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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
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References:
[1] S. A. Adeleke M. A. E. Dummett, P. M. Neumann: On a question of Frege’s about right-ordered groups. Bull. London Math. Soc, 19 (1987), 513-521. · Zbl 0632.06022
[2] G. Frege: Die Grundgesetze der Arithmetik. Band II, Jena 1903. · JFM 34.0071.05
[3] V. M. Kopytov: Lattice Ordered Groups. (in Russian), Moscow 1984. · Zbl 0567.06011
[4] S. Varaksin: Semilinear orders on solvable and nilpotent groups. (in Russian), preprint. · Zbl 0797.20028
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