Dual of Lattice Ordering is Lattice Ordering

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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

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