Classes of Ulm type and coding rank-homogeneous trees in other structures

The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelianp-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank.

Turing computable embeddings

In [3]. two different effective versions of Borel embedding are defined. The first, called computable embedding, is based on uniform enumeration reducibility. while the second, called Turing computable embedding, is based on uniform Turing reducibility. While [3] focused mainly on computable embeddings, the present paper considers Turing computable embeddings. Although the two notions are not equivalent, we can show that they behave alike on the mathematically interesting classes chosen for investigation in [3]. We give a “Pull-back Theorem”, saying that if Ф is a Turing computable embedding of K into K′, then for any computable infinitary sentence φ in the language of K′, we can find a computable infinitary sentence φ* in the language of K such that for all A ∈ K A ⊨ φ* iff Φ (A) ⊨ φ and φ* has the same “complexity” as φ (i.e., if φ is computable Σα or computable Πα, for α ≥ 1, then so is φ*). The Pull-back Theorem is useful in proving non-embeddability, and it has other applications as well.