DEFINING TOTALITY IN THE ENUMERATION DEGREES

Abstract. We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. We also obtain a definition of the “c.e. in ” relation of total degrees in the enumeration degrees. The total degrees are the image of the Turing degrees under their natural embed-ding into the enumeration degrees. The question of whether the total degrees are definable in the enumeration degrees as a partial order is almost old as the enumer-ation degrees themselves. There have been a few partial solutions to this problem. Kalimullin [10] gave a definition of the enumeration jump operator. Combining this with Friedberg jump-inversion [5] gives us a definition of the total degrees above 0′e: they are the range of the jump operator. Ganchev and Soskova [7] defined the total degrees in the local structure (i.e., the degrees below 0′e). Putting these together, the total degrees comparable to 0′e are definable. Finally, Soskova [19] gave a def