Duality Principle (Order Theory)/Local Duality

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 $\left({S, \preceq}\right)$ be an ordered set, and let $\left({S, \succeq}\right)$ be its dual.

Then the following are equivalent:


 * $(1): \quad \Sigma$ is true for $\left({S, \preceq}\right)$
 * $(2): \quad \Sigma^*$ is true for $\left({S, \succeq}\right)$

$(1)$ implies $(2)$
By assumption, $\Sigma$ is true for $\left({S, \preceq}\right)$.

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

Hence $\Sigma^*$ is true for $\left({S, \succeq}\right)$.

$(2)$ implies $(1)$
From Dual of Dual Statement (Order Theory), $\left({\Sigma^*}\right)^* = \Sigma$.

From Dual of Dual Ordering, $\left({S, \preceq}\right)$ is the dual of $\left({S, \succeq}\right)$.

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

Also see

 * Dual Statement
 * Duality Principle (Category Theory), a more general duality principle.