Ordered right cosets of a convex lattice subgroup
= Ordered right cosets of a convex lattice subgroup
{title2=$Pg\le Ph\iff g\le ph\text{ for some }p\in P$}
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 $1\le phg^{-1}\le pq\in P$, hence $hg^{-1}\in P$. A <prime convex lattice subgroup> makes this order total even if the subgroup is not normal.