Twisted partial actions and extensions of semilattices of groups by groups

TL;DR

This paper introduces a framework for understanding extensions of semilattices of groups by groups through twisted partial actions, establishing a correspondence with Sieben twisted modules over inverse semigroups.

Contribution

It defines extensions of semilattices of groups by groups and characterizes them via crossed products and twisted partial actions, linking to inverse semigroup modules.

Findings

Classified all such extensions as crossed products by twisted partial actions.

Established a one-to-one correspondence with Sieben twisted modules.

Provided a structural understanding of extensions in this algebraic context.

Abstract

We introduce the concept of an extension of a semilattice of groups $A$ by a group $G$ and describe all the extensions of this type which are equivalent to the crossed products $A*_\Theta G$ by twisted partial actions $\Theta$ of $G$ on $A$. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and so-called Sieben twisted modules over $E$-unitary inverse semigroups.