# Definition:Dual Statement (Order Theory)

## Contents

## 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
- Dual Pairs, which makes it easier to determine dual statements.
- Duality Principle

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