Soare’s low2 supersets lattice conjecture

Determine whether, for every low2 computably enumerable set A, the lattice L*(A) of its computably enumerable supersets modulo finite difference is isomorphic to E*, the lattice of all computably enumerable sets. This resolves the longstanding question of characterizing L*(A) for low2 c.e. sets by establishing or refuting the conjectured isomorphism L*(A) ≅ E*.

Background

The paper studies the lattice L*(A) of computably enumerable supersets of a set A modulo finite difference, and its relationship to the global lattice E* of all c.e. sets. For low c.e. sets (lower jump complexity), Soare proved that L*(A) is effectively isomorphic to E*, but for the broader class of low2 c.e. sets this remains unresolved.

Shoenfield showed that if a degree is not low2, then there exists a c.e. set B in that degree with no maximal superset, implying L*(B) is not isomorphic to E*. Thus any general positive characterization must be confined to low2 degrees. Maass further related effective isomorphisms to the semilow1.5 property, but there are low2 sets that are not semilow1.5, leaving the full low2 case unsettled.

The authors note historical claims of a proof (due to Harrington, Lachlan, Maass, and Soare) but emphasize that no proof has appeared, and they address key test cases (e.g., atomless hyperhypersimple supersets) to support the conjecture without resolving it completely.