zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] L. Fuchs, Partially Ordered Algebraic Systems [Russian translation], Mir, Moscow (1965). · Zbl 0192.09603
[2] A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).
[3] W. B. Powell and C. Tcinakis, ”The distributive lattice free product as a sublattice of the Abelian l-group free product,” J. Austral. Math. Soc.,A34, No. 1, 92–100 (1983). · Zbl 0516.06011 · doi:10.1017/S1446788700019789
[4] V. M. Kopytov, ”On lattice ordered locally nilpotent groups,” Algebra Logika,14, No. 4, 407–413 (1975).
[5] V. M. Kopytov and N. Ya. Medvedev, ”On linearly ordered groups whose system of convex subgroups is central,” Mat. Zametki,19, No. 1, 85–90 (1976). · Zbl 0358.06036
[6] A. I. Kokorin and V. M. Kopytov, Linearly Ordered Groups [in Russian], Nauka, Moscow (1972). · Zbl 0192.36401
[7] W. B. Powell and C. Tcinakis, ”Free products of Abelian l-groups are cardinally indecomposable,” Proc. Am. Math. Soc.,86, No. 3, 385–390 (1982). · Zbl 0516.06012
[8] N.Ya. Medvedev, ”On decomposition of free l-groups into l -direct products,” Sib. Mat. Zh.,21, No. 5, 63–69 (1980). · Zbl 0449.06012
[9] A. I. Budkin, ”On quasiidentities in a free group,” Algebra Logika,15, No. 1, 39–52 (1976).
[10] G. Baumslag, ”On generalized free products,” Math. Z.,78, No. 5, 423–438 (1962). · Zbl 0104.24402 · doi:10.1007/BF01195185
[11] R. D. Bleier, ”The SP-hull of a lattice-ordered group,” Can. J. Math.,26, No. 4, 866–878 (1974). · Zbl 0298.06021 · doi:10.4153/CJM-1974-081-x
[12] K. R. Pierce, ”Amalgamation of lattice-ordered groups,” Trans. Am. Math. Soc.,172, No. 4, 249–260 (1972). · Zbl 0259.06017 · doi:10.1090/S0002-9947-1972-0325488-3
[13] A. W. Glass and Y. Gurevich, ”The word problem for lattice-ordered groups,” Trans. Am. Math. Soc.,280, No. 1, 127–138 (1983). · Zbl 0527.06009
[14] V. P. Belkin and V. A. Gorbunov, ”Lattice filters of quasivarieties of algebraic systems,” Algebra Logika,14, No. 4, 373–392 (1975). · Zbl 0328.08005 · doi:10.1007/BF01668815
[15] G. Birkhoff and R. Pierce, ”Lattice ordered rings,” An. Acad. Brasil. Ci.,28, No. 1, 41–69 (1956). · Zbl 0070.26602
[16] J. R. Isbell, ”Notes on ordered rings,” Algebra Univ.,1, No. 3, 393–399 (1972). · Zbl 0238.06013 · doi:10.1007/BF02944999
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.