Duality Principle (Order Theory)/Global Duality

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This proof is about Global Duality in 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 $\left({S, \preceq}\right)$ be an ordered set, and let $\left({S, \succeq}\right)$ be its dual,

By assumption, $\Sigma$ is true for $\left({S, \succeq}\right)$.

By Local Duality, this implies $\Sigma^*$ is true for $\left({S, \preceq}\right)$.

Since $\left({S, \preceq}\right)$ was arbitrary, the result follows.

$\blacksquare$


$(2)$ implies $(1)$

From Dual of Dual Statement (Order Theory):

$\left({\Sigma^*}\right)^* = \Sigma$

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

$\blacksquare$


Also see