On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

Abstract 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 A S ( G E ) of the graph groupoid G E associated to an arbitrary graph E to be congruence-simple. Motivated by a result of Clark and Sims, we show that the natural homomorphism from the Leavitt path algebra L B ( E ) to the Steinberg algebra A B ( G E ) , where B is the Boolean semifield, is an isomorphism if and only if E is row-finite. Moreover, we establish the Reduction Theorem and Uniqueness Theorems for Leavitt path algebras of row-finite graphs over the Boolean semifield B .