Low$_2$ computably enumerable sets have hyperhypersimple supersets
Peter Cholak, Rodney Downey, Noam Greenberg
TL;DR
The paper addresses the problem of characterizing the lattice ${\mathcal{L}}^{*}(A)$ of c.e. supersets modulo finite for coinfinite $A$, focusing on the low$_2$ boundary. It proves that every coinfinite $\mathrm{low}_2$ c.e. set has an atomless hyperhypersimple superset, and moreover that for any $\Sigma_3$-Boolean algebra $B$ there exists a c.e. $H\supseteq A$ with ${\mathcal L}^{*}(H)\cong B$. The approach deploys $\Delta_3$-level guessing on an $\omega$-branching priority tree manifested as a pinball machine, and introduces a novel splitting method to achieve atomless hyperhypersimple supersets, complemented by a robust certification mechanism using $\emptyset'$-dominating functions. The results advance the low$_2$ conjecture by providing an explicit, flexible construction that realizes arbitrary $\Sigma_3$-Boolean algebras within supersets, thereby linking lattice-theoretic properties of c.e. sets with the structure of their Turing degrees. This work sharpens our understanding of how low$_2$ degrees constrain lattice behavior and lays groundwork for a broader isomorphism program between ${\mathcal{L}}^{*}(A)$ and $\mathcal{E}^{*}$ in the low$_2$ regime.
Abstract
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 $\mathcal{L}^*(H)\cong B$.