Contrasting the Halves of an Ahmad Pair

TL;DR

This paper characterizes the degrees forming halves of Ahmad pairs within the $ olinebreak oldsymbol{ ext{Sigma}}^0_2$ enumeration degrees, revealing a hierarchy of join irreducibility and separating the properties of the pair's halves.

Contribution

It introduces a hierarchy of join irreducibility notions to characterize the halves of Ahmad pairs and extends previous work to distinguish between different levels of Ahmad n-pairs.

Findings

Left halves are $ ext{low}_2$ and join-irreducible.

Right halves are $ ext{high}_2$, establishing a separation.

Existence of sets that are left halves of Ahmad n-pairs but not of Ahmad (n+1)-pairs.

Abstract

We study Ahmad pairs in the $\Sigma^0_2$ enumeration degrees. $(A,B)$ is an Ahmad pair if $A \not \leq_e B$ and every $Z <_e A$ satisfies $Z \leq_e B$. 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 $\Sigma^0_2$ e-degrees as a partial order and also provide a new $\Pi_3$ definition of $\lowww$ as well as $\highh$.