Essentially indecomposable abelian 𝑝-groups having a filtration of prescribed type

Perfectly Bounded Classes of Abelian Groups

“…This parallels results of Dugas, Fay and Shelah (1987) and Keef (1995) on the behavior of classes of abelian p-groups with elements of infinite height. It also simplifies, clarifies and generalizes a result of Cutler, Mader and Megibben (1989) which states that the p ω+1 -projectives cannot be characterized using filtrations.…”

“…Let R be the class of uncountable regular cardinals, and R f be the collection of finite subsets of R. By an R f -invariant we mean a subclass X ⊆ R f such that (1) if S, T ∈ R f , S ∈ X and S ⊆ T , then T ∈ X ; and (2) the collection of sets in A that are minimal under inclusion is actually a set and not a proper class. Another way to view (2) is that it asserts that there is a cardinal κ such that for all T ∈ R f , T ∈ X iff T ∩ κ ∈ X . We let 0 R = ∅ and 1 R = R f ; so for every R f -invariant X we have 0 R ⊆ X ⊆ 1 R .…”

“…The first example of ap ω+1 -bounded group that is not p ω+1 -projective was essentially contained in [2]. Our approach has several advantages, however.…”

“…In [2] the following question (reportedly posed by L. Fuchs) was addressed: Can the p ω+1 -projective groups be characterized by filtrations? In our notation, this might be phrased as follows: If G is ap ω+1 -bounded group (i.e.…”

“…In other words, is the analogue of Theorem 3.7(e) valid for perfect B/U-pairs of length ω + 1. In [2] this question was answered negatively, using some significant set theory (i.e. the diamond principle in the constructible universe, V = L).…”