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Effective categoricity of Abelian p-groups [article]

W. Calvert, D. Cenzer, V. S. Harizanov, A. Morozov
2008 arXiv   pre-print
The present paper seeks to characterize Δ^0_α categoricity for Abelian p-groups, and results of this kind are given for broad classes of Abelian p-groups and values of α.  ...  Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p.  ...  In Section 3, we examine effective categoricity of Abelian p-groups. We show that every computably categorical Abelian p-group is also relatively computably categorical.  ... 
arXiv:0805.1889v1 fatcat:7vlszshwfvcjbl2mdbpflcvimi

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.  ...  Which computable abelian groups are computably categorical? The unknown cases are torsion groups in which p-components are computably categorical, and mixed groups of finite rank.  ...  Categoricity relative to an oracle. As we have seen before, every abelian p-group that is a direct sum of cyclic and quasi-cyclic summands is ∆ 0 3 -categorical.  ... 
doi:10.1017/bsl.2014.32 fatcat:utyr6wtenjh5bg7vm4kkinsocu

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  ...  We are thankful to Isaac Newton Institute for Mathematical Sciences and, more specifically, Semantics and Syntax: A Legacy of Alan Turing program, for partial support of our project.  ... 
doi:10.1090/s0002-9947-2014-06115-1 fatcat:lqcxxcqnanfrzplblqu7hzycvq

Effectively categorical abelian groups

Rodney Downey, Alexander G. Melnikov
2013 Journal of Algebra  
We study effective categoricity of computable abelian groups of the form i∈ω H, where H is a subgroup of (Q , +). Such groups are called homogeneous completely decomposable.  ...  of primes P with the semi-low complement (within the set of all primes).  ...  Effective categoricity of torsion-free abelian groups We study 0 2 -categorical and 0 3 -categorical torsion-free abelian groups.  ... 
doi:10.1016/j.jalgebra.2012.09.020 fatcat:fr5ae4mhkzhlla66zchc2sz4cq

Abelian p -groups and the Halting problem

Rodney Downey, Alexander G. Melnikov, Keng Meng Ng
2016 Annals of Pure and Applied Logic  
We investigate which effectively presented abelian p-groups are isomorphic relative to the halting problem.  ...  We partially reduce the description of ∆ 0 2 -categorical p-groups of Ulm type 1 to the analogous problem for equivalence structures. Using this reduction, we solve to a problem left open in [5] .  ...  To prove that effective ∆ 0 2 -categoricity of R implies effective ∆ 0 2 -categoricity of R, Categoricity of groups In this section all groups are countable abelian p-groups, where p is any fixed prime  ... 
doi:10.1016/j.apal.2016.04.016 fatcat:3pw72abu5vbp3ond427m26sxum

Computable Ordered Abelian Groups and Fields [chapter]

Alexander G. Melnikov
2010 Lecture Notes in Computer Science  
We apply these codings to study the effective categoricity of linear ordered groups and fields.  ...  We present transformations of linearly ordered sets into ordered abelian groups and ordered fields. We study effective properties of the transformations.  ...  Similar examples are built in the class of abelian p-groups [3] . There are also examples of ∆ 0 n -categorical torsion-free abelian groups for small n [20] .  ... 
doi:10.1007/978-3-642-13962-8_36 fatcat:g4l3m5slonhthpcw7chpviqegy

Totally categorical groups and rings

Walter Baur, Gregory Cherlin, Angus Macintyre
1979 Journal of Algebra  
In the case of &,-categoricity, satisfactory analyses have been given in the following cases: (1) linear orders (Rosenstein [32]); (2) Abelian groups (Rosenstein [33]);  ...  Shelah's notions of superstability and stability are receiving increasing attention, but only in the case of Abelian groups can we give them useful algebraic characterizations at present.  ...  Totally categorical groups are Abelian by jinite (i.e., have a normal Abelian subgroup of finite index).  ... 
doi:10.1016/0021-8693(79)90230-8 fatcat:vugwgrqsejg5bpelpzdshy4sdi

Page 6280 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
The usual group of n-cocycles is replaced by a symmetric cat-group of n-cocycles which has as its objects pairs consisting of an object x of P” and an arrow a:0(x) — 0 in P"*!  ...  Since a symmetric cat-group has, at most, only two nontrivial homotopy groups, its abelian group 1o(@) of connected components, and its abelian group 2;(@) of automor- phisms of 0, they are able to show  ... 

