Effective properties of Marker's extensions

The paper is devoted to the study of Marker’s extensions of sequences of countable structures. In the first part of the paper definability properties of the Marker’s extension are investigated. The results demonstrate that for any sequence of structures the Marker’s extension codes the elements of the sequence so that the n-th structure of the sequence appears positively at the n-th level of the definability hierarchy. In the second part we study the spectra of the Marker’s extensions. The results provide a general method given a sequence of structures, to construct a structure with n-th jump spectrum contained in the spectrum of the n-th member of the sequence. As an application a structure with spectrum consisting of the Turing degrees which are non-lown for all n < ω is obtained.