HNN extensions of torsion-free nilpotent groups are left-orderable, but HNN extensions of left-orderable groups are not always left-orderable.
Button, Topics in infinite groups, Lecture Notes
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Examples of left-orderable and non-left-orderable HNN extensions
HNN extensions of torsion-free nilpotent groups are left-orderable, but HNN extensions of left-orderable groups are not always left-orderable.