Partial right orders on periodic extensions of abelian groups (Q5967041)

It is shown that every semilinearly ordered group that is an extension of an abelian group by a periodic group is right-ordered. A series of new examples of not fully right-orderable groups is presented and, moreover, it is proven that every nonabelian soluble abelian-by-finite right-orderable group is not fully right-orderable. As a corollary, it is shown that the axiomatic closure of the class of fully right-orderable groups does not coincide with the class of right-orderable groups.