# Dual Pairs (Order Theory)

## 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 section in $S$ $T$ is an upper section in $S$ $\struct{S, \vee, \preceq}$ is a join semilattice $\struct{S, \wedge, \preceq}$ is a meet semilattice $\struct{S, \preceq}$ is a complete join semilattice $\struct{S, \preceq}$ is a complete meet semilattice

## 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 Section is Dual to Upper Section
Join Semilattice is Dual to Meet Semilattice
Complete Join Semilattice is Dual to Complete Meet Semilattice

$\blacksquare$