Contrasting the Halves of an Ahmad Pair
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908679897350144 |
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| author | Ravishankar, Karthik |
| author_facet | Ravishankar, Karthik |
| contents | 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 $\highh$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22901 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Contrasting the Halves of an Ahmad Pair Ravishankar, Karthik Logic 03D30 Primary, 03D35 Secondary 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 $\highh$. |
| title | Contrasting the Halves of an Ahmad Pair |
| topic | Logic 03D30 Primary, 03D35 Secondary |
| url | https://arxiv.org/abs/2511.22901 |