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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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