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 complete characterization congruence-simpleness for Steinberg algebras second-countable groupoids, extending well-known characterizations when S is field. apply our congruence-simplicity results to tight groupoids inverse semigroup representations associated self-similar graphs.