Low${}_2$ computably enumerable sets have hyperhypersimple supersets

Abstract

A longstanding question is to characterize the lattice of supersets (modulo finite sets), ${\mathcal{L}}^{\ast }(A)$, of a low${}_2$ computably enumerable (c.e.) set. The conjecture is that ${\mathcal{L}}^{\ast }(A)\cong {\mathcal{E}}^{\ast }$. 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 $\hat{I}{\pounds }_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that ${\mathcal{L}}^{\ast }(H)\cong B$.

Revised due to referee's comments. Thanks!