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. |
$\blacksquare$