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On the theory of varieties of lattice ordered groups. (English. Russian original) Zbl 0679.20022
Algebra Logic 27, No. 3, 153-167 (1988); translation from Algebra Logika 27, No. 3, 249-273 (1988).
Let the natural number $$n=p_ 1^{n_ 1}p_ 2^{n_ 2}...p_ r^{n_ r}$$ be the product of prime numbers, where $$p_ 1,p_ 2,...,p_ r$$ are distinct prime numbers, $$n_ i\geq 1$$ $$(i=1,2,...,r)$$, $$\bar n=n_ 1+n_ 2+...+n_ r+1$$ and $${\mathfrak L}_ n$$ be the $$\ell$$- variety defined by the law $$[x^ n,y^ n]=e$$. In this paper the following main results are proved. 1) $${\mathfrak L}_ n\subseteq ({\mathfrak A}_{\ell})^{\bar n}$$, where $${\mathfrak A}_{\ell}$$ is the $$\ell$$- variety of all abelian $$\ell$$-groups (Theorem 1). 2) Let $${\mathfrak N}$$ be an $$\ell$$-variety and every linearly ordered group from the $$\ell$$- variety $${\mathfrak N}$$ is abelian. Then there exists a natural number $$n=n({\mathfrak N})$$ such that $${\mathfrak N}\subseteq {\mathfrak L}_ n$$ (Theorem 2). 3) The lattice of all $$\ell$$-subvarieties of the $$\ell$$-variety $${\mathfrak L}_ n\wedge ({\mathfrak A}_{\ell})^ 2$$ is described and it is proved: a) every $$\ell$$-variety $${\mathfrak L}\subseteq {\mathfrak L}_ n\wedge ({\mathfrak A}_{\ell})^ 2$$ has a finite basis of identities; b) if the $$\ell$$-variety $${\mathfrak L}$$ has finite basis rank, then the lattice of all $$\ell$$-subvarieties of $${\mathfrak L}$$ is finite (Theorems 6.7). 4) An $$\ell$$-variety of nilpotent $$\ell$$-groups of nilpotency class 3 with finite axiomatic rank and without independent basis of identities is constructed (Theorem 8). 5) The existence of linearly ordered nilpotent groups with the property $$var_{\ell}G\neq var_{\ell}G^*$$, where $$G^*$$ is the Malcev completion of the nilpotent group G, is established (Theorem 9).
Reviewer: N.Medvedev

##### MSC:
 20E10 Quasivarieties and varieties of groups 20F60 Ordered groups (group-theoretic aspects) 06F15 Ordered groups 06B20 Varieties of lattices 20F18 Nilpotent groups 08B15 Lattices of varieties
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##### References:
 [1] J. Martinez, ”Free products in varieties of lattice ordered groups,” Czech. Math. J.,22, No. 4, 535–553 (1972). · Zbl 0247.06022 [2] E. B. Scrimger, ”A large class of small varieties of lattice-ordered groups,” Proc. Am. Math. Soc.,51, No. 2, 301–306 (1975). · Zbl 0312.06010 [3] C. D. Fix, ”On the Scrimger varieties of lattice ordered groups,” Algebra Univ.,16, No. 2, 163–166 (1983). · Zbl 0521.06017 [4] S. A. Gurchenkov, ”Varieties of -groups with the identity [xp, yp] = e are finitely based,” Algebra Logika,23, No. 1, 27–47 (1984). [5] W. C. Holland, A. H. Mekler, and N. R. Reily, ”Varieties of lattice ordered groups in which prime powers commute,” Preprint (1984). [6] S. A. Gurchenkov, ”On varieties of -groups in which linearly ordered groups are Abelian,” 20th All-Union Symp. on Group Theory, Gomel (1986), p. 69. [7] S. A. Gurchenkov, ”On the finite basis property of certain -varieties,” 9th All-Union Symp. on Group Theory, Moscow (1984), p. 194. · Zbl 0545.06008 [8] N. R. Reily, ”Varieties of lattice ordered groups that contain no non-Abelian o-groups are solvable,” Abstr. Am. Math. Sic.,7, No. 2, 223 (1986). [9] V. M. Kopytov and N. Ya. Medvedev, ”On varieties of lattice ordered groups,” Algebra Logika,16, No. 4, 417–423 (1977). · Zbl 0395.17015 [10] T. Feil, ”An uncountable tower of -group varieties,” Algebra Univ.,14, No. 1, 129–131 (1982). · Zbl 0438.06003 [11] N. Ya. Medvedev, ”-varieties without an independent basis of identities,” Math. Slovaca,32, No. 4, 417–425 (1982). · Zbl 0503.06018 [12] N. Ya. Medvedev, ”On coverings in a lattice of -varieties,” Algebra Logika,22, No. 1, 53–60 (1983). [13] S. A. Gurchenkov, ”On varieties of nilpotent lattice ordered groups,” Algebra Logika,21, No. 5, 499–510 (1982). [14] S. A. Gurchenkov, ”On varieties of -groups with infinite axiomatic rank,” Sib. Mat. Zh.,26, No. 1, 66–70 (1985). · Zbl 0576.06018 [15] V. M. Kopytov, Lattice Ordered Groups [in Russian], Nauka, Moscow (1984). · Zbl 0567.06011 [16] M. I. Kargapolov and Yu. I. Merzlyakov, Foundations of Group Theory [in Russian], 3rd ed., Nauka, Moscow (1982). · Zbl 0508.20001 [17] S. A. Gurchenkov, ”On coverings in a lattice of -varieties,” Mat. Zametki,35, No. 5, 677–684 (1984). · Zbl 0545.06008 [18] S. A. Gurchenkov and V. M. Kopytov, ”On a description of coverings of the variety of Abelian lattice ordered groups,” Sib. Mat. Zh.,28, No. 3, 66–69 (1987). · Zbl 0622.06014 [19] K. A. Baker, ”Primitive satisfaction and equational problems for lattices and other algebras,” Trans. Am. Math. Soc.,190, No. 1, 125–150 (1974). · Zbl 0291.08001
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