Partial Orders on Dlab Groups

Abstract

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: $D_H \left( I \right),D_{H_* } \left( I \right)D_{_* H} \left( I \right)$, and $\bar D_H \left( I \right)$ of the unit interval $I = \left[ {0,1} \right]$ and $D_H$ and $D_{H_* }$ of the extended real line ${\bar R}$. More precisely, first, every group that is isomorphically embeddable in one of the above-mentioned Dlab groups lacks non-trivial minimal partial orders; second, $D_H \left( I \right)$ and $D_H$ have 4 maximal isolated partial orders and 4 non-trivial minimal isolated partial orders; third, $D_{H_* } \left( I \right)$, $D_{H_* } \left( I \right)$, and $D_{H_* }$ have 10 maximal partial orders and 8 non-trivial minimal isolated partial orders; fourth, $D_H \left( I \right)$ has 16 non-trivial minimal isolated partial orders and 40 maximal partial orders.