Studying Boolean algebras with distinguished subalgebras, we establish the existence of continuum many simple superatomic Boolean algebras with distinguished subalgebra whose elementary theories are distinct and each of them lacks countably saturated models. The subalgebras coincide with the Boolean algebras modulo their Fréchet ideals.
- Boolean algebra
- Boolean algebra with distinguished subalgebra
- elementary equivalence
- elementary theory
- finitely axiomatizable theory
- local algebra
- solvable theory