Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group $G$ onto a space $R(G:H)$ of right cosets of $G$ w.r.t. some subgroup $H$ of $G$. Examples are constructed showing that the condition of being convex for $H$ in $G$ is insufficient for this. A necessary and sufficient condition (in terms of a subgroup $H$ and a positive cone $P$ of $G$) is specified under which an order of $G$ can be induced onto $R(G:H)$. Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups $G$ for which $R(G:H)$ is partially ordered for every convex subgroup $H$, and properties of the class of groups such that $R(G:H)$ is partially ordered for every partial right order $P$ on $G$ and every subgroup $H$ that is convex under $P$.