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The lattice of quasivarieties of torsion-free metabelian groups. (Russian, English) Zbl 1030.08003
Algebra Logika 42, No. 2, 161-181 (2003); translation in Algebra Logic 42, No. 2, 92-104 (2003).
Let \(M\) be a quasivariety and let \(L_q(M)\) be the lattice of quasivarieties in \(M\). The author denotes by \(F_2(A^2)\) a free metabelian group on two generators and by \(F_2(N_2)\) a free nilpotent group of degree two on two generators.
Theorem 2. If a quasivariety \(M\) contains the groups \(F_2(A^2)\) and \(F_2(N_2)\) then \(L_q(M)\) is not a modular lattice.
Theorem 3. Under the conditions of Theorem 2 the lattice \(L_q(M)\) is infinite.
MSC:
08C15 Quasivarieties
20F18 Nilpotent groups
08B15 Lattices of varieties
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