Universal Boolean algebras, their ideals, and applications to semantic classes of models
TL;DR
The existence and uniqueness of universal Boolean algebras are shown and the classes in which such algebras exist are determined, and the universal algebras up to isomorphism are classified.
Abstract
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.