Partial Ordering/Examples
Jump to navigation
Jump to search
Examples of Partial Orderings
Arbitrary Example
Let $X = \set {x, y, z}$.
Let $\RR = \set {\tuple {x, x}, \tuple {x, y}, \tuple {x, z}, \tuple {y, y}, \tuple {z, z} }$.
Then $\RR$ is a partial ordering on $X$.
The strict partial ordering on $X$ corresponding to $\RR$ is its reflexive reduction:
- $\RR^{\ne} = \set {\tuple {x, y}, \tuple {x, z} }$
Parallel 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 $S$.
Ancestry
Let $P$ denote the set of all people who have ever lived.
Let $\DD$ denote the relation on $P$ defined as:
- $a \mathrel \DD b$ if and only if $a$ is a descendant of or the same person as $b$.
Its dual $\DD^{-1}$ is defined as:
- $a \mathrel {\DD^{-1} } b$ if and only if $a$ is an ancestor of or the same person as $b$.
Then $\DD$ and $\DD^{-1}$ are partial orderings on $P$.