On centrally orderable groups

In [V.V. Bludov, A.M.W. Glass, A.H. Rhemtulla, Ordered groups in which all convex jumps are central, J. Korean Math. Soc. 40 (2003) 225–239] we considered the class of all orderable groups with all orders having central convex jumps and the class of all orderable groups all of whose two generator subgroups belong to . Both these classes contain all locally nilpotent torsion-free groups. We proved that every soluble-by-finite group belonging to must be locally nilpotent, but there is a two-generator metabelian group belonging to (whence it is not locally nilpotent). In this paper we consider the closure of and under elementary equivalence, direct sums and full Cartesian products, homomorphic images, and residual properties. We further show that every metabelian group G which has a central order can be embedded in a metabelian group belonging to so that each central order on G can be extended to . This is achieved by ascending HNN-extensions. We further show that a finitely generated metabelian group has a central order if and only if it is residually torsion-free nilpotent.