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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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