Computable completely decomposable groups

Abstract

Abstract. A completely decomposable group is an abelian group of the form i H i , where H i ≤ (Q, +). We show that every computable completely decomposable group is ∆ 0 5 -categorical. We construct a computable completely decomposable group which is not ∆ 0 4 -categorical, and give an example of a computable completely decomposable group G which is ∆ 0 4 -categorical but not ∆ 0 3 -categorical. We also prove that the index set of computable completely decomposable groups is arithmetical.

OnΔ20-categoricity of equivalence relations

“…Only recently, there has been significant progress in understanding Δ 0 n -categoricity in several specific classes, for small n. It follows from [8,30] that every free (non-abelian) group of rank ω is Δ 0 3 -categorical, and the result cannot be improved to Δ 0 2 . It is also known that every computable completely decomposable group is Δ 0 5 -categorical, and the result is sharp [13]. Every computable homogeneous completely decomposable group is Δ 0 3 -categorical, and a group of this form is Δ 0 2 -categorical if and only if it encodes a semi-low set into its divisibility relation [12].…”

“…The study of Δ 0 n -categorical structures has some independent technical interest as such investigations typically require new ideas and techniques (see, e.g., [2,12,13]). As a consequence of these technical difficulties, our knowledge of Δ 0 n -categorical structures is rather limited even when n = 2.…”

The decomposability problem for torsion-free abelian groups is analytic-complete

“…Downey and Melnikov ( [3], [4]) studied homogeneous completely decomposable groups, in which each summand has the same type (see Definition 2.2). They found that every completely decomposable group is ∆ 0 5 -categorical, but homogenous completely decomposable groups are actually ∆ 0 3 -categorical.…”

“…In their research of completely decomposable groups, Downey and Melnikov [4] found that the index set of completely decomposable groups can be described by a Σ 0 7 formula (though it is not known if this is sharp). The property with which this paper is concerned is a similar one: decomposability.…”

Independence in computable algebra

“…We suspect that our metatheorem has relativized and generalized versions that could be applied to, say, completely decomposable groups [DM14,Khi02] and other structures where the notions of independence are not r.i.c.e. but are relatively intrinsically Σ 0 n for some n > 1.…”