## Covers in the lattice of varieties of $${\ell}$$-groups.(English. Russian original)Zbl 0902.06028

Algebra Logika 37, No. 3, 253-269 (1998); translation in Algebra Logic 37, No. 3, 141-150 (1998).
An example of an $$o$$-approximable variety of $$\ell$$-groups $${\mathcal {V}}$$ which has no covers in the lattice L of $$o$$-approximable varieties of $$\ell$$-groups was given by N. Ya. Medvedev [Czech. Math. J. 34 (109), 6-17 (1984; Zbl 0551.06017)]. In the article under review it is shown that a variety $${\mathcal {V}}$$ has continuum many covers in the lattice L of varieties of $$\ell$$-groups, and that the same is also true of an arbitrary $$o$$-approximable variety $${\mathcal {X}}$$ with the property $${\mathcal {V}} \subseteq {\mathcal {X}}$$. It is proven that every $$o$$-approximable quasivariety $${\mathcal {Q}}$$ of $$\ell$$-groups, for which $${\mathcal {V}} \subseteq {\mathcal {Q}}$$, has continuum many covers in the quasivariety lattice $$\Lambda$$.

### MSC:

 06F15 Ordered groups 08B15 Lattices of varieties 08B05 Equational logic, Mal’tsev conditions 08C15 Quasivarieties

Zbl 0551.06017
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