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Sibirsk. Mat. Zh., 2006, Volume 47, Number 3, Pages 695–706 (Mi smj887)  

This article is cited in 10 scientific papers (total in 10 papers)

On $\Sigma$-subsets of naturals over abelian groups

A. N. Khisamiev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We obtain conditions for the $\Sigma$-definability of a subset of the set of naturals in the hereditarily finite admissible set over a model and for the computability of a family of such subsets. We prove that: for each $e$-ideal $I$ there exists a torsion-free abelian group $A$ such that the family of $e$-degrees of $\Sigma$-subsets of $\omega$ in $\mathbb{HF}(A)$ coincides with $I$ there exists a completely reducible torsion-free abelian group in the hereditarily finite admissible set over which there exists no universal $\Sigma$-function; for each principal $e$-ideal $I$ there exists a periodic abelian group $A$ such that the family of $e$-degrees of $\Sigma$-subsets of $\omega$ in $\mathbb{HF}(A)$ coincides with $I$.

Keywords: admissible set, e-reducibility, computability, $\Sigma$-definability, abelian group

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English version:
Siberian Mathematical Journal, 2006, 47:3, 574–583

Bibliographic databases:

UDC: 512.540, 510.5
Received: 30.06.2004

Citation: A. N. Khisamiev, “On $\Sigma$-subsets of naturals over abelian groups”, Sibirsk. Mat. Zh., 47:3 (2006), 695–706; Siberian Math. J., 47:3 (2006), 574–583

Citation in format AMSBIB
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\by A.~N.~Khisamiev
\paper On $\Sigma$-subsets of naturals over abelian groups
\jour Sibirsk. Mat. Zh.
\yr 2006
\vol 47
\issue 3
\pages 695--706
\mathnet{http://mi.mathnet.ru/smj887}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2251077}
\zmath{https://zbmath.org/?q=an:1115.03038}
\transl
\jour Siberian Math. J.
\yr 2006
\vol 47
\issue 3
\pages 574--583
\crossref{https://doi.org/10.1007/s11202-006-0068-8}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000239228700017}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33744721349}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. N. Khisamiev, “On quasiresolvent periodic abelian groups”, Siberian Math. J., 48:6 (2007), 1115–1126  mathnet  crossref  mathscinet  zmath  isi
    2. A. N. Khisamiev, “$\Sigma$-Bounded algebraic systems and universal functions. I”, Siberian Math. J., 51:1 (2010), 178–192  mathnet  crossref  mathscinet  isi
    3. Khisamiev A.N., “Bounded algebraic systems and universal functions”, Doklady Mathematics, 81:2 (2010), 309–311  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. N. Khisamiev, “$\Sigma$-uniform structures and $\Sigma$-functions. I”, Algebra and Logic, 50:5 (2011), 447–465  mathnet  crossref  mathscinet  zmath  isi
    5. A. N. Khisamiev, “On a universal $\Sigma$-function over a tree”, Siberian Math. J., 53:3 (2012), 551–553  mathnet  crossref  mathscinet  isi
    6. A. N. Khisamiev, “Universal functions and almost $c$-simple models”, Siberian Math. J., 56:3 (2015), 526–540  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. A. N. Khisamiev, “Universal functions over trees”, Algebra and Logic, 54:2 (2015), 188–193  mathnet  crossref  crossref  mathscinet  isi
    8. A. N. Khisamiev, “A class of almost $c$-simple rings”, Siberian Math. J., 56:6 (2015), 1133–1141  mathnet  crossref  crossref  mathscinet  isi  elib
    9. A. N. Khisamiev, “Universal functions and unbounded branching trees”, Algebra and Logic, 57:4 (2018), 309–319  mathnet  crossref  crossref  isi
    10. A. N. Khisamiev, “Universalnye funktsii i $K\Sigma$-struktury”, Sib. matem. zhurn., 61:3 (2020), 703–716  mathnet  crossref
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