Crossed Products by Partial Actions of Inverse Semigroups

TL;DR

This paper introduces a new framework for crossed products by partial actions of inverse semigroups, establishing a correspondence between covariant representations and connecting different types of crossed products.

Contribution

It defines the crossed product for inverse semigroups by partial actions and links covariant representations of inverse semigroups with those of associated inverse semigroups.

Findings

Established a correspondence between covariant representations of G and S_G.

Connected crossed products of inverse semigroup actions with partial actions.

Provided a new construction for crossed products in the context of inverse semigroups.

Abstract

In this work, for a given inverse semigroup we will define the crossed product of an inverse semigroup by a partial action. Also, we will associate to an inverse semigroup $G$ an inverse semigroup $S_G$, and we will prove that there is a correspondence between the covariant representation of $G$ and covariant representation of $S_G$. Finally, we will explore a connection between crossed products of an inverse semigroup actions and crossed products by partial actions of inverse semigroups.