Isomorphy Types of Expanded Superatomic Ershov Algebras

Countable superatomic Ershov algebras with distinguished ideal are studied. A classification is given of isomorphy types of countable superatomic Ershov algebras with distinguished ideal and ordinal type at most $\omega$. It is proven that, for a sufficiently wide class of elementary theories $T$, there exist continuum many nonisomorphic countable Ershov algebras with distinguished ideal and countable ordinal type of a given theory $T$.