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Effective categoricity of Abelian p-groups
[article]
2008
arXiv
pre-print
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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
2014
Bulletin of Symbolic Logic
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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
2014
Transactions of the American Mathematical Society
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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
2013
Journal of Algebra
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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
2016
Annals of Pure and Applied Logic
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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]
2010
Lecture Notes in Computer Science
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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
1979
Journal of Algebra
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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
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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]
2017
arXiv
pre-print
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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
2018
Transactions of the American Mathematical Society
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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
2007
Journal of the Australian Mathematical Society
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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
2007
Journal of Pure and Applied Algebra
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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
1998
Topology and its Applications
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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
1973
Journal of Algebra
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., 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
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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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