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

Abstract

<jats:p> We explore numbered Boolean algebras over classes <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">Ξ</mml:mi> </mml:math> </jats:inline-formula> of arithmetical and analytical hierarchies. We show the existence and uniqueness (up to computable isomorphism) of universal Boolean <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">Ξ</mml:mi> </mml:math> </jats:inline-formula> -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 <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ω</mml:mi> </mml:math> </jats:inline-formula> -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 <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> , <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> , and <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> , respectively, and the fourth one is a countable atomic Boolean <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> -algebra whose quotient modulo the Frèchet ideal is a <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>4</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> -universal Boolean <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msubsup> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mn>4</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:math> </jats:inline-formula> -algebra. </jats:p>