Hasse Diagram/Examples/Parallel Lines

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Example of Hasse Diagram

Recall this partial ordering on the set of lines:

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 $S$.


Hasse-Diagram-Parallel-Lines.png

This Hasse diagram illustrates the restriction of $\LL$ to the set of all infinite straight lines in the cartesian plane which are parallel to and one unit away from either the $x$-axis or the $y$-axis.


Source of Name

This entry was named for Helmut Hasse.


Sources