On some subgroups of semilinearly ordered groups

Let $G$ be a semilinearly ordered group with a positive cone $P$. Denote by $\mathbf n(G)$ the greatest convex directed normal subgroup of $G$, by $\mathbf o(G)$ the greatest convex right-ordered subgroup of $G$, and by $\mathbf r(G)$ 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 $\mathbf r(G)$ is a convex right-ordered subgroup of $G$, and $\mathbf n(G)\subset\mathbf r(G)\subset\mathbf o(G)$. Here, we establish a new property of $\mathbf r(G)$ and show that the inequalities in the given system of inclusions are, generally, strict.