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Enumerations and Completely Decomposable Torsion-Free Abelian Groups

Alexander G. Melnikov
2009 Theory of Computing Systems  
We study possible degree spectra of completely decomposable torsion-free Abelian groups. Partially supported by President grant of Scientific School NSh-4413.2006.1. Definition 9.  ...  We say that an Abelian group A, + is torsion-free if for all elements a = 0 and any natural n = 0, na = 0.  ...  A torsion-free Abelian group is completely decomposable if it can be presented as the direct sum of groups having rank 1.  ... 
doi:10.1007/s00224-009-9175-9 fatcat:bl7kiysjsrbglexetdkgeafmd4

Page 1920 of Mathematical Reviews Vol. , Issue 82e [page]

1982 Mathematical Reviews  
All groups considered are torsion free abelian groups of finite rank.  ...  In this note, a number of structural properties of the p-adic completion of torsion-free abelian groups are proved.  ... 

COMPUTABLE ABELIAN GROUPS

ALEXANDER G. MELNIKOV
2014 Bulletin of Symbolic Logic  
We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.  ...  In Subsection 4.3 we finally approach computable torsion-free abelian groups that are not necessarily completely decomposable.  ...  The isomorphism problem for computable torsion-free abelian groups is Σ 1 1 -complete.  ... 
doi:10.1017/bsl.2014.32 fatcat:utyr6wtenjh5bg7vm4kkinsocu

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

Kyle Riggs
2013 arXiv   pre-print
We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is Σ^0_3-complete.  ...  However, when we consider groups of infinite rank, it becomes Σ^1_1-complete, so it cannot be characterized by a first-order formula in the language of arithmetic.  ...  We will then shift our focus to groups of infinite rank and show that: 1) the index set of computable decomposable torsion-free abelian groups is Σ 1 1complete, and 2) the set of decomposable torsion-free  ... 
arXiv:1311.1865v1 fatcat:hryinzpu2jeidlkqeeig6v4hfq

Effectively categorical abelian groups

Rodney Downey, Alexander G. Melnikov
2013 Journal of Algebra  
We prove that a homogeneous completely decomposable group of infinite rank is 0 2 -categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set  ...  We apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.  ...  Definition 2.8 (Completely decomposable group). A torsion-free abelian group is called completely decomposable if G is a direct sum of groups each having rank 1.  ... 
doi:10.1016/j.jalgebra.2012.09.020 fatcat:fr5ae4mhkzhlla66zchc2sz4cq

Some ugly aleph_1-free abelian groups [article]

Saharon Shelah, Lutz Strüngmann
2001 arXiv   pre-print
Given an aleph_1-free abelian group G we characterize the class C_G of all torsion abelian groups T satisfying Ext(G,T)=0 assuming the continuum hypothesis CH.  ...  Moreover, in Godel's constructable universe we prove that this characterizes C_G for arbitrary torsion-free abelian G. It follows that there exist some ugly aleph_1-free abelian groups.  ...  In [SW] satisfactory characterizations of T C(G) were obtained for countable torsion-free abelian groups and for completely decomposable groups.  ... 
arXiv:math/0107208v1 fatcat:7gojirzf5zdqni67t3aplwhbma

Page 39 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews  
Given an abelian group of the form G = T @ A, where T is torsion and A a torsion-free group, the authors describe certain subrings of the endomorphism ring of A.  ...  They show that these groups share properties analo- gous to those of the usual torsion-free completely decomposable groups. B. Goldsmith (Dublin) 2000j:20105 20K30 16850 20K21 Krylov, P.  ... 

On a paper of Szele and Szendrei on groups with commutative endomorphism rings

Ph. Schultz
1973 Acta Mathematica Hungarica  
Soc. 122, 3 (1994), 961-964. (33) The near endomorphism ring of an almost completely decomposable group, Abelian Groups and Modules, Eds.  ...  (23) Finite extensions of torsion-free groups , in Abelian Group Theory, Gordon and Breach, New York, (1986), 333-350.  ... 
doi:10.1007/bf01894610 fatcat:6qkijen4cjflzdb6d6xzu5qe2e

Two problems onℵ0-indecomposable Abelian Groups

Saharon Shelah, Alexander Soifer
1986 Journal of Algebra  
Theorem 3 proves Conjecture 1, and thus completely describes torsion parts of countable K,-indecomposable abelian groups of finite torsion-free rank in terms of their Ulm-Kaplansky invariants.  ...  K,-indecomposable abelian group, then already some subgroup of finite torsion-free rank H of G with the same torsion part as G is Ha-indecomposable.  ... 
doi:10.1016/0021-8693(86)90033-5 fatcat:74fa22ittnetbo4flts6whgsci

Decompositions of torsion-free abelian groups [article]

Gabor Braun. Phill Schultz, Lutz Struengmann
2018 arXiv   pre-print
It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism.  ...  We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.  ...  This defines ϕ on the linearly independent generators v i and x n of H, from which it uniquely extends to H. By the definition it is obvious that the image of ϕ is B v 0 ,T,α .  ... 
arXiv:1810.09644v1 fatcat:s65mrjr5szclnczs74y23wynui

Page 137 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
A torsion-free group G admits an No-prebalanced chain if and only if in its balanced projective resolution 0 — B —- C —+ G —0 (where C is completely decomposable), B is a B- group.  ...  (SA-OFS; Bloemfontein) Completely decomposable pure subgroups of completely decomposable abelian groups. Rend. Sem. Mat. Univ. Padova 92 (1994), 63-69.  ... 

Computable completely decomposable groups

Rodney Downey, Alexander G. Melnikov
2014 Transactions of the American Mathematical Society  
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  ...  Many thanks to André Nies, Asher Kach and Kyle Riggs for pointing numerous typos and minor mathematical problems in the early draft of the paper.  ... 
doi:10.1090/s0002-9947-2014-06115-1 fatcat:lqcxxcqnanfrzplblqu7hzycvq

On decomposable pseudofree groups

Dirk Scevenels
1999 International Journal of Mathematics and Mathematical Sciences  
Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum.  ...  In particular, using sequential representations, we give a new proof of a result by Baer, stating that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian  ...  Recall that a torsion-free Abelian group A of rank is called completely decomposable if A = i=1 A i , (2.3) where each A i is torsion-free of rank one.  ... 
doi:10.1155/s0161171299226178 fatcat:szotdowcjvgqfi6accflj5hjgi

Decidability and Computability of Certain Torsion-Free Abelian Groups

Rodney G. Downey, Sergei S. Goncharov, Asher M. Kach, Julia F. Knight, Oleg V. Kudinov, Alexander G. Melnikov, Daniel Turetsky
2010 Notre Dame Journal of Formal Logic  
We study completely decomposable torsion-free abelian groups of the form G S := ⊕ n∈S Qp n for sets S ⊆ ω.  ...  We show that G S has a decidable copy if and only if S is Σ 0 2 and has a computable copy if and only if S is Σ 0 3 .  ...  However, not as much is known about the effective properties of completely decomposable torsion-free abelian groups.  ... 
doi:10.1215/00294527-2010-006 fatcat:nivejewytzanxak4ki4awl4pma

Page 9081 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
All groups considered in this paper are countable torsion-free abelian groups. Let G be a completely decomposable group in the sense that it is a direct sum of groups G; of rank 1.  ...  G is called strongly decomposable if there exists a strongly constructive enumeration v of the group G such that there is a recursively enumerable system of nontrivial elements 5; € G;, in (G,v).  ... 
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