On decomposition of commutative Moufang groupoids

On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the well-known characterizations when S is a field or a commutative ring. We also provide a criterion for the Steinberg algebra of the graph groupoid associated to an arbitrary graph to be congruence-simple. Motivated by a result of Clark and Sims, we show that, over the Boolean semifield, the natural homomorphism from the Leavitt path algebra to the Steinberg algebr…

Noncrossed Product Matrix Subrings and Ideals of Graded Rings

We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced …

部分群胚作用的伽罗瓦对应

Galois Correspondence for Partial Groupoid Actions

本文证明了群胚在交换环上正交部分作用的伽罗瓦对应定理，并将结果推广到非正交情形，分别建立了强伽罗瓦部分群胚作用与全局群胚作用的两种对应关系。

Chain conditions on étale groupoid algebras with applications to Leavitt path algebras and inverse semigroup algebras

The author has previously associated to each commutative ring with unit $R$ and étale groupoid $\mathscr G$ with locally compact, Hausdorff and totally disconnected unit space an $R$-algebra $R\mathscr G$. In this paper we characterize when $R\mathscr G$ is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda~Pino and Siles~Molina of finite dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelma…

Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed subspaces of their unit spaces are all effective. For such a groupoid, we completely describe the ideal lattice of the associated Steinberg algebra over any commutative ring with identity. Our results are new even for the special case of Leavitt path algebras; so we describe explicitly what they say in this context, and give two concrete examples.

艾达尔群胚代数的链条件

Chain Conditions for Etale Groupoid Algebras

本文针对艾达尔群胚代数，刻画了其范畴诺特/阿廷、局部诺特/阿廷及半单性质，推广了Leavitt路代数的相关链条件研究结果。

Sheffer operation in relational systems

The concept of a Sheffer operation known for Boolean algebras and orthomodular lattices is extended to arbitrary directed relational systems with involution. It is proved that to every such relational system there can be assigned a Sheffer groupoid and also, conversely, every Sheffer groupoid induces a directed relational system with involution. Hence, investigations of these relational systems can be transformed to the treaty of special groupoids which form a variety of algebras. If the Sheffer operation is also commutative then the induced binary relation is antisymmetric. Moreover, commutat…

Representative functions on discrete groupoids and duality with Hopf algebroids

The aim of this paper is to establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids in the sense of P. Deligne and A. Bruguières. In one direction we have the usual contravariant functor which assigns to each Hopf algebroid its characters groupoid (the fiber groupoid at the ground field). In the other direction we construct the contravariant functor which associated to each discrete groupoid its Hopf algebroid of representative functions. This duality extends the well known duality between discrete groups and commu…

Ideals of etale groupoid algebras and Exel's Effros-Hahn conjecture

We extend to arbitrary commutative base rings a recent result of Demeneghi that every ideal of an ample groupoid algebra over a field is an intersection of kernels of induced representations from isotropy groups, with a much shorter proof, by using the author's Disintegration Theorem for groupoid representations. We also prove that every primitive ideal is the kernel of an induced representation from an isotropy group; however, we are unable to show, in general, that it is the kernel of an irreducible induced representation. If each isotropy group is finite (e.g., if the groupoid is principal)…