Contrasting the Halves of an Ahmad Pair

We study Ahmad pairs in the ${\mathrm{\Sigma }}_2^0$ enumeration degrees. $(A,B)$ is an Ahmad pair if $A\mathrm{̸}{\le }_{e}B$ 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 $\text{lowww}$ and join irreducible. We then show that the right half has to be $\text{highh}$ 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 $\text{lowww}$ and $n$-join irreducible, while the right halves are $\text{highh}$. 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 $\mathrm{\forall }\mathrm{\exists }$-theory of the ${\mathrm{\Sigma }}_2^0$ e-degrees as a partial order and also provide a new ${\mathrm{\Pi }}_3$ definition of $\text{lowww}$ as well as $\text{highh}$.