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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.
Reviewer: P.F.Conrad

MSC:
06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
20E10 Quasivarieties and varieties of groups
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References:
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