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:

$b \in a^\prec$, the strict lower closure of $a$

$b \in a^\succ$, the strict upper closure of $a$

Proof

By definition of strict lower closure:

$b \in a^\prec$

if and only if

$b$ 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:

$b \in a^\succ$

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

$\blacksquare$

Also see

• Duality Principle (Order Theory)