The P-T-degrees of the recursive sets: lattice embeddings, extensions of embeddings and the two quantifier theory

It has been shown in the literature that the two basic nondistributive lattices can be embedded in R_pt, the polynomial-time turing degrees of the recursive sets. The authors introduce more general techniques to extend these results to show that every recursive lattice can be embedded in R_pt. In addition to lattice theoretic representation theorems, they use the scheme of priority style arguments coupled with looking back techniques. They also generalize the density-type results to settle the full extension of the embeddings problem for R_pt. Combined with the logical analysis of sentences with one alternation of quantifiers, these results suffice to decide the full ∀∃-theory of R_pt. They also give a strong nonhomogeneity result