Some subgroups of semilinearly ordered groups. (Russian, English) Zbl 1051.06013

Kopytov, V. M.

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)

Keywords:

order normalizer; convex closure; positive cone; quotient group; semilinearly ordered group