Lattice Ordering/Examples/Parallel Lines
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Example of Ordering which is not Lattice Ordering
Recall the partial ordering on the set of straight 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 $S$.
$S$ is not a lattice ordering.
Proof
Two straight lines admit a supremum and infimum when parallel.
Hence, by definition, $S$ is not a lattice ordering.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Example $14.2$