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:


 * {| style="text-align:center"


 * $b \preceq a$
 * width = "20px" |
 * $a \preceq b$
 * $a$ 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 ${\bar\downarrow} a$ of $a$
 * the weak upper closure $\mathop{\bar \uparrow} a$ of $a$
 * the strict down-set ${\dot\downarrow} a$ of $a$
 * the strict up-set ${\dot\uparrow} a$ of $a$
 * the strict down-set ${\dot\downarrow} T$ of $T$
 * the strict up-set ${\dot\uparrow} T$ 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 $S$
 * }
 * $a$ is the smallest element
 * the weak lower closure ${\bar\downarrow} a$ of $a$
 * the weak upper closure $\mathop{\bar \uparrow} a$ of $a$
 * the strict down-set ${\dot\downarrow} a$ of $a$
 * the strict up-set ${\dot\uparrow} a$ of $a$
 * the strict down-set ${\dot\downarrow} T$ of $T$
 * the strict up-set ${\dot\uparrow} T$ 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 $S$
 * }
 * the strict up-set ${\dot\uparrow} T$ 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 $S$
 * }
 * $T$ 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 Down-Set is Dual to Strict Up-Set
 * Join is Dual to Meet
 * Lower Set is Dual to Upper Set

Also see

 * Duality Principle