Contrasting the Halves of an Ahmad Pair

TL;DR

This work advances the understanding of the local structure of $Σ^0_2$ enumeration degrees by providing a complete characterization of the left halves of Ahmad pairs as precisely the $low_3$ and join-irreducible degrees, while showing that the right halves must be $high_2$ and are thus fundamentally different. It extends the framework to Ahmad $n$-pairs, introducing $n$-join irreducibility and establishing a proper hierarchy where left halves can be left of Ahmad $(n+1)$-pairs but not of Ahmad $n$-pairs. The paper then derives definability results (including a $Π_3$ definition of $low_3$ and $high_2$) and significant consequences for the $orallorall ext{-} orall ext{-} ext{AE}$-theory, improving decidability statements about embeddings in the local structure. Overall, the results separate the roles of left and right halves in a robust, hierarchically organized way and clarify the impact of Ahmad pairs on extension problems for embeddings in $ ext{Σ}^0_2$ e-degrees.

Abstract

We study Ahmad pairs in the $Σ^0_2$ 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 $Σ^0_2$ e-degrees as a partial order and also provide a new $Π_3$ definition of $\lowww$ as well as $\highh$.

Theorems & Definitions (88)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Lemma 2.2