Definition:Dual Statement (Order Theory)

Definition

Let $\struct {S, \preceq}$ be an ordered set.

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

Let $\Sigma$ be any statement pertaining to $\struct {S, \preceq}$ (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 if and only if 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:

$\paren {a \preceq c} \land \paren {b \preceq c}$

$\forall d: \paren {a \preceq d} \land \paren {b \preceq d} \implies c \preceq d$

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

Their dual statements are seen to be:

$\paren {a \succeq c} \land \paren {b \succeq c}$

$\forall d: \paren {a \succeq d} \land \paren {b \succeq d} \implies c \succeq d$

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

$\paren {c \preceq a} \land \paren {c \preceq a}$

$\forall d: \paren {d \preceq a} \land \paren {d \preceq b} \implies d \preceq c$

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

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

Also see

• Dual Ordering
• Dual Pairs, which makes it easier to determine dual statements.
• Duality Principle

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