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Izv. Akad. Nauk SSSR Ser. Mat., 1988, Volume 52, Issue 4, Pages 875–881 (Mi izv1208)  

This article is cited in 2 scientific papers (total in 2 papers)

On filters in the lattice of quasivarieties of groups

A. I. Budkin


Abstract: Suppose $\mathfrak M$ is a quasivariety of groups. Assume there exist finitely presented groups $A$ and $B$ such that $A\notin\mathfrak M$, $B\in\mathfrak M$, and $B$ is not contained in the quasivariety generated by $A$. It is proved that the principal filter generated in the lattice of quasivarieties of groups by has the cardinality of the continuum. In particular, the principal filters generated by a) the quasivariety generated by all proper varieties of groups, b) the quasivariety of all $RN$-groups and c) the quasivariety generated by all periodic groups, have the cardinality of the continuum. It is shown that the smallest quasivariety of groups containing all proper quasivarieties of groups in which nontrivial quasi-identities of the form
$$ (\forall x_1)…(\forall x_n)(f(x_1,…,x_n)=1\to g(x_1,…,x_n)=1), $$
are true, where $f$ and $g$ are terms of group signature, does not coincide with the class of all groups.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1989, 33:1, 201–207

Bibliographic databases:

UDC: 512.543
MSC: 20E10
Received: 26.11.1986

Citation: A. I. Budkin, “On filters in the lattice of quasivarieties of groups”, Izv. Akad. Nauk SSSR Ser. Mat., 52:4 (1988), 875–881; Math. USSR-Izv., 33:1 (1989), 201–207

Citation in format AMSBIB
\Bibitem{Bud88}
\by A.~I.~Budkin
\paper On~filters in the lattice of quasivarieties of groups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1988
\vol 52
\issue 4
\pages 875--881
\mathnet{http://mi.mathnet.ru/izv1208}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=966988}
\zmath{https://zbmath.org/?q=an:0668.20024|0656.20032}
\transl
\jour Math. USSR-Izv.
\yr 1989
\vol 33
\issue 1
\pages 201--207
\crossref{https://doi.org/10.1070/IM1989v033n01ABEH000821}


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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, “Semivarieties of nilpotent groups”, Algebra and Logic, 49:5 (2010), 389–399  mathnet  crossref  mathscinet  zmath  isi  elib
    2. A. V. Kravchenko, A. M. Nurakunov, M. V. Schwidefsky, “Structure of Quasivariety Lattices. I. Independent Axiomatizability”, Algebra and Logic, 57:6 (2019), 445–462  mathnet  crossref  crossref  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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