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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.

06F15 Ordered groups
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