  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.
