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Ordered groups in which all convex jumps are central. (English) Zbl 1031.20036
A linearly ordered group $$G$$ is centrally ordered if $$[G,D]\subseteq C$$ for every convex jump $$C\prec D$$ in $$G$$ or equivalently, if $$f^{-1}gf\leq f^2$$ for all $$f,g$$ with $$g>1$$. It is proved that if every order on every two-generator subgroup of a locally soluble orderable group is central, then $$G$$ is locally nilpotent. An example of a non-nilpotent two-generator metabelian orderable group in which all orders are central is constructed.

##### MSC:
 20F60 Ordered groups (group-theoretic aspects) 06F15 Ordered groups 20F19 Generalizations of solvable and nilpotent groups 20F14 Derived series, central series, and generalizations for groups 20E26 Residual properties and generalizations; residually finite groups
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