Abelian groups with partial decomposition bases in 𝐿_{∞𝜔}^{𝛿}, Part II

Jacoby, Carol; Loth, Peter

doi:10.1090/conm/576/11348

Cited by 2 publications

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References 8 publications

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“…Lettingŵ c p e G = min w c p e G , we have the following theorem. 11]). If G has a partial decomposition basis C, thenŵ c p e G is the largest integer n, if it exists, such that there are X ∈ C and x 1 x n ∈ X such that U x i ∈ c and U p x i ∈ e for all i = 1 n. If no such n exists,ŵ c p e G = .…”

Section: Preliminariesmentioning

confidence: 99%

“…Generalizing the concept of decomposition basis, the first author [8] introduced the notion of partial decomposition basis in order to extend Barwise and Eklof's [1] classification of torsion groups in L to Warfield groups. Using modifications of the Ulm and Warfield invariants, abelian groups with partial decomposition bases have been completely classified in L , or equivalently, up to partial isomorphism ( [9], [10]), as well as in L ( [14], [11], [12]). …”

Section: Introductionmentioning

confidence: 99%

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Partial Decomposition Bases and Global Warfield Groups

Jacoby¹,

Loth

2016

Communications in Algebra

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The class of abelian groups with partial decomposition bases was developed by the first author in order to generalize Barwise and Eklof's classification of torsion groups in L . In this article, we continue to explore algebraic characteristics of this class and establish a uniqueness theorem, extending our previous work on mixed p-local groups to the global case. It is shown that groups with partial decomposition bases are characterized in terms of Warfield groups and k-groups of Hill and Megibben. In fact, we prove that the class of groups with partial decomposition bases is identical to the class of k-groups, and, as such, closed under direct summands, and that every finitely generated subgroup of a k-group is locally nice. Also, we introduce and explore subgroups possessing a partial subbasis. As an application, it is shown that isotype k-subgroups of abelian groups are k-groups.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Partial Decomposition Bases and Global Warfield Groups

Jacoby¹,

Loth

2016

Communications in Algebra

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The classification of $${\mathbb {Z}}_p$$ Z p -modules with partial decomposition bases in $$L_{\infty \omega }$$ L ∞ ω

Jacoby¹,

Loth

2016

Arch. Math. Logic

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Ulm's Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω-equivalence. In this paper, we extend this classification to a class of mixed Zp-modules which includes all Warfield modules and is closed under L∞ω-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in L∞ω using invariants deduced from the classical Ulm and Warfield invariants.

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Abelian groups with partial decomposition bases in 𝐿_{∞𝜔}^{𝛿}, Part II

Cited by 2 publications

References 8 publications

Partial Decomposition Bases and Global Warfield Groups

Partial Decomposition Bases and Global Warfield Groups

The classification of $${\mathbb {Z}}_p$$ Z p -modules with partial decomposition bases in $$L_{\infty \omega }$$ L ∞ ω

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