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Filters in the lattice of quasivarieties of metabelian groups. (English. Russian original) Zbl 0905.20017

Sib. Math. J. 39, No. 1, 57-62 (1998); translation from Sib. Mat. Zh. 39, No. 1, 67-73 (1998).
The author gives a negative answer to a question posed by A. I. Budkin: Is every nontrivial filter in the lattice of quasivarieties of torsion-free metabelian groups countable [The Kourovka notebook. Unsolved problems in group theory, 13th ed., Institute of Mathematics, Novosibirsk (1995; Zbl 0838.20001)]. The author proves that each nontrivial filter in the lattice of quasivarieties of torsion-free metabelian groups has the power of the continuum.

MSC:

20E10 Quasivarieties and varieties of groups
08C15 Quasivarieties
08B15 Lattices of varieties
20F16 Solvable groups, supersolvable groups

Citations:

Zbl 0838.20001
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References:

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