On the embedding of α-recursive presentable lattices into the α-recursive degrees below 0′

Abstract

An important problem, widely treated in the analysis of the structure of degree orderings, is that of partial order and lattice embeddings. Thus for example we have the results on embeddings of all countable partial orderings in the Turing degrees by Kleene and Post [3] and in the r.e. T-degrees by Sacks [10]. For lattice embeddings the work on T-degrees culminated in the characterization of countable initial segments by Lachlan and Lebeuf [4]. For the r.e. T-degrees there has been a continuing line of progress on this question. (See Soare [20] and Lerman, Shore, and Soare [8].) Similar projects have been undertaken for the T-degrees below 0′ (Kleene and Post [3], Lerman [6]) as well as for most other degree orderings. The results have been used not only to analyse individual orderings but also to distinguish between them (Shore [16], [19], [17]).The situation for α-jecursive theory, the study of recursion in (admissible) ordinals, is similar to, though not as well developed as, that for Turing degrees. All afinite partial orderings have been embedded even in the α-r.e. degrees (see Lerman [5]). Lattice embedding results are somewhat fragmentary however. In terms of initial segments even the question of the existence of a minimal α-degree has not been settled for all admissibles. (See Shore [12] for a proof for Σ2-admissible ordinals, however.) Results on more complicated lattices have only reached to the finite distributive ones for Σ3-admissible ordinals (see Dorer [1]).