On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over commutative semiring S. In particular, we obtain complete characterization congruence-simpleness for such Steinberg algebras, extending well-known characterizations when S is field or ring. also provide criterion AS(GE) graph groupoid GE associated to an arbitrary E be congruence-simple. Motivated result Clark and Sims, show that natural homomorphism from Leavitt path LB(E) AB(GE), where B Boolean semifield, isomorphism if only row-finite. Moreover, establish Reduction Theorem Uniqueness Theorems algebras row-finite graphs semifield B.