Lower Set is Dual to Upper Set

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Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$.


The following are dual statements:

$T$ is a lower set in $S$
$T$ is an upper set in $S$


Proof

By definition, $T$ is a lower set in $S$ iff:

$\forall t \in T: \forall s \in S: s \preceq t \implies s \in T$

The dual of this statement is:

$\forall t \in T: \forall s \in S: t \preceq s \implies s \in T$

by Dual Pairs (Order Theory).


By definition, this means $T$ is an upper set in $S$.


The converse follows from Dual of Dual Statement (Order Theory).

$\blacksquare$


Also see