Theory of Boolean algebras with a locally finite group of automorphisms

Abstract

In this note we consider the class of model K(G)= { ,] B is a Boolean algebra in the signature , ~ is an automorphism of B from the group G}. It has been shown in [i] that if G is a finite solvable group, then Th(K(G)) is decidable. It has been independently proved in [2, 3] that if G is an arbitrary finite group, then Th(K(G)) is decidable. It has been proved in [3, 4] that if G is not a locally finite group, then Th(K(G)) is undecidable. In [3] a conjecture is proposed that if G is a locally finite recursive group, then Th(K(G)) is decidable. In the present article a family of locally finite recursive groups F is constructed such that Th(K(G)) is hereditarily undecidable for all G.~ F. The proof of undecidability relies on [5].