Globalization of Partial Actions of Ordered Groupoids on Rings
Wesley G. Lautenschlaeger, Thaísa Tamusiunas
TL;DR
The paper addresses when a partial ordered action of an ordered groupoid on a ring can be globalized in an ordered sense and when such globalization is unique. It proves that globalization exists precisely for unital actions and develops a minimal globalization framework for strong, preunital actions on pseudoassociative groupoids, with uniqueness up to equivalence. The results yield a Morita context between partial and global crossed products and extend to inverse semigroup actions via the Ehresmann-Schein-Nambooripad correspondence, establishing globalization criteria and Morita equivalences under finiteness assumptions. Overall, the work unifies partial-to-global transitions in the ordered setting, linking globalization, Morita theory, and ESN-based inverse semigroup actions in noncommutative ring theory.
Abstract
We provide a necessary and sufficient condition to the existence of an ordered globalization of a partial ordered action of an ordered groupoid on a ring and we also present criteria to obtain uniqueness. Furthermore, we apply those results to obtain a Morita context and to show that an inverse semigroup partial action has a globalization (unique up to isomorphism) if, and only if, it is unital.