Galois correspondence for group-type partial actions of groupoids

TL;DR

This paper establishes a Galois correspondence linking subgroupoids of a finite groupoid to subrings of a commutative ring under a partial group-type action, extending known results and providing illustrative examples.

Contribution

It introduces a Galois correspondence for partial group-type actions of finite groupoids on rings, generalizing existing global action results.

Findings

Galois correspondence between subgroupoids and subrings

Extension of known global groupoid action results

Multiple examples illustrating the correspondence

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.