Strong Enumeration Reducibilities

Abstract

We investigate strong versions of enumeration reducibility, the most important one being s-reducibility. We prove that every countable distributive lattice is embeddable into the local structure \(L(\mathfrak D_s)\) of the s-degrees. However, \(L(\mathfrak D_s)\) is not distributive. We show that on \(\Delta^{0}_{2}\) sets s-reducibility coincides with its finite branch version; the same holds of e-reducibility. We prove some density results for \(L(\mathfrak D_s)\). In particular \(L(\mathfrak D_s)\) is upwards dense. Among the results about reducibilities that are stronger than s-reducibility, we show that the structure of the \(\Delta^{0}_{2}\) bs-degrees is dense. Many of these results on s-reducibility yield interesting corollaries for Q-reducibility as well.