T-degrees, jump classes, and strong reducibilities

Abstract

It is shown that there exist r.e. degrees other than 0 and 0 ′ \mathbf {0}’ which have a greatest r.e. 1 1 -degree. This solves an old question of Rogers and Jockusch. We call such degrees 1 1 -topped. We show that there exist incomplete 1 1 -topped degrees above any low r.e. degree, but also show that no nonzero low degree is 1 1 -topped. It then follows by known results that all incomplete 1 1 -topped degrees are low 2 _{2} but not low. We also construct cappable nonzero 1 1 -topped r.e. degrees and examine the relationships between 1 1 -topped r.e. degrees and high r.e. degrees. Finally, we give an analysis of the “local” relationships of r.e. sets under various strong reducibilities. In particular, we analyze the structure of r.e. wtt- {\text {wtt-}} and tt {\text {tt}} -degrees within a single r.e. T {\text {T}} -degree. We show, for instance, that there is an r.e. degree which contains a greatest r.e. wtt- {\text {wtt-}} -degree and a least r.e. tt {\text {tt}} -degree yet does not consist of a single r.e. wtt {\text {wtt}} -degree. This depends on a new construction of a nonzero r.e. T {\text {T}} -degree with a least tt {\text {tt}} -degree, which proves to have several further applications.