Defining totality in the enumeration degrees

Abstract

We show that if $A$ and $B$ form a nontrivial $\mathcal {K}$-pair, then there is a semi-computable set $C$ such that $A\leq _e C$ and $B\leq _e \overline {C}$. 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 $\mathcal {K}$-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.