A first-order theory of Ulm type [article]

Matthew Harrison-Trainor
2017 arXiv   pre-print
The class of abelian p-groups are an example of some very interesting phenomena in computable structure theory.  ...  We will give an elementary first-order theory T_p whose models are each bi-interpretable with the disjoint union of an abelian p-group and a pure set (and so that every abelian p-group is bi-interpretable  ...  Each abelian p-group is effectively bi-interpretable with a model of T p . Each model of T p is ∆ c 2 -bi-interpretable with the disjoint union of an abelian p-group and a pure set.  ... 
arXiv:1702.06586v1 fatcat:ybgqmnkjkzhatlhcotqll67a5q

Computable topological groups and Pontryagin duality

Alexander Melnikov
2018 Transactions of the American Mathematical Society  
The well-known Pontryagin Duality (classically) reduces the study of compact abelian groups to the algebraic theory of discrete abelian groups.  ...  We believe that most of these applications are fundamental to the emerging systematic theory of computable Polish groups.  ...  Let P be a profinite abelian group. (1) P is recursive iff P is constructive. (2) P is co-r.e. iff P is c.e.-presented. (3) P is computably categorical iff P is computably categorical.  ... 
doi:10.1090/tran/7355 fatcat:inuvz2vrvjcphblamqm2fic5m4

Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups

Anatolij Dvurečenskij
2007 Journal of the Australian Mathematical Society  
We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups.  ...  Moreover, we show that we have a categorical equivalence of the category of perfect effect algebras and the category of Abelian interpolation directed po-groups.  ...  APVT-51-032002 and by the Center of Excellence SAS -Physics of Information -1/2/2005, Bratislava, Slovakia.  ... 
doi:10.1017/s1446788700016025 fatcat:irwjskea5bhmnfo4suuqrmccbi

Homotopy classification of braided graded categorical groups

A.M. Cegarra, E. Khmaladze
2007 Journal of Pure and Applied Algebra  
classes of braided G-graded categorical groups are established.  ...  For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the "abelian cohomology of G-modules".  ...  In effect, for any abelian group homomorphism f : P 0 (M) → A, which verifies f (R 1 (M)) = 0, the mapf : M → A given byf (x) = f ([x] ⊗ [1]) is a group homomorphism satisfyingf ∂ 0 = f sincẽ f (x + y)  ... 
doi:10.1016/j.jpaa.2006.06.011 fatcat:lvsnfcnyefb4fj4gqvq4v4f5he

Recent advances in minimal topological groups

Dikran Dikranjan
1998 Topology and its Applications  
The following are the main direction: (a) permanence properties of minimal groups; (b) complete minimal groups; (c) countably compact minimal groups; (d) algebraic structure of minimal abelian groups.  ...  We pay special attention to the case of abelian groups of free-rank < c resolved by Schinkel (1990, Dissertation) and answer an open question of his regarding the case of torsion-free groups of large free-ranks  ...  Trigos for a careful reading of the manuscript.  ... 
doi:10.1016/s0166-8641(97)00141-7 fatcat:kwavxpoyrrfdxpwmuuvvxo7st4

ℵ0-Categoricity of groups

Joseph G Rosenstein
1973 Journal of Algebra  
., with the &-categorical Abelian groups).  ...  It might seem, at first glance, that a presentation of a group is in effect a list of first order properties of the group.  ... 
doi:10.1016/0021-8693(73)90092-6 fatcat:mwrzyptjenb4rj5eplarbcdaa4

Page 3879 of Mathematical Reviews Vol. , Issue 81J [page]

1981 Mathematical Reviews  
He notes that two periodic abelian groups are program-equivalent if and only if for any prime p their p-compo- nents are program-equivalent.  ...  The author gives an algebraic criterion for equivalence of two periodic abelian groups in program logic. Here he uses ideas of J. Barwise and P. Eklof [Ann. Math.  ... 
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