Congruence-simplicity of Steinberg algebras of non-Hausdorff ample groupoids over semifields

We investigate the algebra of an ample groupoid, introduced by Steinberg, over a semifield S. In particular, we obtain a complete characterization of congruence-simpleness for Steinberg algebras of second-countable ample groupoids, extending the well-known characterizations when S is a field. We apply our congruence-simplicity results to tight groupoids of inverse semigroup representations and then specialize to groups acting on graphs.