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Algebra Logika, 2005, Volume 44, Number 2, Pages 238–251 (Mi al108)  

This article is cited in 15 scientific papers (total in 15 papers)

Lattices of Dominions in Quasivarieties of Abelian Groups

S. A. Shakhova


Abstract: Let $\mathcal M$ be any quasivariety of Abelian groups, $\operatorname{dom}^{\mathcal M}_G(H)$ be the dominion of a subgroup $H$ of a group $G$ in $\mathcal M$, and $L_q(\mathcal M)$ be the lattice of subquasivarieties of $\mathcal M$. It is proved that $\operatorname{dom}^{\mathcal M}_G(H)$ coincides with a least normal subgroup of the group $G$ containing $H$, the factor group with respect to which is in $\mathcal M$. Conditions are specified subject to which the set $L(G,H,\mathcal M)=\{\operatorname{dom}^{\mathcal N}_G(H)\mid\mathcal N\in L_q(\mathcal M)\}$ forms a lattice under set-theoretic inclusion and the map $\varphi\colon L_q(\mathcal M)\rightarrow L(G,H,\mathcal M)$ such that $\varphi (\mathcal N)=\operatorname{dom}^{\mathcal N}_G(H)$ for any quasivariety $\mathcal N\in L_q(\mathcal M)$ is an antihomomorphism of the lattice $L_q(\mathcal M)$ onto the lattice $L(G,H,\mathcal M)$.

Keywords: quasivariety, dominion, lattice, group

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English version:
Algebra and Logic, 2005, 44:2, 132–139

Bibliographic databases:

UDC: 512.54.01
Received: 20.04.2004

Citation: S. A. Shakhova, “Lattices of Dominions in Quasivarieties of Abelian Groups”, Algebra Logika, 44:2 (2005), 238–251; Algebra and Logic, 44:2 (2005), 132–139

Citation in format AMSBIB
\Bibitem{Sha05}
\by S.~A.~Shakhova
\paper Lattices of Dominions in Quasivarieties of Abelian~Groups
\jour Algebra Logika
\yr 2005
\vol 44
\issue 2
\pages 238--251
\mathnet{http://mi.mathnet.ru/al108}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2170698}
\zmath{https://zbmath.org/?q=an:1104.08005}
\transl
\jour Algebra and Logic
\yr 2005
\vol 44
\issue 2
\pages 132--139
\crossref{https://doi.org/10.1007/s10469-005-0014-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-18244371661}


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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. S. A. Shakhova, “Distributivity Conditions for Lattices of Dominions in Quasivarieties of Abelian Groups”, Algebra and Logic, 45:4 (2006), 277–285  mathnet  crossref  mathscinet  zmath
    2. A. I. Budkin, “Lattices of dominions of universal algebras”, Algebra and Logic, 46:1 (2007), 16–27  mathnet  crossref  mathscinet  zmath  isi
    3. A. I. Budkin, “Dominions of universal algebras and projective properties”, Algebra and Logic, 47:5 (2008), 304–313  mathnet  crossref  mathscinet  zmath  isi
    4. A. I. Budkin, “Dominions in quasivarieties of metabelian groups”, Siberian Math. J., 51:3 (2010), 396–401  mathnet  crossref  mathscinet  zmath  isi
    5. Budkin A.I., “O dominione polnoi podgruppy metabelevoi gruppy”, Izv. Altaiskogo gos. un-ta, 2010, no. 1-2, 15–19  elib
    6. Shakhova S.A., “Ob odnom svoistve operatsii peresecheniya v reshetkakh dominionov kvazimnogoobrazii abelevykh grupp”, Izv. Altaiskogo gos. un-ta, 2010, no. 1-1, 41–43  elib
    7. Budkin A.I., “O dominionakh konechnykh podgrupp”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 2011, no. 1-2, 15–18  elib
    8. Shakhova S.A., “O suschestvovanii reshetki dominionov v kvazimnogoobraziyakh abelevykh grupp”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 69:1-1 (2011), 31–33  elib
    9. A. I. Budkin, “Dominions in Abelian subgroups of metabelian groups”, Algebra and Logic, 51:5 (2012), 404–414  mathnet  crossref  mathscinet  zmath  isi
    10. A. I. Budkin, “Absolute closedness of torsion-free Abelian groups in the class of metabelian groups”, Algebra and Logic, 53:1 (2014), 9–16  mathnet  crossref  mathscinet  isi
    11. A. I. Budkin, “On the closedness of a locally cyclic subgroup in a metabelian group”, Siberian Math. J., 55:6 (2014), 1009–1016  mathnet  crossref  mathscinet  isi
    12. S. A. Shakhova, “Absolutely Closed Groups in the Class of $2$-Step Nilpotent Torsion-Free Groups”, Math. Notes, 97:6 (2015), 946–950  mathnet  crossref  crossref  mathscinet  isi  elib
    13. A. I. Budkin, “Dominions in solvable groups”, Algebra and Logic, 54:5 (2015), 370–379  mathnet  crossref  crossref  mathscinet  isi
    14. A. I. Budkin, “On $2$-closedness of the rational numbers in quasivarieties of nilpotent groups”, Siberian Math. J., 58:6 (2017), 971–982  mathnet  crossref  crossref  isi  elib
    15. A. I. Budkin, “On dominions of the rationals in nilpotent groups”, Siberian Math. J., 59:4 (2018), 598–609  mathnet  crossref  crossref  isi  elib
  • Алгебра и логика Algebra and Logic
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