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Some subgroups of semilinearly ordered groups. (Russian, English) Zbl 1051.06013
Algebra Logika 39, No. 4, 465-479 (2000); translation in Algebra Logic 39, No. 4, 268-275 (2000).
Summary: Let $$G$$ be a semilinearly ordered group with a positive cone $$P$$. Denote by $$n(G)$$ the greatest convex directed normal subgroup of $$G$$, by $$o(G)$$ the greatest convex right-ordered subgroup of $$G$$, and by $$r(G)$$ a set of all elements $$x$$ of $$G$$ such that $$x$$ and $$x^{-1}$$ are comparable with any element of $$P^\pm$$ (the collection of all group elements comparable with an identity element). Previously, it was proved that $$r(G)$$ is a convex right-ordered subgroup of $$G$$, and $$n(G)\subset r(G)\subset o(G)$$. Here, we establish a new property of $$r(G)$$ and show that the inequalities in the given system of inclusions are, generally, strict.
##### MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects)
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