Quasi-Engel varieties of lattice-ordered groups

Abstract

We show that any ordered group satisfying the identity \([x_1^{k_1}, \ldots , x_n^{k_n}] = e\) must be weakly abelian and that when \(x_i \not = x_1\) for \(2 \le i \le n\), \(\ell \)-groups satisfying the identity \([x_1^n, \ldots , x_k^n] = e\) also satisfy the identity \((x \vee e)^{y^n} \le (x \vee e)^2\). These results are used to study the structure of \(\ell \)-groups satisfying identities of the form \([x_1^{k_1}, x_2^{k_2}, x_3^{k_3}] = e\).