Free products of \(\ell\)-groups. (English. Russian original) Zbl 0577.06014
Algebra Logic 23, 336-346 (1984); translation from Algebra Logika 23, No. 5, 493-511 (1984).
From the author’s introduction: The article contains the following main results: It is proved that sublattices of free products in varieties of nilpotent \(\ell\)-groups generated by the free factors are free products of \(D_{\ell}\)-lattices (Theorem 1); it is proved that the free products of finitely generated \(\ell\)-groups in arbitrary varieties of \(\ell\)- groups contained in the variety of \(\ell\)-groups with subnormal jumps are irreducible into an \(\ell\)-direct product (Theorem 2); it is proved that the lattice of quasivarieties of \(\ell\)-groups is not modular (Theorem 3); examples of varieties of associative lattice-ordered rings having no finite basis of identities are constructed (Propositions 1 and 2).
Reviewer: F.Šik
MSC:
| 06F15 | Ordered groups |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20F18 | Nilpotent groups |
| 20F60 | Ordered groups (group-theoretic aspects) |
| 20E10 | Quasivarieties and varieties of groups |
| 06B20 | Varieties of lattices |
| 08B15 | Lattices of varieties |
| 08C15 | Quasivarieties |
Keywords:
sublattices of free products; varieties of nilpotent \(\ell \)-groups; free products of finitely generated \(\ell \)-groups; varieties of \(\ell \)- groups; lattice of quasivarieties; basis of identitiesReferences:
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