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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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