Galois Theory under inverse semigroup actions
Wesley G. Lautenschlaeger, Thaísa Tamusiunas
TL;DR
This work extends classical Galois theory from finite groups to finite inverse semigroup actions on commutative rings, notably removing the orthogonality constraint that limited prior approaches. It constructs an invariant trace by passing to the quotient $G=S/\sigma$ and leveraging a partial group action, together with a Morita context that characterizes Galois extensions. For finite $E$-unitary inverse semigroups, it proves a precise Galois correspondence between $eta$-complete subsemigroups and $eta$-strong, $A^eta$-separable subalgebras via explicit maps $B o S_B$ and $T o A^{eta|_T}$, and then extends the framework to general inverse semigroups and to semigroups with zero using reductions to $E$-unitary cases and zero-compatible notions. Collectively, the results provide a robust, general Galois theory for inverse semigroup actions, with a coherent bridge between semigroup structure, invariant traces, and separable subalgebras, and they unify and extend partial-groupoid approaches to a broader non-orthogonal setting.
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.