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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

Citations:

Zbl 0551.06017
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