RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

 Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 4, Pages 19–36 (Mi izv250)

Embedding lattices in lattices of varieties of groups

M. I. Anokhin

M. V. Lomonosov Moscow State University

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.

DOI: https://doi.org/10.4213/im250

Full text: PDF file (1630 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 1999, 63:4, 649–665

Bibliographic databases:

MSC: 20E10, 20F16, 08B15, 20F05

Citation: M. I. Anokhin, “Embedding lattices in lattices of varieties of groups”, Izv. RAN. Ser. Mat., 63:4 (1999), 19–36; 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. RAN. Ser. Mat. \yr 1999 \vol 63 \issue 4 \pages 19--36 \mathnet{http://mi.mathnet.ru/izv250} \crossref{https://doi.org/10.4213/im250} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1717677} \zmath{https://zbmath.org/?q=an:0966.20015} \transl \jour Izv. Math. \yr 1999 \vol 63 \issue 4 \pages 649--665 \crossref{https://doi.org/10.1070/im1999v063n04ABEH000250} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000084502900002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746825577}