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Izvestiya: Mathematics, 1999, Volume 63, Issue 4, Pages 649–665
DOI: https://doi.org/10.1070/im1999v063n04ABEH000250
(Mi im250)
 

Embedding lattices in lattices of varieties of groups

M. I. Anokhin

M. V. Lomonosov Moscow State University
References:
Abstract: If $\mathfrak V$ is a variety of groups and $\mathfrak U$ is a subvariety, then the symbol $\langle\mathfrak U,\mathfrak V\rangle$ denotes the complete lattice of varieties $\mathfrak X$ such that $\mathfrak U\subseteq \mathfrak X\subseteq \mathfrak V$. Let $\Lambda=\mathrm C\prod_{n=1}^\infty\Lambda_n$, where $\Lambda_n$ is the lattice of subspaces of the $n$-dimensional vector space over the field of two elements, and let $\mathrm C\prod$ be the Cartesian product operation. A non-empty subset $K$ of a complete lattice $M$ is called a complete sublattice of $M$ if $\sup_MX\in K$ and $\inf_MX\in K$ for any non-empty $X\subseteq K$.
We prove that $\Lambda$ is isomorphic to a complete sublattice of $\langle\mathfrak A_2^4, \mathfrak A_2^5\rangle$. On the other hand, it is obvious that $\langle\mathfrak U,\mathfrak A_2\mathfrak U\rangle$ is isomorphic to a complete sublattice of $\Lambda$ for any locally finite variety $\mathfrak U$. We deduce criteria for the existence of an isomorphism onto a (complete) sublattice of $\langle\mathfrak U,\mathfrak A_2\mathfrak U\rangle$ for some locally finite variety $\mathfrak U$. We also prove that there is a sublattice $\langle\mathfrak A_2^4,\mathfrak A_2^5\rangle$ generated by four elements and containing an infinite chain.
Received: 09.06.1997
Bibliographic databases:
Language: English
Original paper language: Russian
Citation: M. I. Anokhin, “Embedding lattices in lattices of varieties of groups”, Izv. Math., 63:4 (1999), 649–665
Citation in format AMSBIB
\Bibitem{Ano99}
\by M.~I.~Anokhin
\paper Embedding lattices in lattices of varieties of groups
\jour Izv. Math.
\yr 1999
\vol 63
\issue 4
\pages 649--665
\mathnet{http://mi.mathnet.ru//eng/im250}
\crossref{https://doi.org/10.1070/im1999v063n04ABEH000250}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1717677}
\zmath{https://zbmath.org/?q=an:0966.20015}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000084502900002}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746825577}
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  • https://doi.org/10.1070/im1999v063n04ABEH000250
  • https://www.mathnet.ru/eng/im/v63/i4/p19
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Russian version PDF:183
    English version PDF:10
    References:62
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