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Nearly abelian, nilpotent, and Engel lattice-ordered groups. (English) Zbl 0856.06011
From the authors’ abstract: “In this paper, we examine those classes of lattice-ordered groups in which every substitution produces a group element comparable to the group identity, and, under certain natural conditions, obtain a description of the structure of such lattice-ordered groups in terms of the radical of the corresponding $$\ell$$-variety. We especially concentrate on those sets of words which produce the $$\ell$$-varieties of Abelian, nilpotent, Engel, and solvable lattice-ordered groups.”
In more detail: An $$\ell$$-group $$G$$ is nearly-$$\nu$$ with respect to an equational basis $$\{w_n(\vec x)\}$$ of an $$\ell$$-variety $$\nu$$ of $$\ell$$-groups if for any substitution $$\vec x\to \vec g$$ into $$G$$, $$w_n(\vec g)\diamondsuit e$$ ($$a\diamondsuit b$$ denotes that $$a$$ is comparable to $$b$$). An $$\ell$$-group word $$w(\vec x)$$ is balanced if any $$\ell$$-group $$G$$ satisfies the condition that for any substitution $$x_{ijk}\to g_{ijk}$$ into $$G$$ with $$w(\vec g)> e$$ there exists a substitution $$x_{ijk}\to h_{ijk}$$ into the $$\ell$$-subgroup generated by $$\{g_{ijk}\}$$ such that $$w(\vec h)< e$$, and vice versa. An $$\ell$$-group word that is not balanced will be called unbalanced. An $$\ell$$-variety $$\nu$$ has a balanced basis if there exists an equational basis of balanced $$\ell$$-group words for $$\nu$$. It is not known whether every $$\ell$$-variety has a balanced basis. The following theorem holds: Let $$\nu$$ be a normal-valued $$\ell$$-variety of lattice-ordered groups with equational basis $$\{w_\lambda(\vec x)\}$$. Let $$G$$ be nearly-$$\nu$$ with respect to $$\{w_\lambda(\vec x)\}$$ and $$\Delta$$ be a normal plenary subset of $$\Gamma(G)$$. Then $$G$$ can be $$\ell$$-embedded into a special-valued $$\ell$$-group $$H$$ that is also nearly-$$\nu$$ with respect to $$\{w_\lambda(\vec x)\}$$.
Let us denote by $$A$$, $$N_k$$, $$E_k$$ and $$A_k$$, the following $$\ell$$-varieties: Abelian, nilpotent of class $$k$$, Engel of bound $$k$$, solvable of rank $$k$$, respectively, and by $$L_k$$ the powers of the Abelian $$\ell$$-varieties $$A^k$$. For these $$\ell$$-varieties let us refer to the following as their canonical bases: $$A: [x, y]= e$$; $$N_k: [x_1,\dots, x_{k+ 1}]= e$$; $$E_k: [x, y, \dots, y]_k= e$$ (the repeated commutator with $$k$$ occurrences of $$y$$); $$L_k: [x^k, y^k]= e$$. The equational bases for $$A^k$$ are built up recursively by letting $$w_1(y, z)= [x_1, x_2]$$ and $w_{k+ 1} (x_{2k- 1}, x_{2k}, w_k(\vec y), w_k(\vec z))= [|x_{2k- 1}|\wedge |w_k(\vec y)|, |x_{2k}|\wedge |w_k(\vec z)|];$ then the canonical basis for $$A^k$$ is $$w_k(\vec x)= e$$. An obviously similar recursion exists for an equational basis for $$\ell$$-groups that are solvable of rank $$k$$.
Theorem. The canonical bases of the $$\ell$$-varieties $$N_k$$, $$L_k$$, $$A^k$$, $$A_k$$ (for a positive integer $$k$$) and for the Engel $$\ell$$-varieties $$E_2$$ and $$E_3$$ are balanced. It is not known whether the canonical basis for $$E_k$$, $$k> 3$$, is balanced. We only know that an $$\ell$$-group $$G$$ is representable if $$[a, b,\dots, b]_n\diamondsuit e$$ for all $$a, b\in G$$.
There are some more details and an example in which it is shown: If $$\ell$$-group words $$w_1(\vec x)$$ and $$w_2(\vec x)$$ generate the same $$\ell$$-variety $$\nu$$, then an $$\ell$$-group $$G$$ that is nearly-$$\nu$$ with respect to $$w_1(\vec x)$$ need not be nearly-$$\nu$$ with respect to $$w_2(\vec x)$$, even when both $$\ell$$-group words $$w_1(\vec x)$$ and $$w_2(\vec x)$$ are balanced.
##### MSC:
 06F15 Ordered groups 20F18 Nilpotent groups 20F45 Engel conditions 20F60 Ordered groups (group-theoretic aspects)