Contrasting the Halves of an Ahmad Pair

Abstract

We study Ahmad pairs in the ${\Sigma }_2^0$ enumeration degrees. $(A,B)$ is an Ahmad pair if $A/{\le }_{e}B$ and every $Z{<}_{e}A$ satisfies $Z{\le }_{e}B$. We characterize the degrees that are the left halves of an Ahmad pair as those that are $\lowww$ and join irreducible. We then show that the right half has to be $\highh$ giving a natural separation between the two halves which is a significant strengthening of previous work. We define a hierarchy of join irreducibility notions using which we characterize the left halves of Ahmad $n$-pairs as those that are $\lowww$ and $n$-join irreducible, while the right halves are $\highh$. This allows us to extend and clarify previous work to show that for any $n$, there is a set $A$ which is the left half of an Ahmad $n$-pair but not of an Ahmad $(n+1)$-pair. These results have new implications about the $\forall \exists$-theory of the ${\Sigma }_2^0$ e-degrees as a partial order and also provide a new ${\Pi }_3$ definition of $\lowww$ as well as $\highh$.

26 pages