Recently, researchers in the foundations of mathematics have become interested in the distributive lattices Pw and Ps, the study of which is a major part of the field of "mass problems." In general, a mass problem U is a subset of ωω (Baire space). We define U ≤s V if there is an index e for a computable functional so that for all f ∈ V, Φef ∈ U. If we do not require the same e for every f, we get a "weak version": U &lew V if for all f ∈ V, there is an e so that Φef ∈ U.  The relations ≤s and ≤w naturally induce equivalence relations ≡s and ≡w. We define Ps to be the collection of ≡s degrees of nonempty Π01 subsets of 2ω (Cantor space); similarly, we define Pw to be the collection of ≡w degrees of nonempty Π01 subsets of 2ω.  An important open question is whether Pw is dense. The result of Chapter 2 is that the embedding of the free distributive lattice on countably many generators into Ps can be done densely. The way it is done gives indirect evidence that the kinds of priority arguments that show the density of Ps are probably not strong enough to show the density of Pw. The result of Chapter 3 applies these priority arguments to show the decidability of the elementary ∀∃-theory of Ps as a partial order. The result of Chapter 5 is that certain index sets related to Pw are Π11-complete. This leads to a conjecture that the Turing degree of its elementary theory is as high as possible.