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On coverings in the lattice of \(\ell\)-varieties. (English. Russian original) Zbl 0545.06008

Math. Notes 35, 356-360 (1984); translation from Mat. Zametki 35, No. 5, 677-684 (1984).
The author proves that there exists a covering of any proper \(\ell\)- variety V in the lattice of all \(\ell\)-varieties. Also, if \(V\neq N\), E, where N is the \(\ell\)-variety of all representable \(\ell\)-groups and E the trivial \(\ell\)-variety, then there exist infinite many coverings of V. To any variety of solvable \(\ell\)-groups there exists a solvable covering.
Reviewer: F.Šik

MSC:

06F15 Ordered groups
08B15 Lattices of varieties
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References:

[1] N. Ya. Medvedev, ?On the lattice of varieties of lattice ordered groups and Lie algebras,? Algebra Logika,16, No. 1, 40-45 (1977).
[2] E. B. Scrimger, ?A large class of small varieties of lattice ordered groups,? Proc. Am. Math. Soc.,51, 301-306 (1975). · Zbl 0312.06010
[3] A. I. Kokorin and V. M. Kopytov, Linearly Ordered Groups [in Russian] (1972). · Zbl 0192.36401
[4] A. M. Glass, W. C. Holland, and S. H. McCleary, ?The structure of group varieties,? Algebra Univ.,10, 1-20 (1980). · Zbl 0439.06013
[5] C. Holland, ?Each variety ofl-groups is a torsion class,? Czech. Math. J.,29, 11-12 (1979). · Zbl 0432.06011
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