Definition:Dual Statement (Order Theory)

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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