Strict Lower Closure is Lower Section

Theorem

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

Let $p \in S$.

Let $p^\prec$ denote the strict lower closure of $p$.

Then $p^\prec$ is a lower section.

Proof 1

Let $l \in p^\prec$.

Let $s \in S$ with $s \preceq l$.

Then by the definition of strict lower closure:

$l \prec p$

Thus by Extended Transitivity:

$s \prec p$

So by the definition of strict lower closure:

$s \in p^\prec$

Since this holds for all such $l$ and $s$, $p^\prec$ is a lower section.

$\blacksquare$

Proof 2

By Dual Pairs (Order Theory):

strict upper closure is dual to strict lower closure

Upper section is dual to lower section

Thus the theorem holds by Strict Upper Closure is Upper Section and the duality principle.

$\blacksquare$

Also see

• Lower Closure is Lower Section
• Strict Upper Closure is Upper Section