Low$_2$ computably enumerable sets have hyperhypersimple supersets

TL;DR

This paper proves that every low$_2$ computably enumerable set has a hyperhypersimple superset, advancing understanding of the lattice of supersets and addressing a key open problem in computability theory.

Contribution

It demonstrates that low$_2$ c.e. sets have hyperhypersimple supersets and can realize any $ ext{Sigma}_3$-Boolean algebra as their lattice of supersets.

Findings

Low$_2$ c.e. sets have atomless hyperhypersimple supersets.

Any $ ext{Sigma}_3$-Boolean algebra can be realized as the lattice of supersets of such a set.

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 $\Sigma_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that $\mathcal{L}^*(H)\cong B$.