Lattice Ordering/Examples
Examples of Lattice Orderings
Power Set is Lattice
Let $S$ be a set.
Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the subset relation $\subseteq$.
Then $\struct {\powerset S, \subseteq}$ is a lattice.
Parallel Lines
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.
Ancestry
Recall the partial ordering on the set of people:
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 $P$.
$D$ is not a lattice ordering.