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
$a$ is a join irreducible element		$a$ is a meet irreducible element
$a$ is a join prime element		$a$ is a meet prime element
$T$ is a filter of $\struct {S, \preceq}$		$T$ is an ideal of $\struct {S, \preceq}$
$T$ is a completely prime ideal of $\struct{S, \preceq}$		$T$ is a completely prime filter of $\struct{S, \preceq}$

This article is complete as far as it goes, but it could do with expansion. In particular: $T^\preceq$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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

Join Irreducible Element is Dual of Meet Irreducible Element

Join Prime Element is Dual of Meet Prime Element

Ideal is Dual of Filter (Order Theory)

Completely Prime Ideal is Dual of Completely Prime Filter

This theorem requires a proof. In particular: For $T^\prec$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$\blacksquare$

Also see

• Duality Principle