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Algebra Logika, 2008, Volume 47, Number 5, Pages 541–557 (Mi al374)  

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

Dominions of universal algebras and projective properties

A. I. Budkin


Abstract: Let $A$ be a universal algebra and $H$ its subalgebra. The dominion of $H$ in $A$ (in a class $\mathcal M$) is the set of all elements $a\in A$ such that every pair of homomorphisms $f,g\colon A\to M\in\mathcal M$ satisfies the following: if $f$ and $g$ coincide on $H$, then $f(a)=g(a)$. A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras $H$ whose dominions coincide with $H$. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup $H$ is closed in each group $\langle H,a\rangle$ generated by one element modulo $H$.

Keywords: universal algebra, dominion, closed algebra, projective property, nilpotent group.

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English version:
Algebra and Logic, 2008, 47:5, 304–313

Bibliographic databases:

UDC: 512.57
Received: 19.03.2008
Revised: 03.09.2008

Citation: A. I. Budkin, “Dominions of universal algebras and projective properties”, Algebra Logika, 47:5 (2008), 541–557; Algebra and Logic, 47:5 (2008), 304–313

Citation in format AMSBIB
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\paper Dominions of universal algebras and projective properties
\jour Algebra Logika
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\issue 5
\pages 541--557
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\jour Algebra and Logic
\yr 2008
\vol 47
\issue 5
\pages 304--313
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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. I. Budkin, “Dominions in quasivarieties of metabelian groups”, Siberian Math. J., 51:3 (2010), 396–401  mathnet  crossref  mathscinet  zmath  isi
    2. Budkin A.I., “O dominione polnoi podgruppy metabelevoi gruppy”, Izv. Altaiskogo gos. un-ta, 2010, no. 1-2, 15–19  elib
    3. Budkin A.I., “O dominionakh konechnykh podgrupp”, Izvestiya Altaiskogo gosudarstvennogo universiteta, 2011, no. 1-2, 15–18  elib
    4. A. I. Budkin, “Dominions in Abelian subgroups of metabelian groups”, Algebra and Logic, 51:5 (2012), 404–414  mathnet  crossref  mathscinet  zmath  isi
    5. 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
    6. 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
    7. A. I. Budkin, “Dominions in solvable groups”, Algebra and Logic, 54:5 (2015), 370–379  mathnet  crossref  crossref  mathscinet  isi
    8. 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
    9. Campercholi M., “Dominions and Primitive Positive Functions”, J. Symb. Log., 83:1 (2018), 40–54  crossref  mathscinet  zmath  isi  scopus
    10. 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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