Globalization of partial actions of semigroups

TL;DR

This paper introduces two universal methods for extending partial semigroup actions to global actions, generalizing existing constructions and revealing differences between monoid and group cases.

Contribution

It presents two new universal constructions for globalization of partial semigroup actions, one generalizing previous work and the other involving Hom-sets, with distinct outcomes in monoid versus group cases.

Findings

One construction generalizes strong partial action globalization.

The other involves Hom-sets and is novel in monoid context.

In groups, both constructions yield isomorphic globalizations.

Abstract

We propose two universal constructions of globalization of a partial action of a semigroup on a set, satisfying certain conditions which arise in Morita theory of semigroups. One of the constructions is based on the tensor product of a partial semigroup act with the semigroup and generalizes the globalization construction of strong partial actions of monoids due to Megrelishvili and Schr\"oder. It produces the initial object in an appropriate caterory of globalizations of a given partial action. The other construction involves ${\mathrm{Hom}}$-sets and is novel even in the monoid setting. It produces the terminal object in an appropriate category of globalizations. While in the group case the results of the two constructions are isomorphic, they can be far different in the monoid case.