Definition:Dual Statement (Order Theory)

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $\succeq$ be the dual ordering to $\preceq$.

Let $\Sigma$ be any statement pertaining to $\left({S, \preceq}\right)$ (be it in natural language or a formal language).

The dual statement of $\Sigma$, denoted $\Sigma^*$, is the statement obtained from replacing every reference to $\preceq$ in $\Sigma$ with a reference to its dual $\succeq$.

This dual statement may then be turned into a statement about $\preceq$ again by applying the equivalences on Dual Pairs (Order Theory).

Duality
The fact that a dual statement may be interpreted as a statement about the original ordering $\preceq$ again gives rise to a meta-concept referred to as duality.

Duality (for ordered sets) states that a theorem about ordered sets is true iff its dual is true.

A precise interpretation of this claim, and its proof, are found on Duality Principle (Order Theory).

Example
Consider the following statements:


 * $a \preceq c \land b \preceq c$
 * $\forall d: a \preceq d \land b \preceq d \implies c \preceq d$

expressing that $c$ is the supremum of $\left\{{a, b}\right\}$.

Their dual statements are seen to be:


 * $a \succeq c \land b \succeq c$
 * $\forall d: a \succeq d \land b \succeq d \implies c \succeq d$

which by expanding the definition of the dual ordering $\succeq$, can be written as:


 * $c \preceq a \land c \preceq a$
 * $\forall d: d \preceq a \land d \preceq b \implies d \preceq c$

These statements express precisely that $c$ is the infimum of $\left\{{a, b}\right\}$.

Thus, the dual statement to $c = \sup \left\{{a, b}\right\}$ is $c = \inf \left\{{a, b}\right\}$.

Also see

 * Dual Ordering
 * Duality Principle (Order Theory)


 * Dual Statement (Category Theory), a more general approach to duality