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:


 * $b \in \mathop \downarrow \left({a}\right)$, the strict lower closure of $a$
 * $b \in \mathop \uparrow \left({a}\right)$, the strict upper closure of $a$

Proof
By definition of strict lower closure, $b \in \mathop \downarrow \left({a}\right)$ iff:


 * $b$ strictly precedes $a$

The dual of this statement is:


 * $b$ strictly succeeds $a$

by Dual Pairs (Order Theory).

By definition of strict upper closure, this means $b \in \mathop \uparrow \left({a}\right)$.

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

Also see

 * Duality Principle (Order Theory)