Lower Section is Dual to Upper Section

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $T \subseteq S$.

The following are dual statements:

$T$ is a lower section in $S$

$T$ is an upper section in $S$

Proof

By definition, $T$ is a lower section in $S$ if and only if:

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

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 section in $S$.

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

$\blacksquare$

Also see

• Duality Principle (Order Theory)