Galois correspondence for group-type partial actions of groupoids

Abstract

Let $\operatorname{G}$ be a finite groupoid and $\alpha=(S_g,\alpha_g)_{g\in \operatorname{G}}$ a unital partial action of group-type of $\operatorname{G}$ on a commutative ring $S=\oplus_{y\in\operatorname{G}_0}S_y$. We shall prove a Galois correspondence between a class of wide subgroupoids of $\operatorname{G}$ and a class of subrings of $S$. We recover known results for global groupoid actions and we give several examples to illustrate the correspondence.