The branching theorem and computable categoricity in the Ershov hierarchy

Computable categoricity in the Ershov hierarchy is studied. We consider $F_a$- and $G_a$-categorical structures. These were introduced by B. Khoussainov, F. Stephan, and Y. Yang for $a$, which is a notation for a constructive ordinal. A generalization of the branching theorem is proved for $F_a$-categorical structures. As a consequence we obtain a description of $F_a$-categorical structures for classes of Boolean algebras and Abelian $p$-groups. Furthermore, it is shown that the branching theorem cannot be generalized to $G_a$-categorical structures.