Minimality and effectiveness of the groupoid associated to a self-similar ultragraph

The notion of a self-similar ultragraph $(G,\mathcal{U},\phi )$ and its $C^{\ast }$-algebra ${\mathcal{O}}_{G,\mathcal{U}}$ were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such $C^{\ast }$-algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph $(G,\mathcal{U},\phi )$. In particular, we obtain a result for simplicity of the $C^{\ast }$-algebras ${\mathcal{O}}_{G,\mathcal{U}}$ in a certain case.