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 algebra is an isomorphism if and only if the associated graph is row-finite. Moreover, we establish the Reduction Theorem and Uniqueness Theorems for Leavitt path algebras of row-finite graphs over the Boolean semifield.

中文翻译：

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关于交换半环上 Hausdorff 充足群群的 Steinberg 代数

我们研究了 Steinberg 在交换半环 S 上引入的 Hausdorff 充足群群的代数。特别是，我们获得了此类 Steinberg 代数的同余-简单性的完整表征，当 S 是一个域或一个域时，我们扩展了众所周知的表征交换环。我们还提供了一个标准，使与任意图相关联的图群群的 Steinberg 代数是同余简单的。受 Clark 和 Sims 的启发，我们证明，在布尔半域上，从 Leavitt 路径代数到 Steinberg 代数的自然同态是同构当且仅当关联图是行有限的。此外，我们建立了布尔半域上行有限图的Leavitt路径代数的约简定理和唯一性定理。