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Effective categoricity of Abelian p-groups
[article]

2008
*
arXiv
*
pre-print

The present paper seeks to characterize Δ^0_α

arXiv:0805.1889v1
fatcat:7vlszshwfvcjbl2mdbpflcvimi
*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*. ...##
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COMPUTABLE ABELIAN GROUPS

2014
*
Bulletin of Symbolic Logic
*

We provide an introduction to methods and recent results on infinitely generated

doi:10.1017/bsl.2014.32
fatcat:utyr6wtenjh5bg7vm4kkinsocu
*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*. ...##
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Computable completely decomposable groups

2014
*
Transactions of the American Mathematical Society
*

A completely decomposable

doi:10.1090/s0002-9947-2014-06115-1
fatcat:lqcxxcqnanfrzplblqu7hzycvq
*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. ...##
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Effectively categorical abelian groups

2013
*
Journal of Algebra
*

We study

doi:10.1016/j.jalgebra.2012.09.020
fatcat:fr5ae4mhkzhlla66zchc2sz4cq
*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*. ...##
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Abelian p -groups and the Halting problem

2016
*
Annals of Pure and Applied Logic
*

We investigate which

doi:10.1016/j.apal.2016.04.016
fatcat:3pw72abu5vbp3ond427m26sxum
*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 ...##
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Computable Ordered Abelian Groups and Fields
[chapter]

2010
*
Lecture Notes in Computer Science
*

We apply these codings to study the

doi:10.1007/978-3-642-13962-8_36
fatcat:g4l3m5slonhthpcw7chpviqegy
*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] . ...##
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Totally categorical groups and rings

1979
*
Journal of Algebra
*

In the case

doi:10.1016/0021-8693(79)90230-8
fatcat:vugwgrqsejg5bpelpzdshy4sdi
*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). ...##
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Page 6280 of Mathematical Reviews Vol. , Issue 94k
[page]

1994
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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 ...##
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A first-order theory of Ulm type
[article]

2017
*
arXiv
*
pre-print

The class

arXiv:1702.06586v1
fatcat:ybgqmnkjkzhatlhcotqll67a5q
*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. ...##
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Computable topological groups and Pontryagin duality

2018
*
Transactions of the American Mathematical Society
*

The well-known Pontryagin Duality (classically) reduces the study

doi:10.1090/tran/7355
fatcat:inuvz2vrvjcphblamqm2fic5m4
*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*. ...##
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Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups

2007
*
Journal of the Australian Mathematical Society
*

We show that the category

doi:10.1017/s1446788700016025
fatcat:irwjskea5bhmnfo4suuqrmccbi
*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. ...##
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Homotopy classification of braided graded categorical groups

2007
*
Journal of Pure and Applied Algebra
*

classes

doi:10.1016/j.jpaa.2006.06.011
fatcat:lvsnfcnyefb4fj4gqvq4v4f5he
*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) ...##
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Recent advances in minimal topological groups

1998
*
Topology and its Applications
*

The following are the main direction: (a) permanence properties

doi:10.1016/s0166-8641(97)00141-7
fatcat:kwavxpoyrrfdxpwmuuvvxo7st4
*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. ...##
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ℵ0-Categoricity of groups

1973
*
Journal of Algebra
*

., with the &-

doi:10.1016/0021-8693(73)90092-6
fatcat:mwrzyptjenb4rj5eplarbcdaa4
*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*. ...##
###
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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