Strict Lower Closure is Dual to Strict Upper Closure

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

Let $a, b \in S$.

The following are dual statements:


 * $b \in {\dot\downarrow} a$, the strict down-set of $a$
 * $b \in {\dot\uparrow} a$, the strict up-set of $a$

Proof
By definition of strict down-set, $b \in {\dot\downarrow} a$ iff:


 * $b$ strictly precedes $a$

The dual of this statement is:


 * $b$ strictly succeeds $a$

by Dual Pairs (Order Theory).

By definition of strict up-set, this means $b \in {\dot\uparrow} a$.

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

Also see

 * Duality Principle (Order Theory)