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

TL;DR

This paper characterizes when Steinberg algebras of second-countable ample groupoids over semifields are congruence-simple, extending known results from field cases and applying to inverse semigroup representations of self-similar graphs.

Contribution

It provides a complete characterization of congruence-simpleness for Steinberg algebras over semifields, generalizing previous field-based results.

Findings

Characterization of congruence-simpleness for second-countable ample groupoids

Extension of known results from fields to semifields

Application to inverse semigroup representations of self-similar graphs

Abstract

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 associated to self-similar graphs.