LOW $_2$ COMPUTABLY ENUMERABLE SETS HAVE HYPERHYPERSIMPLE SUPERSETS

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 $\mathcal {L}^*(H)\cong B$ .