On centrally orderable groups

In this paper we consider the closure of C and C2 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 G˜ belonging to C so that each central order on G can be extended to G˜. 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.