Sums of Complexes in Torsion Free Abelian Groups

1969 ◽  
Vol 12 (4) ◽  
pp. 479-480 ◽  
Author(s):  
H. Heilbronn ◽  
P. Scherk
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Preceding Paper ◽  
Torsion Free Abelian Group ◽  
Linearly Independent ◽  
Free Abelian Groups ◽  
Image Position ◽  
Torsion Free

Let A, B, denote two non-void finite complexes (= subsets) of the torsion free abelian group G,Let d(A),… denote the maximum number of linearly independent elements of A,… and let n = n(A, B) denote the number of elements of A + B whose representation in the form a + b is unique. In the preceding paper, Tarwater and Entringer [1] proved that n ≥ d(A).

On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank

2001 ◽  
Vol 7 (3) ◽  
pp. 329-344 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Classification Problem ◽  
Dimensional Vector Space ◽  
Torsion Free Abelian Group ◽  
Linearly Independent ◽  
Free Abelian Groups ◽  
Image Position ◽  
Torsion Free

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚn) of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx(p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function

scholarly journals Type radicals and quasi-decompositions of torsion-free abelian groups

1992 ◽  
Vol 52 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Oteo Mutzbauer
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractA composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

Torsion-free abelian groups with optimal Scott families

2018 ◽  
Vol 18 (01) ◽  
pp. 1850002
Author(s):  
Alexander G. Melnikov
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Symbolic Logic ◽  
Technical Result ◽  
Torsion Free Abelian Group ◽  
Scott Family ◽  
Free Abelian Groups ◽  
Torsion Free ◽  
Labeled Tree

We prove that for any computable successor ordinal of the form [Formula: see text] [Formula: see text] limit and [Formula: see text] there exists computable torsion-free abelian group [Formula: see text]TFAG[Formula: see text] that is relatively [Formula: see text] -categorical and not [Formula: see text] -categorical. Equivalently, for any such [Formula: see text] there exists a computable TFAG whose initial segments are uniformly described by [Formula: see text] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [Formula: see text]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) [Formula: see text]-categorical TFAGs for arbitrarily large [Formula: see text] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [Formula: see text]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.

scholarly journals Hypertypes of torsion-free abelian groups of finite rank

1989 ◽  
Vol 39 (1) ◽  
pp. 21-24 ◽  
Author(s):  
H.P. Goeters ◽  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

Let G be a torsion-free abelian group of finite rank n and let F be a full free subgroup of G. Then G/F is isomorphic to T1 ⊕ … ⊕ Tn, where T1 ⊆ T2 ⊆ … ⊆ Tn ⊆ ℚ/ℤ. It is well known that type T1 = inner type G and type Tn = outer type G. In this note we give two characterisations of type Ti for 1 < i < n.

scholarly journals Jump degrees of torsion-free abelian groups

2012 ◽  
Vol 77 (4) ◽  
pp. 1067-1100 ◽  
Author(s):  
Brooke M. Andersen ◽  
Asher M. Kach ◽  
Alexander G. Melnikov ◽  
Reed Solomon
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe show, for each computable ordinal α and degree a > 0(α), the existence of a torsion-free abelian group with proper αth jump degree a.

Sums of Complexes in Torsion-Free Abelian Groups

1969 ◽  
Vol 12 (4) ◽  
pp. 475-478 ◽  
Author(s):  
J. D. Tarwater ◽  
R. C. Entringer
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Multiple Representations ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

The number of elements in the sum A + B of two complexes A and B of a group G which have multiple representations a + b = a '+ b' has been investigated by Scherk and Kemperman [1]. Kemperman [2] appealed to transfinite techniques (to order G) to prove:If G is a torsion-free abelian group with finite subsets A and B with | B | ≥ 2, then at least two elements c of A + B admit exactly one representation c = a + b.

A note on rank and direct decompositions of torsion-free Abelian groups. II

1969 ◽  
Vol 66 (2) ◽  
pp. 239-240 ◽  
Author(s):  
A. L. S. Corner
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Direct Sums ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Direct Decompositions ◽  
Torsion Free

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

Cellular covers of ℵ1-free abelian groups

2015 ◽  
Vol 14 (10) ◽  
pp. 1550139 ◽  
Author(s):  
José L. Rodríguez ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Countable Rank ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Exact Sequences ◽  
Torsion Free ◽  
Cellular Cover

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence [Formula: see text] such that the map G → H is a cellular cover over H and the rank of H is exactly n. In particular, the free abelian group of infinite countable rank is the kernel of a cellular exact sequence of co-rank 2 which answers an open problem from Rodríguez–Strüngmann [J. L. Rodríguez and L. Strüngmann, Mediterr. J. Math.6 (2010) 139–150]. Moreover, we give a new method to construct cellular exact sequences with prescribed torsion free kernels and cokernels. In particular we apply this method to the class of ℵ1-free abelian groups in order to complement results from the cited work and Göbel–Rodríguez–Strüngmann [R. Göbel, J. L. Rodríguez and L. Strüngmann, Fund. Math.217 (2012) 211–231].

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

2011 ◽  
Vol 21 (08) ◽  
pp. 1463-1472 ◽  
Author(s):  
GÁBOR BRAUN ◽  
LUTZ STRÜNGMANN
Keyword(s):  
Abelian Group ◽  
Finite Automaton ◽  
Free Abelian Group ◽  
Finite Automata ◽  
Additive Group ◽  
Rational Numbers ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

In [Todor Tsankov, The additive group of the rationals does not have an automatic presentation, May 2009, arXiv:0905.1505v1], it was shown that the group of rational numbers is not FA-presentable, i.e. it does not admit a presentation by a finite automaton. More generally, any torsion-free abelian group that is divisible by infinitely many primes is not of this kind. In this article we extend the result from [13] and prove that any torsion-free FA-presentable abelian group G is an extension of a finite rank free group by a finite direct sum of Prüfer groups ℤ(p∞).

Dynamical entropy for Bogoliubov actions of torsion-free abelian groups on the CAR-algebra

2000 ◽  
Vol 20 (4) ◽  
pp. 1111-1125 ◽  
Author(s):  
VALENTIN YA. GOLODETS ◽  
SERGEY V. NESHVEYEV
Keyword(s):  
Abelian Group ◽  
Unitary Representation ◽  
Free Abelian Group ◽  
Singular Part ◽  
Torsion Free Group ◽  
Torsion Free Abelian Group ◽  
Car Algebra ◽  
Dynamical Entropy ◽  
Free Abelian Groups ◽  
Torsion Free

We compute dynamical entropy in Connes, Narnhofer and Thirring sense for a Bogoliubov action of a torsion-free abelian group $G$ on the CAR-algebra. A formula analogous to that found by Størmer and Voiculescu in the case $G={\Bbb Z}$ is obtained. The singular part of a unitary representation of $G$ is shown to give zero contribution to the entropy. A proof of these results requires new arguments since a torsion-free group may have no finite index proper subgroups. Our approach allows us to overcome these difficulties, it differs from that of Størmer–Voiculescu.

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