Duality Principle (Order Theory)/Global Duality

This proof is about Global 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 $\Sigma$.

The following are equivalent:

$(1): \quad \Sigma$ is true for all ordered sets

$(2): \quad \Sigma^*$ is true for all ordered sets

Proof

$(1)$ implies $(2)$

Let $\struct {S, \preceq}$ be an ordered set.

Let $\struct {S, \succeq}$ be its dual.

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

By Local Duality, this implies $\Sigma^*$ is true for $\struct {S, \preceq}$.

Since $\struct {S, \preceq}$ was arbitrary, the result follows.

$\blacksquare$

$(2)$ implies $(1)$

From Dual of Dual Statement (Order Theory):

$\paren {\Sigma^*}^* = \Sigma$

The result thus follows from applying the other implication to $\Sigma^*$.

$\blacksquare$

Also see

• Definition:Dual Statement (Order Theory)
• Duality Principle (Category Theory), a more general duality principle.