In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ([Formula: see text]-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever [Formula: see text] is a ∧-preaction of a discrete inverse semigroupoid [Formula: see text] on an ample (possibly non-Hausdorff) groupoid [Formula: see text], the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of [Formula: see text] by [Formula: see text]. This is a far-reaching generalization of analogous results which had been proven in particular cases.

中文翻译：

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半群丛的截面代数

在本文中，我们使用半群来描述代数丛的概念，主要由 Fell（[公式：见文本]-代数）丛以及与它们相关的截面代数推动。作为主要的激励例子，Steinberg 代数可以被视为平凡（直积）丛的截面代数。一些与几何和代数构造相关的定理——通过截面代数的构造——被广泛推广： 半群的直积束对应于代数的张量积；丛的半直积对应于代数的“朴素”叉积；分级丛的斜积对应分级代数的粉碎积；商丛对应于商代数。此外，大多数结果都适用于非 Hausdorff 设置。在这项工作的过程中，我们将粉碎积的定义推广到群梯度代数。作为一个应用程序，我们证明只要 [Formula: see text] 是离散逆半群 [Formula: see text] 在一个宽泛（可能是非 Hausdorff）群 [Formula: see text] 上的 ∧-preaction，Steinberg 代数相关的细菌群的 与 [公式：参见文本] 的斯坦伯格代数的叉积自然同构。这是对已在特定情况下得到证明的类似结果的深远概括。相关细菌群的 Steinberg 代数自然同构于 [Formula: see text] 的 Steinberg 代数 by [Formula: see text] 的叉积。这是对已在特定情况下得到证明的类似结果的深远概括。相关细菌群的 Steinberg 代数自然同构于 [Formula: see text] 的 Steinberg 代数 by [Formula: see text] 的叉积。这是对已在特定情况下得到证明的类似结果的深远概括。