by giving the concepts of α -topological groupoid and α -topological subgroupoid. Furthermore, we show the role of the density condition to allow α -topological subgroupoid inherited properties from α -topological groupoid and the irresoluteness property for the structure maps in α -topological groupoid is studied. We also give some results about the fibers of α -topological groupoids.

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks faithful representations., Comment: 20 pages

In this paper we prove that if S is a commutative semigroup acting on an ordered groupoid G, then there exists a commutative semigroup S̃ acting on the ordered groupoid G̃:=(G × S)/ρ̄ in such a way that G is embedded in G̃. Moreover, we prove that if a commutative semigroup S acts on an ordered groupoid G, and a commutative semigroup S̄ acts on an ordered groupoid Ḡ in such a way that G is embedded in S̄, then the ordered groupoid G̃ can be also embedded in Ḡ. We denote by ρ̄ the equivalence relation on G × S which is the intersection of the quasi-order ρ (on G × S) and its inverse ρ −1.

The monoids of powers in varieties of f-idempotent groupoids and commutative f-idempotent groupoids, where f is an irreducible groupoid power with a length at least 3, are constructed. It is shown that these monoids are free over a countable set of irreducible groupoid powers. Mathematics Subject Classification: 08B20; 03C05

In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural ℝ + × {\mathbb{R}^{\times}_{+}} -action. Specifically, a properly supported semiregular distribution on M × M {M\times M} is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the ℝ + × {\mathbb{R}^{\times}_{+}} -action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

Ideals of implication groupoids are considered. Given a subset of a distributive implication groupoid, the smallest ideal containing it is constructed. A characterization of ideals in distributive implication groupoid using upper sets is given.

For a locally compact higher rank graph ¤, we construct a two-sided path space ¤c with shift homeomorphism ¾ and its corresponding path groupoid i. Then we -nd equivalent conditions of aperiodicity, co-nality and irreducibility of ¤ in (¤c; ¾), i, and the groupoid algebra C¤(i).

We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G\rr P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$. As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.

We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlister's P-theorem for incompressible ordered groupoids.

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid., Comment: 23 pages, a few minor changes, including a correction to Lemma 6.6

Let G be a second-countable locally-compact Hausdorff groupoid with a Haar system, and let {x_n} be a sequence in the unit space of G. We show that the notions of strength of convergence of {x_n} in the orbit space and measure-theoretic accumulation along the orbits are equivalent ways of realising multiplicity numbers associated to a sequence of induced representation of the groupoid C*-algebra., Comment: 34 pages

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of etale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.