Defining totality in the enumeration degrees
Mingzhong Cai, Hristo Ganchev, Steffen Lempp, Joseph S. Miller, Mariya I. Soskova · Journal of the American Mathematical Society · 2015
We show that if A A and B B form a nontrivial K \mathcal {K} -pair, then there is a semi-computable set C C such that A ≤ e C A\leq _e C and B ≤ e C ¯ B\leq _e \overline {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 \mathcal {K} -pair. This answers a long-standing question of Hartley Rogers, Jr. We also obtain a definition of the “c.e. in” relation for total degrees in the enumeration degrees.