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

Abstract

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is Σ 0 3complete. However, when we consider computable groups of infinite rank, it becomes Σ 1 1 -complete (and Σ 1 1 -complete for groups of infinite rank in general), so it cannot be characterized by a first-order formula in the language of arithmetic.Σ 1 1 -COMPLETENESS OF DECOMPOSABLE GROUPS 3

Automatic and Polynomial-Time Algebraic Structures

“…This low syntactical complexity reflects the algorithmic nature of freeness; recall that Nielson transformations [49] can be used to "calculate" a basis of a group (if the basis exists). In contrast, Riggs [58] has shown that the index set I DirDecom of directly decomposable abelian groups is Σ 1 1complete. Recall that a group is directly decomposable if it can be split into the direct sum of its nontrivial subgroups; this brute-force definition is naturally Σ 1 1 .…”

Characterizations of cancellable groups

“…Second, Riggs's [Rig15] produced for each tree T a group G T such that G T decomposes as a non-trivial direct sum if and only if T has an infinite path. We will modify this argument to produce a group H T so that if T has no infinite path then H T is indecomposable and cancellable, and so that if T has an infinite path, then H T decomposes as A⊕B ⊕B ⊕B ⊕· · · so that G T is not cancellable.…”

“…This means that there is no simpler definition than one which uses a universal quantifier over countable sets. The proof uses methods from Riggs's [Rig15] theorem that indecomposability is Π 1 1 m-complete. We conjecture that: (a) if a group cancels with every countable group, then it cancels with every groups, and so the index set is Π 1 2 ; and (b) the class is Π 1 2 m-hard, i.e., that (a) is the best characterization of the cancellation property for infinite-rank groups.…”

Computable Abelian Groups

“…The index set of directly indecomposable abelian groups is Π 1 1 -complete. Although the construction of Riggs [95] uses Definition 4.22 and is similar to those discussed in Subsection 4.3.4, its verification is less tricky since it uses decomposability analysis (e.g., Fuchs [45]) rather than definability analysis. The construction itself is more involved though.…”

“…The construction itself is more involved though. When restricted to finite rank groups, this index set becomes arithmetical (Riggs [95]). §5.…”