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Convex directed subgroups of right ordered tree groups. (English) Zbl 0803.06020
An ordered set $$(T,\leq)$$ is called a tree if (1) $$\forall a,b\in T$$, $$\exists c\in T$$; $$a,b\leq c$$; (2) $$\forall a,x,y\in T$$; $$a\leq x$$, $$y\Rightarrow x\leq y$$ or $$y\leq x$$. If $$G$$ is a right ordered group such that $$(G,\leq)$$ is a tree, then $$G$$ is called a $$tr$$-group. A $$tr$$-group $$G$$ is called an $$str$$-group if $$a\leq b$$ implies $$ca\leq cb$$ for all $$a,b\in G$$ and $$c$$ in the positive cone of $$G$$. In the paper, some structure properties of $$tr$$-groups and $$str$$-groups are studied. A typical result: If $$G$$ is a $$tr$$-group such that the normal system of convex up-directed subgroups of $$G$$ is solvable, then $$G$$ is right linearly ordered.

##### MSC:
 06F15 Ordered groups
##### Keywords:
tree group; right ordered group
##### References:
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