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Nilpotent, weakly Abelian and Hamiltonian lattice ordered groups. (English) Zbl 0553.06020
Let $${\mathcal C}$$ be the variety of lattice ordered groups defined by the law $$[x,x^ y]=1$$ and let $${\mathcal W}$$ be the weakly abelian variety defined by $$(x\vee 1)^ y\leq (x\vee 1)^ 2$$. Let $${\mathcal H}$$ be the class of all lattice ordered groups for which each convex $$\ell$$-subgroup is normal - this Hamiltonian class is not a variety. For a variety $${\mathcal V}$$ of $$\ell$$-groups it is shown that $${\mathcal V}={\mathcal W}$$ iff $${\mathcal V}\leq {\mathcal H}$$, so $${\mathcal W}$$ is the largest variety of Hamiltonian $$\ell$$-groups. Each nilpotent $$\ell$$-group belongs to $${\mathcal W}$$. If G is an o-group with only a finite number of regular subgroups and $$G\in {\mathcal W}$$ then G is nilpotent. The variety $${\mathcal C}$$ is the variety of nilpotent groups of class at most two.
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