# Dual of Lattice Ordering is Lattice Ordering

## Theorem

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

Let $\preccurlyeq$ be the lattice ordering on $\struct {S, \preccurlyeq}$.

Then its dual ordering $\succcurlyeq$ is also a lattice ordering.

## Proof

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

It is to be shown that:

- for all $x, y \in S$, the ordered set $\struct {\set {x, y}, \succcurlyeq}$ admits both a supremum and an infimum.

Let $x, y \in S$.

Then $\struct {\set {x, y}, \preccurlyeq}$ admits both a supremum and an infimum.

Let $c = \map \sup {\set {x, y}, \preccurlyeq}$.

Then by definition of supremum:

- $\forall s \in \set {x, y}: s \preccurlyeq c$
- $\forall d \in S: c \preccurlyeq d$

where $d$ is an upper bound of $\struct {\set {x, y}, \preccurlyeq} \subseteq S$.

Hence by definition of dual ordering:

- $\forall s \in \set {x, y}: c \succcurlyeq s$
- $\forall d \in S: d \succcurlyeq c$

where $d$ is an upper bound of $\struct {\set {x, y}, \preccurlyeq} \subseteq S$.

By Upper Bound is Lower Bound for Inverse Ordering, $d$ is a lower bound of $\struct {\set {x, y}, \succcurlyeq} \subseteq S$.

So by definition of infimum:

- $c = \map \inf {\set {x, y}, \succcurlyeq}$

That is, $\struct {\set {x, y}, \succcurlyeq}$ admits an infimum.

$\Box$

Let $c = \map \inf {\set {x, y}, \preccurlyeq}$.

Then by definition of infimum:

- $\forall s \in \set {x, y}: c \preccurlyeq s$
- $\forall d \in S: d \preccurlyeq c$

where $d$ is a lower bound of $\struct {\set {x, y}, \preccurlyeq} \subseteq S$.

Hence by definition of dual ordering:

- $\forall s \in \set{x, y}: s \succcurlyeq c$
- $\forall d \in S: c \succcurlyeq d$

where $d$ is a lower bound of $\struct {\set {x, y}, \preccurlyeq} \subseteq S$.

By Lower Bound is Upper Bound for Inverse Ordering, $d$ is an upper bound of $\struct {\set {x, y}, \succcurlyeq} \subseteq S$.

So by definition of supremum:

- $c = \map \sup {\set {x, y}, \succcurlyeq}$

That is, $\struct {\set {x, y}, \succcurlyeq}$ admits a supremum.

$\Box$

Hence $\struct {\set {x, y}, \succcurlyeq}$ admits both a supremum and an infimum.

That is, $\succcurlyeq$ is a lattice ordering.

$\blacksquare$

## Sources

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