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Semilinearly and semilattice right ordered groups. (English) Zbl 0955.06009
A right \(\vee \)-group is a right partially ordered group \((G,\cdot , \leq , e)\) such that \((G,\leq)\) is an upper semilattice. \(G\) is semilinear if it is updirected and for all \(a, x, y \in G\), \(a\leq x\) and \(a\leq y\) imply \(x\leq y\) or \(y\leq x\). In this paper it is proved that any semilinear right \(\vee \)-group \(G\) is isolated, i.e., for all \(n\in \mathbb N\) and \(a\in G\), \(a^n\geq e\) implies \(a\geq e\). It is also proved that any convex \(\vee \)-subgroup of a semilinear \(\vee \)-group is 2-isolated.
06F15 Ordered groups
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