an:03880782
Zbl 0553.06020
Reilly, Norman R.
Nilpotent, weakly Abelian and Hamiltonian lattice ordered groups
EN
Czech. Math. J. 33(108), 348-353 (1983).
00150800
1983
j
06F15 20F60 20E10
variety of lattice ordered groups; weakly abelian variety; convex \(\ell \)-subgroup; Hamiltonian class; variety of Hamiltonian \(\ell \)-groups; nilpotent \(\ell \)-group; regular subgroups; variety of nilpotent groups
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.
P.F.Conrad