Jump inversions of algebraic structures and Σ-definability

It is proved that for every countable structure A and a computable successor ordinal α there is a countable structure A−α which is (Formula presented.) -least among all countable structures C such that A is Σ-definable in the αth jump C(α). We also show that this result does not hold for the limit ordinal α = ω. Moreover, we prove that there is no countable structure A with the degree spectrum (Formula presented.) for (Formula presented.).