# Definition:Lattice Ordering

## Definition

Let $\struct {S, \preceq}$ be a lattice.

Then the ordering $\preceq$ is referred to as a **lattice ordering**.

## Examples

### 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**.

## Also see

- Results about
**lattice theory**can be found**here**.

## Sources

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