The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

Abstract

Computable categoricity in the Ershov hierarchy is studied. We consider F a -categorical 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.Research on computable categoricity of structures was initiated by B. L. van der Waerden who studied the question whether there exists an effective algorithm for constructing an isomorphism between two different algebraic closures of a field built in different ways. Subsequently, starting with 1950s, computable categoricity of structures was explored by A. I. Mal'tsev, A. Fröhlich, J. Shepherdson, S. S. Goncharov, J. Remmel, C. Ash, J. Knight, and many other authors.In investigating computably categorical structures, one of the most important problems is to gain an insight into computable categoricity relative to classes of known hierarchies of sets. In particular, of special interest is the Ershov hierarchy introduced in [1-3]. There are quite many works dealing with the Ershov hierarchy. (Among these, it is worth mentioning resent papers [4-10] that focus primarily on numberings in the Ershov hierarchy.) Research on computable categoricity relative to levels of the Ershov hierarchy was started in [11]. There, for an arbitrary a, which is *