Duality Principle (Order Theory)/Local Duality

This proof is about Local Duality in the context of Order Theory. For other uses, see Duality Principle.

Theorem

Let $\Sigma$ be a statement about ordered sets (whether in natural or a formal language).

Let $\Sigma^*$ be the dual statement of $\Sigma$.

Let $\struct {S, \preceq}$ be an ordered set, and let $\struct {S, \succeq}$ be its dual.

Then the following are equivalent:

$(1): \quad \Sigma$ is true for $\struct {S, \preceq}$

$(2): \quad \Sigma^*$ is true for $\struct {S, \succeq}$

By assumption, $\Sigma$ is true for $\struct {S, \preceq}$.

By Dual of Dual Ordering, the dual statement $\Sigma^*$ applied to $\struct {S, \succeq}$ is the same as $\Sigma$ applied to $\struct {S, \preceq}$.

Hence $\Sigma^*$ is true for $\struct {S, \succeq}$.

$\Box$

From Dual of Dual Statement (Order Theory), $\paren {\Sigma^*}^* = \Sigma$.

From Dual of Dual Ordering, $\struct {S, \preceq}$ is the dual of $\struct {S, \succeq}$.

The result thus follows from applying the other implication to $\Sigma^*$ and $\struct {S, \succeq}$.

$\blacksquare$