The first order theory of Boolean algebras with a distinguished group of automorphisms

Abstract

We will let B6 denote such an algebra, and B the reduct to the Boolean operations. Thus Be is a Boolean algebra B with a group G of automorphisms acting on it. The only result on the theory of ~sg(G) that we know of is due to Wolf ([7], 1975): I f G is a finite solvable group then ~sg(G) has a decidable theory. His proof is based on Arens and Kaplansky's theorem that a finite solvable group of homeomorphisms acting on the Boolean space of a countable Boolean algebra has a fundamental domain. We take another approach, introducing monadic algebras into the study, and arrive at a more general result.