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:

$a \mathrel \LL b$ if and only if:

$a$ 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:

$a \mathrel {\LL^{-1} } b$ if and only if:

$a$ 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 $S$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings