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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.01939 |
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| _version_ | 1866909845351825408 |
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| author | Cholak, Peter Downey, Rodney Greenberg, Noam |
| author_facet | Cholak, Peter Downey, Rodney Greenberg, Noam |
| contents | A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal{L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal{L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e.\ $A$ is low$_2$ then $A$ has an atomless hyperhypersimple superset. In fact, if $A$ is c.e.\ and low$_2$, then for any $Σ_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that $\mathcal{L}^*(H)\cong B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01939 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Low$_2$ computably enumerable sets have hyperhypersimple supersets Cholak, Peter Downey, Rodney Greenberg, Noam Logic 03D25 A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal{L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal{L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e.\ $A$ is low$_2$ then $A$ has an atomless hyperhypersimple superset. In fact, if $A$ is c.e.\ and low$_2$, then for any $Σ_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that $\mathcal{L}^*(H)\cong B$. |
| title | Low$_2$ computably enumerable sets have hyperhypersimple supersets |
| topic | Logic 03D25 |
| url | https://arxiv.org/abs/2412.01939 |