Partial Orders on Dlab Groups

For every subgroup $H$ of rank 1 in a multiplicative group of positive reals, complete descriptions are furnished for maximal partial orders and for minimal isolated partial orders on the following Dlab groups: $D_H(\mathbf I)$, $D_{H*}(\mathbf I)$, $D_{*H}(\mathbf I)$, and ${\bar D}_H(\mathbf I)$ of the unit interval ${\mathbf I}=[0,1]$ and $D_{H}$ and $D_{H*}$ of the extended real line $\bf\bar R$. More precisely, first, every group that is isomorphically embeddable in one of the above-mentioned Dlab groups lacks non-trivial minimal partial orders; second, $D_H(\mathbf I)$ and $D_H$ have 4 maximal isolated partial orders and 4 non-trivial minimal isolated partial orders; third, $D_{H*}(\mathbf I)$, $D_{*H}(\mathbf I)$, and $D_{H*}$ have 10 maximal partial orders and 8 non-trivial minimal isolated partial orders; fourth, ${\bar D}_H(\mathbf I)$ has 16 non-trivial minimal isolated partial orders and 40 maximal partial orders.