# Partial actions and proper extensions of two-sided restriction semigroups

## TL;DR

This paper characterizes proper extensions of two-sided restriction semigroups using partial actions, extending previous results for monoids and inverse semigroups, and establishes categorical equivalences between these structures.

## Contribution

It introduces new classes of partial actions for two-sided restriction semigroups and proves an adjunction linking proper extensions to these partial actions.

## Key findings

- Established an adjunction between proper extensions and partial actions.
- Defined and studied classes of partial actions generalizing previous concepts.
- Proved categorical equivalences and subcategory structures for these actions.

## Abstract

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category ${\mathcal{P}}(S)$ of proper extensions of a restriction semigroup (or, in particular, an inverse semigroup) $S$ and a category ${\mathcal{A}}(S)$ of partial actions of $S$ subject to certain conditions going back to the work of O'Carroll. In the category ${\mathcal{A}}(S)$, we specify two isomorphic subcategories, one being reflective and the other one coreflective, each of which is equivalent to the category ${\mathcal{P}}(S)$.