Geodesic
On one-sided orders in groups with ascending central series
Algebra i logika, Tome 6 (1967) no. 2, pp. 77-88.

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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. 
@article{AL_1967_6_2_a6,
     author = {D. M. Smirnov},
     title = {On one-sided orders in groups with ascending central series},
     journal = {Algebra i logika},
     pages = {77--88},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_1967_6_2_a6/}
}
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D. M. Smirnov. On one-sided orders in groups with ascending central series. Algebra i logika, Tome 6 (1967) no. 2, pp. 77-88. http://geodesic.mathdoc.fr/item/AL_1967_6_2_a6/