Pi10 classes and strong degree spectra of relations

We study the weak truth-table and truth-table de- grees of the images of subsets of computable structures under iso- morphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear order. Countable 0 1 subsets of 2 ! and Kolmogorov complexity play a major role in the proof.