Nearly abelian, nilpotent, and Engel lattice-ordered groups.

*(English)*Zbl 0856.06011From 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.

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.

Reviewer: František Šik (Brno)

##### MSC:

06F15 | Ordered groups |

20F18 | Nilpotent groups |

20F45 | Engel conditions |

20F60 | Ordered groups (group-theoretic aspects) |