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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

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

Bibliographic databases:

MSC: 20E10, 20F16, 08B15, 20F05
Received: 09.06.1997

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
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