It is proved, that for the right-ordered $Z-A$-group $Q$ the following four properties are equivalent: 1 ) the group $Q$ is archimedean, 2 ) the group $Q$ has no proper convex subgroups, 3 ) in the group $Q$ all abelian subgroups are archimedean, 4) the group $Q$ has the archimedean embedded centre $Z$, i.e . $(\forall q\in Q, \forall z\in Z)\ q>z>1\to (\exists n>0)\ z^n>q$. In the paper [1] it was demonstrated the example of the right-ordered metabelian group, which has the properties 2) and 3), but is not archimedean.