Linearly ordered groups whose system of convex subgroups is central

The order $P$ on a group $G$ is called rigid if for $p\in P$ the relation $p|[x,p]|^\varepsilon\in P$ holds for every $x\in G$, $\varepsilon=\pm1$ In this note we give criteria for the existence of a rigid linear order, for the extendability of a rigid partial order to a rigid linear order, and for the extendability of each rigid partial order to a rigid linear order on a group. It is proved that the class of groups each of whose rigid partial orders can be extended to a rigid linear order is closed with respect to direct products. A new proof of the theorem of M. I. Kargapolov which states that if a group $G$ can be approximated by finite $p$-groups for infinite number of primes $p$, then it has a central system of subgroups with torsion-free factors is presented.