Ordered right cosets of a convex lattice subgroup ($Pg\le Ph⟺g\le ph\text{ for some }p\in P$)

The displayed relation makes the right cosets into a lattice, with $Pg\vee Ph=P(g\vee h)$ and $Pg\wedge Ph=P(g\wedge h)$. Antisymmetry uses convexity: $g\le ph$ and $h\le qg$ give $1\le ph{g}^{-1}\le pq\in P$, hence $h{g}^{-1}\in P$. A prime convex lattice subgroup makes this order total even if the subgroup is not normal.