# Duality Principle (Order Theory)

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*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

## Also see

- Dual Statement
- Dual Pairs, which allows for easier determination of dual statements.
- Duality Principle (Category Theory), a more general duality principle.