Universal Boolean algebras, their ideals, and applications to semantic classes of models

We explore numbered Boolean algebras over classes Ξ of arithmetical and analytical hierarchies. We show the existence and uniqueness (up to computable isomorphism) of universal Boolean Ξ -algebras, determine the classes in which such algebras exist, and classify the universal algebras up to isomorphism. As applications, we characterize Tarski–Lindenbaum algebras of four semantic classes of models of a given finite rich signature, namely, the class of all countable saturated models having decidable ω -stable theories, the class of prime models with decidable theories and first-order definable elements, the class of models with decidable non-finitely axiomatizable theories, and the class of models with finitely axiomatizable theories; it is shown that the first three of these algebras are universal Boolean algebras over hierarchy classes Σ 1 1 , Σ 2 0 , and Σ 3 0 , respectively, and the fourth one is a countable atomic Boolean Π 3 0 -algebra whose quotient modulo the Frèchet ideal is a Σ 4 0 -universal Boolean Σ 4 0 -algebra.