# Dual Pairs (Order Theory)

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## Theorem

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

Let $a, b \in S$, and let $T \subseteq S$.

Then the following phrases about, and concepts pertaining to $\left({S, \preceq}\right)$ are dual to one another:

$b \preceq a$ $a \preceq b$ $a$ succeeds $b$ $a$ precedes $b$ $a$ strictly succeeds $b$ $a$ strictly precedes $b$ $a$ is an upper bound for $T$ $a$ is a lower bound for $T$ $a$ is a supremum for $T$ $a$ is an infimum for $T$ $a$ is a maximal element of $T$ $a$ is a minimal element of $T$ $a$ is the greatest element $a$ is the smallest element the weak lower closure $a^\preceq$ of $a$ the weak upper closure $a^\succeq$ of $a$ the strict lower closure $a^\prec$ of $a$ the strict upper closure $a^\succ$ of $a$ the strict lower closure $T^\prec$ of $T$ the strict upper closure $T^\succ$ of $T$ the join $a \vee b$ of $a$ and $b$ the meet $a \wedge b$ of $a$ and $b$ $T$ is a lower set in $S$ $T$ is an upper set in $S$

## Proof

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

By definition of dual statement:

- $b \preceq a$

is dual to:

- $b \succeq a$

and by definition of dual ordering, this is equivalent to:

- $a \preceq b$

By virtue of Dual of Dual Statement (Order Theory), the converse follows.

The other claims are proved on the following pages, in order:

- Succeed is Dual to Precede
- Strictly Succeed is Dual to Strictly Precede
- Upper Bound is Dual to Lower Bound
- Supremum is Dual to Infimum
- Maximal Element is Dual to Minimal Element
- Greatest Element is Dual to Smallest Element
- Weak Lower Closure is Dual to Weak Upper Closure
- Strict Lower Closure is Dual to Strict Upper Closure
- Join is Dual to Meet
- Lower Set is Dual to Upper Set

$\blacksquare$