Galois Theory under inverse semigroup actions

TL;DR

This paper extends Galois theory to commutative rings acted upon by finite inverse semigroups, establishing key equivalences and a correspondence theorem, including cases with zero elements.

Contribution

It introduces a novel Galois theory framework for inverse semigroup actions on rings, generalizing classical Galois theory.

Findings

Established equivalences for Galois extensions under inverse semigroup actions

Proved a Galois correspondence theorem in this context

Analyzed the behavior of the theory with inverse semigroups containing zero

Abstract

We develop a Galois theory of commutative rings under actions of finite inverse semigroups. We present equivalences for the definition of Galois extension as well as a Galois correspondence theorem. We also show how the theory behaves in the case of inverse semigroups with zero.