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

Also see

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