Reflectors and globalizations of partial actions of groups

TL;DR

This paper investigates how partial group actions can be extended to global actions through reflectors, focusing on algebraic structures and conditions for globalization, with applications to semigroups and ideals.

Contribution

It introduces a method to construct reflectors of partial actions in the global category and characterizes when these are globalizations, especially for algebraic structures.

Findings

Reduction of globalization problem to embeddability of generalized amalgams

Characterization of globalizable partial actions on semigroups with ideals

Construction of reflectors for partial actions in algebraic categories

Abstract

Given a partial action $\theta$ of a group on a set with an algebraic structure, we construct a reflector of $\theta$ in the corresponding subcategory of global actions and study the question when this reflector is a globalization. In particular, if $\theta$ is a partial action on an algebra from a variety ${\sf V}$, then we show that the problem reduces to the embeddability of certain generalized amalgam of ${\sf V}$-algebras associated with $\theta$. As an application, we describe globalizable partial actions on semigroups, whose domains are ideals.