Geodesic
Embedding lattices in lattices of varieties of groups
Izvestiya. Mathematics , Tome 63 (1999) no. 4, pp. 649-665.

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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. 
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     author = {M. I. Anokhin},
     title = {Embedding lattices in lattices of varieties of groups},
     journal = {Izvestiya. Mathematics },
     pages = {649--665},
     publisher = {mathdoc},
     volume = {63},
     number = {4},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a1/}
}
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M. I. Anokhin. Embedding lattices in lattices of varieties of groups. Izvestiya. Mathematics , Tome 63 (1999) no. 4, pp. 649-665. http://geodesic.mathdoc.fr/item/IM2_1999_63_4_a1/

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