Partial Ordering/Examples/Parallel Lines

Example of Partial Ordering
Let $S$ denote the set of all infinite straight lines embedded in a cartesian plane.

Let $\LL$ denote the relation on $S$ defined as:


 * $a \mathrel \LL b$ :
 * $a$ is parallel $b$
 * if $a$ is not parallel to the $y$-axis, then coincides with or lies below $b$
 * but if $b$ is parallel to the $y$-axis, then $a$ coincides with or lies to the right of $b$

Its dual $\LL^{-1}$ is defined as:


 * $a \mathrel {\LL^{-1} } b$ :
 * $a$ is parallel $b$
 * if $a$ is not parallel to the $y$-axis, then coincides with or lies above $b$
 * but if $b$ is parallel to the $y$-axis, then $a$ coincides with or lies to the left of $b$.

Then $\LL$ and $\LL^{-1}$ are partial orderings on $S$.