Atomless \(r\)-maximal sets (Q1961349)

Let \(A\) be a computably enumerable set, and \(L(A)\) be the lattice of its supersets under the inclusion relation. It was proved earlier that for many sets \(A\), the filters \(L(A)\) are isomorphic. For example, for all low \(A\), by a result of R. Soare, \(L(A)\) are isomorphic to the lattice of all computably enumerable degrees, and, therefore, are isomorphic to each other. The last can also be said about the lattices \(L(A)\) for maximal sets \(A\). The authors decide a longstanding problem and show that there exist infinitely many pairwise incomparable sets \(L(A)\).