Duality Principle (Order Theory)

This proof is about the Duality Principle 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 $\Sigma$.

Local Duality

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 $\left({S, \succeq}\right)$

Global Duality

The following are equivalent:

$(1): \quad \Sigma$ is true for all ordered sets
$(2): \quad \Sigma^*$ is true for all ordered sets