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Quasi-Engel varieties of lattice-ordered groups. (English) Zbl 07673629

Summary: We show that any ordered group satisfying the identity \([x_1^{k_1}, \dots, x_n^{k_n}] = e\) must be weakly abelian and that when \(x_i \neq x_1\) for \(2 \leq i \leq n\), \(\ell\)-groups satisfying the identity \([x_1^n, \dots, x_k^n] = e\) also satisfy the identity \((x \vee e)^{y^n} \leq (x \vee e)^2\). These results are used to study the structure of \(\ell\)-groups satisfying identities of the form \([x_1^{k_1}, x_2^{k_2}, x_3^{k_3}] = e\).

MSC:

06F15 Ordered groups
20F19 Generalizations of solvable and nilpotent groups
20F45 Engel conditions
20F60 Ordered groups (group-theoretic aspects)
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