In ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In an annex of [E6], Lesieur’s axioms have been simplified. In this article, we suppose that the basis is central; in that case, we prove that a specific sub-C∗ algebra is, in a sense, invariant under all the data which define the measured quantum group, which allow us to prov...

We analyse extensions $\Sigma$ of groupoids G by bundles A abelian groups. describe a pushout construction for such extensions, and use it to the extension group given groupoid bundle A. There is natural action Sigma on dual A, yielding corresponding transformation groupoid. The this map from fibre product with its Cartesian circle twist over resulting prove that full C*-algebra isomorphic $\Si...

In this work we prove, in a self contained way, that any finite dimensional C∗-quantum groupoid can be deformed in order that the square of the antipode is the identity on the base. We also prove that for any C∗-quantum groupoid with non abelian base, there is uncountably many C∗-quantum groupoids with the same underlying algebra structure but which are not isomorphic to it. In fact, the C∗quan...

Heckenberger introduced the Weyl groupoid of a finite-dimensional Nichols algebra diagonal type. We replace matrix its braiding by higher tensor and present construction which yields further groupoids. Abelian cohomology theory gives evidence for existence associated to such tensor.

We consider circuits and expressions whose gates carry out multiplication in a non-associative groupoid such as loop. We deene a class we call the polyabelian groupoids, formed by iterated quasidirect products of Abelian groups. We show that a loop can express arbitrary Boolean functions if and only if it is not polyabelian, in which case its Expression Evaluation and Circuit Value problems are...

For a finitely presented discrete group $\Gamma$, we introduce two generalizations of the orbifold Euler characteristic and $\Gamma$-orbifold to class proper topological groupoids large enough include all cocompact Lie groupoids. The $\Gamma$-Euler is defined as an integral with respect over orbit space groupoid, $\Gamma$-inertia usual associated groupoid. A key ingredient application o-minimal...

The category of modules over a fixed quasigroup in the category of all quasigroups is equivalent to the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. The corresponding equivalent category of coverings, and the generalization to the right quasigroup case, are also described.

We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties structures discussed by their decomposition theorems: (1) L1/R1-transposition equivalent to each other, they can be decomposed the union disjoint Abelian subgroups; (2) LR-transposition an example is given illustrate that ...

We present definitions of homology groups Hn(p), n ≥ 0, associated to a complete type p. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H2(p) for strong types in stable theories and show that any profinite abelian group can occur as the group H2(p). The work described in this paper was originally inspired by Hrushovski’s ...