An elementary Green imprimitivity theorem for inverse semigroups

TL;DR

This paper extends Green's imprimitivity theorem to inverse semigroups, establishing Morita equivalence between certain crossed products, thus broadening the understanding of algebraic structures associated with inverse semigroups.

Contribution

It introduces a Morita equivalence for crossed products by inverse semigroups, generalizing Green's theorem from group actions to inverse semigroup actions.

Findings

Morita equivalence between $A times H$ and $B times G$ for inverse semigroup actions

Extension of Green's imprimitivity theorem to inverse semigroups

Framework applicable to finite sub-inverse semigroups

Abstract

A Morita equivalence similar to that found by Green for crossed products by groups will be established for crossed products by inverse semigroups. More precisely, let $G$ be an inverse semigroup, $H$ a finite sub-inverse semigroup of $G$ and $A$ a $G$-algebra or a $H$-algebra. Then the crossed product $A \rtimes H$ is Morita equivalent to a certain crossed product $B \rtimes G$.