# Duality Principle (Order Theory)

This proof is about Duality Principle 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 $\Sigma$.

### Local Duality

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 $\struct {S, \succeq}$

### Global Duality

The following are equivalent:

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