Category:Dual Pairs (Order Theory)

This category contains pages concerning Dual Pairs (Order Theory):

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 $\struct{S, \preceq}$