Gurchenkov, S. A. Varieties of \(\ell\)-groups with infinite axiomatic rank. (English. Russian original) Zbl 0576.06018 Sib. Math. J. 26, 50-54 (1985); translation from Sib. Mat. Zh. 26, No. 1(149), 66-70 (1985). The author constructs the variety of nilpotent metabelian lattice ordered groups of the degree 3 with infinite axiomatic rank. It is shown that the lattice of varieties of nilpotent \(\ell\)-groups of degree n (n\(\geq 3)\) has the cardinality of the continuum. There are introduced some corollaries for \(\ell\)-ordered Lie algebras. Reviewer: B.Smarda Cited in 1 Document MSC: 06F15 Ordered groups 08B15 Lattices of varieties 17B30 Solvable, nilpotent (super)algebras 06B20 Varieties of lattices Keywords:variety of nilpotent metabelian lattice ordered groups; infinite axiomatic rank; lattice of varieties; \(\ell \)-ordered Lie algebras PDFBibTeX XMLCite \textit{S. A. Gurchenkov}, Sib. Math. J. 26, 50--54 (1985; Zbl 0576.06018); translation from Sib. Mat. Zh. 26, No. 1(149), 66--70 (1985) Full Text: DOI References: [1] A. Yu. Ol’shanskii, ?On the problem of a finite basis of identities in groups,? Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 2, 316-384 (1970). [2] M. R. Vaughan-Lee, ?Uncountable many varieties of groups,? Bull. London Math. Soc.,2, No. 6, 280-286 (1970). · Zbl 0216.08401 · doi:10.1112/blms/2.3.280 [3] V. M. Kopytov and N. Ya. Medvedev, ?Varieties of lattice-ordered groups,? Algebra Logika,16, No. 4, 417-423 (1977). · Zbl 0395.17015 · doi:10.1007/BF01670004 [4] J. Martinez, ?Varieties of lattice-ordered groups,? Math. Z.,137, No. 4, 265-284 (1974). · Zbl 0281.20023 · doi:10.1007/BF01214370 [5] M. I. Kargapolov and Yu. I. Merzlyakov, Basic Group Theory [in Russian], Nauka, Moscow (1972). [6] A. I. Kokorin and V. M. Kopytov Linearly Ordered Groups [in Russian], Nauka, Moscow (1972). [7] A. M. Glass, W. C. Holland, and S. H. McCleary, ?The structure of Z-group varieties,? Algebra Univ.,10, 1-20 (1980). · Zbl 0439.06013 · doi:10.1007/BF02482885 [8] V. M. Kopytov, ?Lattice-ordered Lie algebras,? Sib. Mat. Zh.,18, No. 3, 595-607, (1977). · Zbl 0365.06010 [9] V. M. Kopytov, ?Ordered Lie algebras,? Algebra Logika,11, No. 3, 295-325 (1972). · Zbl 0253.17006 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.