A note on lattice orderability of groups
In this elegant paper the author proves the Theorem: Lattice orderability is not a residual property; i.e., there exists a group G containing a family of normal subgroups \(\{N_ i: i\in I\}\) such that (i) \(\cap \{N_ i: i\in I\}=\{1\}\), (ii) \(G/N_ i\) is a lattice orderable group for all \(i\in I\), but (iii) G is not a lattice orderable group. The proof hinges on the obvious fact that in any lattice orderable group \(x^ 2=y^ 2\) implies \((x\vee y)x=y(x\vee y)\); so any group G which satisfies (i) \& (ii) but for which there are x \& y which are not conjugate yet \(x^ 2=y^ 2\) will be an example. The author constructs one. Note that \textit{A. A. Vinogradov} and \textit{A. A. Vinogradov} [Algebra Logika 8, 636- 639 (1969; Zbl 0264.06013)] had previously proved lattice orderability is not a local property; i.e., all finitely generated subgroups are lattice orderable but the entire group is not.
scientific article; zbMATH DE number 3898953